Page curve of a semi-infinite radiation segment

Quit[];

The evaporating black hole solution of JT EOM

f[x_, k_]:= (1/(Pi T1))(1/(BesselI[1, 2Pi (T1/k)] BesselK[0, 2Pi (T1/k)(E^(-k (x/2)))]+BesselK[1, 2Pi (T1/k)] BesselI[0, 2Pi (T1/k)(E^(-k (x/2)))]))(BesselI[0, 2Pi (T1/k)] BesselK[0, 2Pi (T1/k)(E^(-k (x/2)))]- BesselK[0, 2Pi (T1/k)] BesselI[0, 2Pi (T1/k)(E^(-k (x/2)))]);
(*first derivative*)
fd[x_, k_]:=(k^2)/(4 (π^2) (T1^2) (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^2));
(*second derivative*)
fdd[x_, k_]:=(((E^(-((k x)/2))) (k^2) (BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]-BesselI[1,((2 π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)]))/(2 π T1 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^3)))
(*simplified form of the dilaton*)
ϕ[xp_,ym_,k_]:= ϕr (ffdd[ym,k]/ffd[ym,k]+2 ffd[ym,k]/(xp-ff[ym,k]))

Setup for Numerics

(*Numerical input for parameters in the model*)

$AssertFunction=k>0&&T1>0&&u>0&&t>0&&xp>0&&xm∈Reals&&yp>0&&ym∈Reals;
vvk=(1/1024);vvT1=(1/π);vvT0=(509/(512 π));vvϵ=1;vvc=1024;vvϕr=(1/(1024^2));vvGN=(3/(1024^4));vvEs=((vvc π)/(12vvk))((vvT1^2)-(vvT0^2));vvm=12 π (vvEs/(vvc vvϵ));
(*f and derivatives of f used in numerical calculations*)
F[u_,T1_,k_]:=(-BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[0,((2 π T1)/k)]+BesselI[0,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)])/(π T1 (BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))
Fprime[u_,T1_,k_]:=((k^2)/(4 (π^2) (T1^2) (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^2)))
Fpprime[x_,T1_,k_]:=(((E^(-((k x)/2))) (k^2) (BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]-BesselI[1,((2 π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)]))/(2 π T1 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^3)))
Fppprime[x_,T1_,k_]:=-(((E^(-k x)) (k^2) ((((BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)])^2)-3 ((BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)])^2)) ((BesselK[1,((2 π T1)/k)])^2)+2 BesselI[1,((2 π T1)/k)] BesselK[1,((2 π T1)/k)] (BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+3 BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)])+((BesselI[1,((2 π T1)/k)])^2) (((BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)])^2)-3 ((BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)])^2))))/(2 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^4)))
NF[u_?NumberQ,T1_,k_]:=(-BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[0,((2 π T1)/k)]+BesselI[0,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)])/(π T1 (BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))//N[#,500]&
NFprime[u_?NumberQ,T1_,k_]:=((k^2)/(4 (π^2) (T1^2) (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^2)))//N[#,500]&
NFpprime[x_?NumberQ,T1_,k_]:=(((E^(-((k x)/2))) (k^2) (BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]-BesselI[1,((2 π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)]))/(2 π T1 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^3)))//N[#,500]&
NFppprime[x_?NumberQ,T1_,k_]:=-(((E^(-k x)) (k^2) ((((BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)])^2)-3 ((BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)])^2)) ((BesselK[1,((2 π T1)/k)])^2)+2 BesselI[1,((2 π T1)/k)] BesselK[1,((2 π T1)/k)] (BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+3 BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)])+((BesselI[1,((2 π T1)/k)])^2) (((BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)])^2)-3 ((BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)])^2))))/(2 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^4)))//N[#,500]&
tinfinity[T1_,k_]:=((BesselI[0,((2 π T1)/k)])/(π T1 BesselI[1,((2 π T1)/k)]))
Ntinfinity[T1_,k_]:=N[((BesselI[0,((2 π T1)/k)])/(π T1 BesselI[1,((2 π T1)/k)])),300]

Entropy for [σ1=3,∞[ υ (Subscript[QM, L])

Clear[vvσ1];
Clear[vvϕ0];

vvϕ0=0;
vvσ1=3;

Quench saddle

Squench[xqm_, xqp_, σ1_,u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[2m (u+σ1)((( ff[u-σ1,k]))/(Sqrt[ffd[u-σ1,k]]))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))

(*ff and ffd are placeholders for the function f and its derivative*)

(*xqm and xqp are (SuperMinus[(Subscript[x, QQ])]) and (SuperPlus[(Subscript[x, QQ])]) respectively*)

(*Extemizing the generalised entropy defined above, gives the following QES equations*)

eq1=-(((-1+(π^2) (T0^2) (xqp^2)) ϕr)/(2 GN (((xqm-xqp))^2)));
eq2=(((-1+(π^2) (T0^2) (xqm^2)) ϕr)/(2 GN (((xqm-xqp))^2)));

(*Notice that the solution is just the bifurcation surface*)

(*Numerical computation of (Subscript[S, Q])*)

QESquench={(-1+(π^2) (T0^2) (xqp^2)) ϕr,
(-1+(π^2) (T0^2) (xqm^2)) ϕr};
NSolveQESquench[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESquench[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm},QESquench[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm}},{xxm,xxp},WorkingPrecision->105]]
NSgenquench[UU_?NumberQ,σ1_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[2m(u+σ1)((( F[u-σ1,T1,k]))/(Sqrt[Fprime[u-σ1,T1,k]]))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))/.NSolveQESquench[u,σ1,T0,T1,k][[3]]]
tableSquench=Table[{U,NSgenquench[U,vvσ1,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,3,31,(1/10)}];
ListPlot[{tableSquench}]

Scrambling saddle

Sscrambling[xqm_, xqp_, σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[2m xqm(u+σ1)((( xqp-ff[u-σ1,k]))/((xqp-xqm)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))

(*xqm and xqp are (SuperMinus[(Subscript[x, QS])]) and (SuperPlus[(Subscript[x, QS])]) respectively*)

(*Extemizing the generalised entropy defined above, gives the following QES equations*)

eq1=(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2)));
eq2= (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp)) ff[u-σ1,k]);

(*Numerical computation of the QES location and the (Subscript[S, S])*)

QESscrambling={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscrambling[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscrambling[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm},QESscrambling[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm}},{xxm,xxp},WorkingPrecision->105]]
NSgenscrambling[UU_?NumberQ,σ1_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[-2m xxm(u+σ1)((( xxp-F[u-σ1,T1,k]))/((xxp-xxm)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))/.NSolveQESscrambling[u,σ1,T0,T1,k][[3]]]
tableSscrambling=Table[{U,NSgenscrambling[U,vvσ1,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,3,31,(1/10)}];
ListPlot[{tableSscrambling}]

Late saddle

Slate[xqp_, yqm_, σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[(2/ϵ)( (u+σ1)-yqm)(((xqp-ff[u-σ1,k])/(-ff[yqm,k]+xqp)))((Sqrt[ffd[yqm,k]])/(Sqrt[ffd[u-σ1,k]]))]+(((ϕ0+ϕ[xqp,y1m,k]))/(4 GN))

(*xqp and yqm are (SuperPlus[(Subscript[x, QL])]) and (SuperMinus[(Subscript[y, QL])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, L])*)

QESlate={(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),(((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k]))+((2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2))-(((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2))+((2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k])))+((k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/(NFprime[yqm,T1,k])))};
NSolveQESlate[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlate[[1]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym},QESlate[[2]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym}},{yym,-((Log[((8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk))])/(2 (π TT1))),0,uu},{xxp,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],(1/(π TT0))},WorkingPrecision->105,MaxIterations->200]]
NSgenlate[UU_?NumberQ,σ1_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},((ϕ0)/(4GN))+((ϕr)/(4GN))( (((Fpprime[yym,T1,k])/(Fprime[yym,T1,k]))+2 ((Fprime[yym,T1,k])/(xxp- F[yym,T1,k])))+2k Log[Abs[((2(xxp-F[u-σ1,T1,k])(u+σ1-yym))/((xxp-F[yym,T1,k]) ϵ))(Sqrt[((Fprime[yym,T1,k])/(Fprime[u-σ1,T1,k]))])]])/.NSolveQESlate[u,σ1,T0,T1,k]]
tableSlate=Table[{U,NSgenlate[U,vvσ1,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,7,35,(1/10)}];
ListPlot[{tableSlate}]

Combined plot of entropy saddles

ListPlot[{tableSquench,tableSscrambling,tableSlate},PlotLegends->{"Quench Saddle","Scrambling Saddle","Late Saddle"}]

Entropy for [σ1=3,∞[

Clear[vvσ1];
Clear[vvϕ0];

vvϕ0=0;
vvσ1=3;

Quench saddle

Squench[σ1_,u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[m (u+σ1)((( ff[u-σ1,k]))/(Sqrt[ffd[u-σ1,k]]))]

(*ff and ffd are placeholders for the function f and its derivative*)

(*Numerical computation of (Subscript[S, Q])*)

NSgenquench[UU_?NumberQ,σ1_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[m(u+σ1)((( F[u-σ1,T1,k]))/(Sqrt[Fprime[u-σ1,T1,k]]))]]]
tableSquench=Table[{U,NSgenquench[U,vvσ1,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,3,31,(1/10)}];
ListPlot[{tableSquench}]

Scrambling saddle

Sscrambling[xqm_, xqp_, σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[2m xqm(u+σ1)((( xqp-ff[u-σ1,k]))/((xqp-xqm)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]

(*xqm and xqp are (SuperMinus[(Subscript[x, QS])]) and (SuperPlus[(Subscript[x, QS])]) respectively*)

(*Extemizing the generalised entropy defined above, gives the following QES equations*)

eq1=(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2)));
eq2= (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp)) ff[u-σ1,k]);

(*Numerical computation of the QES location and the (Subscript[S, S])*)

QESscrambling={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscrambling[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscrambling[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm},QESscrambling[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm}},{xxm,xxp},WorkingPrecision->105]]
NSgenscrambling[UU_?NumberQ,σ1_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[-2m xxm(u+σ1)((( xxp-F[u-σ1,T1,k]))/((xxp-xxm)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.NSolveQESscrambling[u,σ1,T0,T1,k][[3]]]
tableSscrambling=Table[{U,NSgenscrambling[U,vvσ1,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,3,31,(1/10)}];
ListPlot[{tableSscrambling}]

Late saddle

Slate[xqp_, yqm_, σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[(2/ϵ)( (u+σ1)-yqm)(((xqp-ff[u-σ1,k])/(-ff[yqm,k]+xqp)))((Sqrt[ffd[yqm,k]])/(Sqrt[ffd[u-σ1,k]]))]+(((ϕ0+ϕ[xqp,yqm,k]))/(4 GN))+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]

(*xqp and yqm are (SuperPlus[(Subscript[x, QL])]) and (SuperMinus[(Subscript[y, QL])]) respectively*)

(*Extemizing the generalised entropy defined above, gives the following QES equations*)

eq1=((-((2 k ϕr)/(σ1+u-yqm))+((2 k ϕr ffd[yqm,k])/(xqp-ff[yqm,k]))+((k ϕr ffdd[yqm,k])/(ffd[yqm,k]))+Derivative[0,1,0][ϕ][xqp,yqm,k])/(4 GN));
eq2=((xqp^2) Derivative[1,0,0][ϕ][xqp,yqm,k]-ff[yqm,k] (2 k ϕr+xqp Derivative[1,0,0][ϕ][xqp,yqm,k])+ff[-σ1+u,k] (2 k ϕr+(-xqp+ff[yqm,k]) Derivative[1,0,0][ϕ][xqp,yqm,k]))/(4 GN (xqp-ff[-σ1+u,k]) (xqp-ff[yqm,k]));

(*Numerical computation of the QES location and the (Subscript[S, L])*)

QESlate={(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),(((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k]))+((2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2))-(((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2))+((2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k])))+((k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/(NFprime[yqm,T1,k])))};
NSolveQESlate[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlate[[1]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym},QESlate[[2]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym}},{yym,-((Log[((8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk))])/(2 (π TT1))),0,uu},{xxp,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],(1/(π TT0))},WorkingPrecision->105,MaxIterations->200]]
NSgenlate[UU_?NumberQ,σ1_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},((ϕ0)/(4GN))+((ϕr)/(4GN))( (((Fpprime[yym,T1,k])/(Fprime[yym,T1,k]))+2 ((Fprime[yym,T1,k])/(xxp- F[yym,T1,k])))+2k Log[Abs[((2(xxp-F[u-σ1,T1,k])(u+σ1-yym))/((xxp-F[yym,T1,k]) ϵ))(Sqrt[((Fprime[yym,T1,k])/(Fprime[u-σ1,T1,k]))])]])+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.NSolveQESlate[u,σ1,T0,T1,k]]
tableSlate=Table[{U,NSgenlate[U,vvσ1,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,7,35,(1/10)}];
ListPlot[{tableSlate}]

Combined plot of entropy saddles

ListPlot[{tableSquench,tableSscrambling,tableSlate},PlotLegends->{"Quench Saddle","Scrambling Saddle","Late Saddle"}]

Combined plot of entropy saddles for very late times

Asymptotics of Squench and Slate

Squench[c_,Es_,ϵ_,u_,σ1_,k_,T1_]:=(1/6) c Log[((6 Es (Ntinfinity[T1,k]-2 (E^((4 π (-T1+(Sqrt[(E^(-k u)) (T1^2)])))/k)) Ntinfinity[T1,k]) (Sqrt[π]) (u+σ1))/(c (Sqrt[(E^((4 π (-T1+(Sqrt[(E^(-k u)) (T1^2)])))/k)) Ntinfinity[T1,k] (Sqrt[(E^(-k u)) (T1^2)])]) ϵ))];

Slate[ϕ0_,ϕr_,T0_,T1_,u_,σ1_,k_,GN_,c_,ϵ_]:=((12 ϕ0-3 (E^(k u)) k ϕr+12 π (T0+(Sqrt[(E^(-k u)) (T1^2)])) ϕr+4 c GN Log[-(((E^(k u)) k (k-8 π) Log[((8 π T1)/(3 k))])/(4 (Sqrt[6]) (π^(5/2)) (T1^2) (Sqrt[(k/(Sqrt[(E^(-k u)) (T1^2)]))]) ϵ))])/(24 GN))+2 k ((ϕr)/(4GN))Log[2];

tableSquench=Table[{U,Squench[vvc,vvEs,vvϵ,U,vvσ1,vvk,vvT1]},{U,3,3500,(1/10)}];

tableSlate=Table[{U,Slate[vvϕ0,vvϕr,vvT0,vvT1,U,vvσ1,vvk,vvGN,vvc,vvϵ]},{U,7,3500,(1/10)}];

ListPlot[{tableSquench,tableSlate},PlotLegends->{"Quench Saddle","Late Saddle"}]

Including  Sscrambling (figure 6)

Sscrambling[xqm_, xqp_, σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[2m xqm(u+σ1)((( xqp-ff[u-σ1,k]))/((xqp-xqm)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]

eq1=(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2)));
eq2= (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp)) ff[u-σ1,k]);

(*Now let us do some numerics to make a plot*)

(*The equations that we ame to solve*)

QESscrambling={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscrambling[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscrambling[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm},QESscrambling[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm}},{xxm,xxp},WorkingPrecision->105]]
NSgenscrambling[UU_?NumberQ,σ1_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[-2m xxm(u+σ1)((( xxp-F[u-σ1,T1,k]))/((xxp-xxm)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.NSolveQESscrambling[u,σ1,T0,T1,k][[3]]]
tableSscrambling=Table[{U,NSgenscrambling[U,vvσ1,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,3,3500,(1/10)}];
ListPlot[{tableSquench,tableSlate,tableSscrambling},PlotLegends->{"Quench Saddle","Scrambling Saddle","Late Saddle"}]

Appendix: derivation of the asymptotic expressions for Slate and Squench

(*Squench*)

Series[(c/6)Log[12 (Es/(c ϵ)) ((Sinh[π Tb (u+σ1)])/Tb)((Ntinfinity-2 Ntinfinity (E^(-4(π/k)(T1-((((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))^(1/2))-Tb Log[((T1+Tb)/(Tb+((((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))^(1/2))))]+k(((u-σ1))/2)Tb))))/(((2π ((((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))^(1/2))(2 Ntinfinity (E^(-4(π/k)(T1-((((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))^(1/2))-Tb Log[((T1+Tb)/(Tb+((((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))^(1/2))))]+k(((u-σ1))/2)Tb))))))^(1/2)))],{Tb,0,0}]//Simplify

(*Slate*)

Series[((2ϕ0+2π T0 ϕr)/(4GN))+(1/(4GN))(2π ϕr((((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))^(1/2))-ϕr k (((T1^2)+(Tb^2)-2 Tb ((((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))^(1/2)))/(2 ((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))) )+(c/6)Log[(2/( ϵ)) ((Sinh[π Tb (u+σ1-(u+σ1-(1/(2π ((((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))^(1/2))))Log[((8 π T1)/(k(4-(((1-(Tb/((((Tb^2)+((T1^2)-(Tb^2))(E^(-k (u-σ1)))))^(1/2)))))^2))))]))])/(π Tb))(((((2k((((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))^(1/2)))/(3π ((Tb^2)+((T1^2)-(Tb^2))(E^(-k u)))+2π Tb ((((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))^(1/2))-π (Tb^2)))))^(1/2))(1-(k/(8π ((Tb^2)+((T1^2)-(Tb^2))(E^(-k u)))))(((((((Tb^2)+((T1^2)-(Tb^2))(E^(-k u))))^(1/2))-Tb))^2))],{Tb,0,0}]//Simplify

Page curve of a finite radiation segment

The evaporating black hole solution of JT EOM

f[x_, k_]:= (1/(Pi T1))(1/(BesselI[1, 2Pi (T1/k)] BesselK[0, 2Pi (T1/k)(E^(-k (x/2)))]+BesselK[1, 2Pi (T1/k)] BesselI[0, 2Pi (T1/k)(E^(-k (x/2)))]))(BesselI[0, 2Pi (T1/k)] BesselK[0, 2Pi (T1/k)(E^(-k (x/2)))]- BesselK[0, 2Pi (T1/k)] BesselI[0, 2Pi (T1/k)(E^(-k (x/2)))]);
(*first derivative*)
fd[x_, k_]:=(k^2)/(4 (π^2) (T1^2) (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^2));
(*second derivative*)
fdd[x_, k_]:=(((E^(-((k x)/2))) (k^2) (BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]-BesselI[1,((2 π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)]))/(2 π T1 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^3)))
(*simplified form of the dilaton*)
ϕ[xp_,ym_,k_]:= ϕr (ffdd[ym,k]/ffd[ym,k]+2 ffd[ym,k]/(xp-ff[ym,k]))

Setup for Numerics

(*Numerical input for parameters in the model*)

$AssertFunction=k>0&&T1>0&&u>0&&t>0&&xp>0&&xm∈Reals&&yp>0&&ym∈Reals;
vvk=(1/1024);vvT1=(1/π);vvT0=(509/(512 π));vvϵ=1;vvc=1024;vvϕr=(1/(1024^2));vvGN=(3/(1024^4));vvEs=((vvc π)/(12vvk))((vvT1^2)-(vvT0^2));vvm=12 π (vvEs/(vvc vvϵ));
(*f and derivatives of f used in numerical calculations*)
F[u_,T1_,k_]:=(-BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[0,((2 π T1)/k)]+BesselI[0,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)])/(π T1 (BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))
Fprime[u_,T1_,k_]:=((k^2)/(4 (π^2) (T1^2) (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^2)))
Fpprime[x_,T1_,k_]:=(((E^(-((k x)/2))) (k^2) (BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]-BesselI[1,((2 π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)]))/(2 π T1 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^3)))
Fppprime[x_,T1_,k_]:=-(((E^(-k x)) (k^2) ((((BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)])^2)-3 ((BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)])^2)) ((BesselK[1,((2 π T1)/k)])^2)+2 BesselI[1,((2 π T1)/k)] BesselK[1,((2 π T1)/k)] (BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+3 BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)])+((BesselI[1,((2 π T1)/k)])^2) (((BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)])^2)-3 ((BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)])^2))))/(2 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^4)))
NF[u_?NumberQ,T1_,k_]:=(-BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[0,((2 π T1)/k)]+BesselI[0,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)])/(π T1 (BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))//N[#,500]&
NFprime[u_?NumberQ,T1_,k_]:=((k^2)/(4 (π^2) (T1^2) (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k u)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k u)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^2)))//N[#,500]&
NFpprime[x_?NumberQ,T1_,k_]:=(((E^(-((k x)/2))) (k^2) (BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]-BesselI[1,((2 π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)]))/(2 π T1 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^3)))//N[#,500]&
NFppprime[x_?NumberQ,T1_,k_]:=-(((E^(-k x)) (k^2) ((((BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)])^2)-3 ((BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)])^2)) ((BesselK[1,((2 π T1)/k)])^2)+2 BesselI[1,((2 π T1)/k)] BesselK[1,((2 π T1)/k)] (BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+3 BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)])+((BesselI[1,((2 π T1)/k)])^2) (((BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)])^2)-3 ((BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)])^2))))/(2 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^4)))//N[#,500]&
tinfinity[T1_,k_]:=((BesselI[0,((2 π T1)/k)])/(π T1 BesselI[1,((2 π T1)/k)]))
Ntinfinity[T1_,k_]:=N[((BesselI[0,((2 π T1)/k)])/(π T1 BesselI[1,((2 π T1)/k)])),300]
(*definition of functions to fine-tune k in order for non-trivial behavior to manifest*)
Nkfirstuncorrectedbound[T0_,T1_,σ1_,σ2_,t∞_]:=N[((2 π (T0-T1) (π T0 t∞ σ1-σ2))/( (σ2-π σ1 (T0 t∞+6 T1 σ2)+3 (π^2) T0 T1 t∞ (((σ1)^2)+((σ2)^2))))),300];
Nkfirstbound[T0_,T1_,σ1_,σ2_,t∞_]:=N[((4 π (T0-T1) (π T0 t∞ σ1-σ2))/ (-4 σ2+2 π σ1 (2 T0 t∞-9 T1 σ2)+6 (π^2) T0 T1 t∞ (2 ((σ1)^2)+((σ2)^2))+(-π T0 t∞ σ1+σ2) Log[((8 π T1)/(3 Nkfirstuncorrectedbound[T0,T1,σ1,σ2,t∞]))]-4 (π T0 t∞ σ1-σ2) Log[-((3 (π^(3/2)) T1 (-(T0^2)+(T1^2)) (-1+π T0 t∞) (Nufirstzero[T0,T1,σ1,σ2,t∞,Nkfirstuncorrectedbound[T0,T1,σ1,σ2,t∞]]+σ1))/(8 Nkfirstuncorrectedbound[T0,T1,σ1,σ2,t∞] T0 (Sqrt[T1 t∞])  (4 π T1 σ1+Log[((8 π T1)/(3 Nkfirstuncorrectedbound[T0,T1,σ1,σ2,t∞]))])))])),300];
Nufirstzero[T0_,T1_,σ1_,σ2_,t∞_,k_]:=N[-((2 (T0-T1))/(3 (k T1) ))-((2-6 π T1 σ1+Log[((8 π T1)/(3 k))])/(6 (π T1))),300];

Entropy for [σ1=0,σ2=3]

Clear[vvσ1];
Clear[vvσ2];
Clear[vvϕ0];

vvϕ0=0;
vvσ1=0;
vvσ2=3;

Trivial QES saddles

SEarlyConnected[σ1_, σ2_,ϕr_,ϵ_,GN_,k_]:= 4 k ((ϕr)/(4GN))Log[((σ2-σ1)/ϵ)];
SEarlyDisconnected[σ1_, σ2_,ϕr_,ϵ_,GN_,k_]:= 2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]+2 k ((ϕr)/(4GN))Log[2((σ1)/ϵ)];
SCrossDisconnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, m_,T1_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[m(u+σ1)((F[u-σ1,T1,k])/(Sqrt[Fprime[u-σ1,T1,k]]))]]+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)];
SCrossConnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, m_,T1_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(m/ϵ) (u-σ2)(((σ1-σ2)F[u-σ1,T1, k])/(Sqrt[Fprime[u-σ1,T1, k]]))]];
SFutureConnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_,T1_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(F[u-σ1,T1, k]-F[u-σ2,T1, k])(((σ2-σ1))/((ϵ^2)(Sqrt[Fprime[u-σ1,T1, k]Fprime[u-σ2,T1, k]])))]]

tableSCrossDisconnected=Table[{U,N[SCrossDisconnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN,vvm,vvT1, vvk],105]},{U,1,3,(1/10)}];
ListPlot[{tableSCrossDisconnected}]

tableSCrossConnected=Table[{U,N[SCrossConnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN,vvm,vvT1, vvk],105]},{U,1,3,(1/10)}];
ListPlot[{tableSCrossConnected}]

tableSFutureConnected=Table[{U,N[SFutureConnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN, vvT1,vvk],105]},{U,3,31,(1/10)}];
ListPlot[{tableSFutureConnected}]

Nontrivial QES saddles: intermediate times σ1<u<σ2

ScramblingTrivial saddle

SScramblingTrivial[xqm_, xqp_, σ1_,σ2_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[-2m xqm(u+σ1)((( xqp-ff[u-σ1,k]))/((xqp-xqm)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]+2 k ((ϕr)/(4GN))Log[2]

(*xqm and xqp are (SuperMinus[(Subscript[x, QS])]) and (SuperPlus[(Subscript[x, QS])]) respectively*)

(*Extemizing the generalised entropy defined above, gives the following QES equations*)

eq1=(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2)));
eq2= (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp)) ff[u-σ1,k]);

(*Numerical computation of the QES location and the (Subscript[S, ST])*)

QESscramblingtrivial={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscramblingtrivial[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscramblingtrivial[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm},QESscramblingtrivial[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm}},{xxm,xxp},WorkingPrecision->105]]
NSgenscramblingtrivial3[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[-2m xxm(u+σ1)((( xxp-F[u-σ1,T1,k]))/((xxp-xxm)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))+((ϕ0+2ϕr π T0)/(4 GN))+2 k ((ϕr)/(4GN))Log[2]+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]/.NSolveQESscramblingtrivial[u,σ1,T0,T1,k][[3]]]
NSgenscramblingtrivial4[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[-2m xxm(u+σ1)((( xxp-F[u-σ1,T1,k]))/((xxp-xxm)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))+((ϕ0+2ϕr π T0)/(4 GN))+2 k ((ϕr)/(4GN))Log[2]+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]/.NSolveQESscramblingtrivial[u,σ1,T0,T1,k][[4]]]
tableSscramblingtrivial3=Table[{U,NSgenscramblingtrivial3[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,1,4,(1/10)}];
tableSscramblingtrivial4=Table[{U,NSgenscramblingtrivial4[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,1,4,(1/10)}];
ListPlot[{tableSscramblingtrivial3,tableSscramblingtrivial4},PlotLegends->{"3","4"}]
tableQESx1mscramblingtrivial=Table[{U,NSolveQESscramblingtrivial[U,vvσ1,vvT0,vvT1,vvk][[4,1,2]]},{U,1,4,(1/10)}];
tableQESx1pscramblingtrivial=Table[{U,NSolveQESscramblingtrivial[U,vvσ1,vvT0,vvT1,vvk][[4,2,2]]},{U,1,4,(1/10)}];
ListPlot[{tableQESx1mscramblingtrivial}];
ListPlot[{tableQESx1pscramblingtrivial}];

(* solution 4 has lower entropy compared to solution 3*)

LateTrivial saddle- doesn’t exist since segment is to short

SLateTrivial[yqm_,xqp_, σ1_,σ2_, u_, k_]:=(((ϕ0+ϕ[xqp,yqm,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)(((((u+σ1-yqm) (xqp-ff[u-σ1,k]) ))/(((xqp-ff[yqm,k]) )))) Sqrt[(((ffd[yqm,k] ))/((ffd[u-σ1,k] )))]]]+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]

(*xqp and yqm are (SuperPlus[(Subscript[x, QL])]) and (SuperMinus[(Subscript[y, QL])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, LT])*)

QESlatetrivial={(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),(((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k]))+((2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2))-(((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2))+((2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k])))+((k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/(NFprime[yqm,T1,k])))};
NSolveQESlatetrivial[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_,ϕϕr_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlatetrivial[[1]]/.{T1->TT1,k->kk,ϕr-> ϕϕr,u->uu,σ1->σσ1,xqp->xxp,yqm->yym},QESlatetrivial[[2]]/.{T1->TT1,k->kk,ϕr-> ϕϕr,u->uu,σ1->σσ1,xqp->xxp,yqm->yym}},{yym,-((Log[((8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk))])/(2 (π TT1))),0,uu},{xxp,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],(1/(π TT0))},WorkingPrecision->105,MaxIterations->200]]
NSgenlatetrivial[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},((ϕ0)/(4GN))+((ϕr)/(4 GN)) ( (((Fpprime[yym,T1,k])/(Fprime[yym,T1,k]))+2 ((Fprime[yym,T1,k])/(xxp- F[yym,T1,k])))+2 k  Log[Abs[((2(xxp-F[u-σ1,T1,k])(u+σ1-yym))/(ϵ (xxp-F[yym,T1,k]) ))(Sqrt[((Fprime[yym,T1,k])/(Fprime[u-σ1,T1,k]))])]])+((ϕ0+2ϕr π T0)/(4 GN))+2 k ((ϕr)/(4GN))Log[2]+2 k ((ϕr)/(4 GN)) Log[Abs[2((σ2)/ϵ)]]/.NSolveQESlatetrivial[u,σ1,T0,T1,k,ϕr]]
tableSlatetrivial=Table[{U,NSgenlatetrivial[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN]},{U,5,6,(1/10)}];
ListPlot[{tableSlatetrivial}]
tableQESy1mlatetrivial=Table[{U,NSolveQESlatetrivial[U,vvσ1,vvT0,vvT1,vvk,vvϕr][[1,2]]},{U,5,6,(1/10)}];
tableQESx1platetrivial=Table[{U,NSolveQESlatetrivial[U,vvσ1,vvT0,vvT1,vvk,vvϕr][[2,2]]},{U,5,6,(1/10)}];
ListPlot[{tableQESy1mlatetrivial}];
ListPlot[{tableQESx1platetrivial}];

Nontrivial QES saddles: late times σ1<σ2<u

ScramblingQuench  and ScramblingScrambling saddle

(*First ScramblingScrambling saddle*)
Sscramblingscrambling[x1p_, x1m_,x2p_,x2m_,σ2_ ,σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[2m x1m(u+σ1)((( ff[u -σ1,k]-x1p))/((x1p-x1m)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) x1p x1m))/(x1p-x1m)))/(4 GN))+2 k ((ϕr)/(4GN))Log[2m x2m(u+σ2)((( ff[u -σ2,k]-x2p))/((x2p-x2m)(Sqrt[ffd[u-σ2,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) x2p x2m))/(x2p-x2m)))/(4 GN))

(*x1m and x1p are (SuperMinus[(Subscript[x, 1QS])]) and (SuperPlus[(Subscript[x, 1QS])]) respectively*)
(*x2m and x2p are (SuperMinus[(Subscript[x, 2QS])]) and (SuperPlus[(Subscript[x, 2QS])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, SS])*)

QESscramblingscrambling={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscramblingscrambling1[UU_?NumberQ,σσ1_?NumberQ,σσ2_?NumberQ,TT0_,TT1_,kk_,ϕϕr_,GGN_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscramblingscrambling[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,xqm->xx1m},QESscramblingscrambling[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,xqm->xx1m}},{xx1m,xx1p},WorkingPrecision->105]]
NSolveQESscramblingscrambling2[UU_?NumberQ,σσ1_?NumberQ,σσ2_?NumberQ,TT0_,TT1_,kk_,ϕϕr_,GGN_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscramblingscrambling[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx2p,xqm->xx2m},QESscramblingscrambling[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx2p,xqm->xx2m}},{xx2m,xx2p},WorkingPrecision->105]]
NSgenscramblingscrambling3[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[2m xx1m(u+σ1)((( F[u -σ1,T1,k]-xx1p))/((xx1p-xx1m)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx1p xx1m))/(xx1p-xx1m)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[2m xx2m(u+σ2)((( F[u -σ2,T1,k]-xx2p))/((xx2p-xx2m)(Sqrt[Fprime[u-σ2,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx2p xx2m))/(xx2p-xx2m)))/(4 GN))/.NSolveQESscramblingscrambling1[u,σ1,σ2,T0,T1,k,ϕr,GN][[3]]/.NSolveQESscramblingscrambling2[u,σ1,σ2,T0,T1,k,ϕr,GN][[3]]]
NSgenscramblingscrambling4[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[2m xx1m(u+σ1)((( F[u -σ1,T1,k]-xx1p))/((xx1p-xx1m)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx1p xx1m))/(xx1p-xx1m)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[2m xx2m(u+σ2)((( F[u -σ2,T1,k]-xx2p))/((xx2p-xx2m)(Sqrt[Fprime[u-σ2,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx2p xx2m))/(xx2p-xx2m)))/(4 GN))/.NSolveQESscramblingscrambling1[u,σ1,σ2,T0,T1,k,ϕr,GN][[4]]/.NSolveQESscramblingscrambling2[u,σ1,σ2,T0,T1,k,ϕr,GN][[4]]]
tableSscramblingscrambling3=Table[{U,NSgenscramblingscrambling3[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,3,31,(1/10)}];
tableSscramblingscrambling4=Table[{U,NSgenscramblingscrambling4[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,3,31,(1/10)}];
ListPlot[{tableSscramblingscrambling3,tableSscramblingscrambling4}]
tableQESx1mscramblingscrambling=Table[{U,NSolveQESscramblingscrambling1[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,1,2]]},{U,3,31,(1/10)}];
tableQESx1pscramblingscrambling=Table[{U,NSolveQESscramblingscrambling1[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,2,2]]},{U,3,31,(1/10)}];
tableQESx2mscramblingscrambling=Table[{U,NSolveQESscramblingscrambling2[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,1,2]]},{U,3,31,(1/10)}];
tableQESx2pscramblingscrambling=Table[{U,NSolveQESscramblingscrambling2[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,2,2]]},{U,3,31,(1/10)}];
ListPlot[{tableQESx1mscramblingscrambling,tableQESx2mscramblingscrambling}];
ListPlot[{tableQESx1pscramblingscrambling,tableQESx2pscramblingscrambling}];

(*ScramblingQuench*)
SScramblingQuench[xqp_, xqm_,σ2_ ,σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[2m xqm(u+σ1)((( ff[u -σ1,k]-xqp))/((xqp-xqm)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+((ϕ0+2ϕr π T0)/(4 GN))+2 k ((ϕr)/(4GN))Log[2]+2 k ((ϕr)/(4GN))Log[m(u+σ2)((ff[u-σ2,k])/(Sqrt[ffd[u-σ2,k]]))]

(*xqm and xqp are (SuperMinus[(Subscript[x, QS])]) and (SuperPlus[(Subscript[x, QS])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, SQ])*)

QESscramblingquench={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscramblingquench[UU_?NumberQ,σσ1_?NumberQ,σσ2_?NumberQ,TT0_,TT1_,kk_,ϕϕr_,GGN_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscramblingquench[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,xqm->xx1m},QESscramblingquench[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,xqm->xx1m}},{xx1m,xx1p},WorkingPrecision->105]]
NSgenscramblingquench3[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[2m xx1m(u+σ1)((( F[u -σ1,T1,k]-xx1p))/((xx1p-xx1m)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx1p xx1m))/(xx1p-xx1m)))/(4 GN))+((ϕ0+2ϕr π T0)/(4 GN))+2 k ((ϕr)/(4GN))Log[2]+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1,k]]))]]/.NSolveQESscramblingquench[u,σ1,σ2,T0,T1,k,ϕr,GN][[3]]]
NSgenscramblingquench4[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[2m xx1m(u+σ1)((( F[u -σ1,T1,k]-xx1p))/((xx1p-xx1m)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx1p xx1m))/(xx1p-xx1m)))/(4 GN))+((ϕ0+2ϕr π T0)/(4 GN))+2 k ((ϕr)/(4GN))Log[2]+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1, k]]))]]/.NSolveQESscramblingquench[u,σ1,σ2,T0,T1,k,ϕr,GN][[4]]]
tableSscramblingquench3=Table[{U,NSgenscramblingquench3[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,3,31,(1/10)}];
tableSscramblingquench4=Table[{U,NSgenscramblingquench4[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,3,31,(1/10)}];
ListPlot[{tableSscramblingquench3,tableSscramblingquench4}]
tableQESx1mscramblingquench=Table[{U,NSolveQESscramblingquench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,1,2]]},{U,3,31,(1/10)}];
tableQESx1pscramblingquench=Table[{U,NSolveQESscramblingquench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,2,2]]},{U,3,31,(1/10)}];
ListPlot[{tableQESx1mscramblingquench}];
ListPlot[{tableQESx1pscramblingquench}];

LateQuench and LateScrambling

SLateScrambling[x1p_, y1m_,x2p_,x2m_,σ2_, σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)( (u+σ1)-y1m)(((ff[y1p,k]-ff[u-σ1,k])/(-ff[y1m,k]+ff[y1p,k])))((Sqrt[ffd[y1m,k]])/(Sqrt[ffd[u-σ1,k]]))]]+(((ϕ0+ϕ[x1p,y1m,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[2m x2m(u+σ2)((( ff[u -σ2,k]-x2p))/((x2p-x2m)(Sqrt[ffd[u-σ2,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) x2p x2m))/(x2p-x2m)))/(4 GN))

(*y1m and x1p are (SuperMinus[(Subscript[y, 1QL])]) and (SuperPlus[(Subscript[x, 1QL])]) respectively*)
(*x2m and x2p are (SuperMinus[(Subscript[x, 2QS])]) and (SuperPlus[(Subscript[x, 2QS])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, LS])*)

QESlatescrambling={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k]),(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),(((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k]))+((2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2))-(((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2))+((2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k])))+((k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/(NFprime[yqm,T1,k])))};

NSolveQESlatescrambling1[UU_?NumberQ,σσ1_?NumberQ,σσ2_?NumberQ,TT0_,TT1_,kk_,ϕϕr_,GGN_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlatescrambling[[3]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,yqm->yy1m},QESlatescrambling[[4]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,yqm->yy1m}},{yy1m,-((Log[((8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk))])/(2 (π TT1))),0,uu},{xx1p,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],(1/(π TT0))},WorkingPrecision->105,MaxIterations->200]]
NSolveQESlatescrambling2[UU_?NumberQ,σσ1_?NumberQ,σσ2_?NumberQ,TT0_,TT1_,kk_,ϕϕr_,GGN_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESlatescrambling[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx2p,xqm->xx2m},QESlatescrambling[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx2p,xqm->xx2m}},{xx2m,xx2p},WorkingPrecision->105]]
NSgenlatescrambling3[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},((ϕ0)/(4GN))+2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)( (u+σ1)-yy1m)(((xx1p-F[u-σ1,T1,k])/(-F[yy1m,T1,k]+xx1p)))((Sqrt[Fprime[yy1m,T1,k]])/(Sqrt[Fprime[u-σ1,T1,k]]))]]+((ϕr)/(4GN)) (((Fpprime[yy1m,T1,k])/(Fprime[yy1m,T1,k]))+2 ((Fprime[yy1m,T1,k])/(xx1p- F[yy1m,T1,k])))+2 k ((ϕr)/(4GN))Log[Abs[2m xx2m(u+σ2)((( F[u -σ2,T1,k]-xx2p))/((xx2p-xx2m)(Sqrt[Fprime[u-σ2,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx2p xx2m))/(xx2p-xx2m)))/(4 GN))/.NSolveQESlatescrambling1[u,σ1,σ2,T0,T1,k,ϕr,GN]/.NSolveQESlatescrambling2[u,σ1,σ2,T0,T1,k,ϕr,GN][[3]]]
NSgenlatescrambling4[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},((ϕ0)/(4GN))+2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)( (u+σ1)-yy1m)(((xx1p-F[u-σ1,T1,k])/(-F[yy1m,T1,k]+xx1p)))((Sqrt[Fprime[yy1m,T1,k]])/(Sqrt[Fprime[u-σ1,T1,k]]))]]+((ϕr)/(4GN)) (((Fpprime[yy1m,T1,k])/(Fprime[yy1m,T1,k]))+2 ((Fprime[yy1m,T1,k])/(xx1p- F[yy1m,T1,k])))+2 k ((ϕr)/(4GN))Log[Abs[2m xx2m(u+σ2)((( F[u -σ2,T1,k]-xx2p))/((xx2p-xx2m)(Sqrt[Fprime[u-σ2,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx2p xx2m))/(xx2p-xx2m)))/(4 GN))/.NSolveQESlatescrambling1[u,σ1,σ2,T0,T1,k,ϕr,GN]/.NSolveQESlatescrambling2[u,σ1,σ2,T0,T1,k,ϕr,GN][[4]]]
tableSlatescrambling3=Table[{U,NSgenlatescrambling3[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,5,31,(1/10)}];
tableSlatescrambling4=Table[{U,NSgenlatescrambling4[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,5,31,(1/10)}];
ListPlot[{tableSlatescrambling3,tableSlatescrambling4}]
tableQESy1mlatescrambling=Table[{U,NSolveQESlatescrambling1[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[1,2]]},{U,5,31,(1/10)}];
tableQESx1platescrambling=Table[{U,NSolveQESlatescrambling1[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[2,2]]},{U,5,31,(1/10)}];
tableQESx2mlatescrambling=Table[{U,NSolveQESlatescrambling2[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,1,2]]},{U,5,31,(1/10)}];
tableQESx2platescrambling=Table[{U,NSolveQESlatescrambling2[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,2,2]]},{U,5,31,(1/10)}];
ListPlot[{tableQESy1mlatescrambling}];
ListPlot[{tableQESx2mlatescrambling}];
ListPlot[{tableQESx1platescrambling,tableQESx2platescrambling}];

(*LateQuench*)

SLateQuench[xqp_, yqm_,σ2_ ,σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)( (u+σ1)-yqm)(((xqp-ff[u-σ1,k])/(-ff[yqm,k]+xqp)))((Sqrt[ffd[yqm,k]])/(Sqrt[ffd[u-σ1,k]]))]]+(((ϕ0+ϕ[xqp,yqm,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((ff[u-σ2,k])/(Sqrt[ffd[u-σ2,k]]))]]+((ϕ0+2ϕr π T0)/(4 GN))+2 k ((ϕr)/(4GN))Log[2]

(*yqm and xqp are (SuperMinus[(Subscript[y, QL])]) and (SuperPlus[(Subscript[x, QL])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, LQ])*)

QESlatequench={(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),(((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k]))+((2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2))-(((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2))+((2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k])))+((k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/(NFprime[yqm,T1,k])))};
NSolveQESlatequench[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlatequench[[1]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym},QESlatequench[[2]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym}},{yym,-((Log[((8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk))])/(2 (π TT1))),0,uu},{xxp,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],(1/(π TT0))},WorkingPrecision->105,MaxIterations->200]]
NSgenlatequench[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},((ϕ0)/(4GN))+((ϕr)/(4GN))( (((Fpprime[yym,T1,k])/(Fprime[yym,T1,k]))+2 ((Fprime[yym,T1,k])/(xxp- F[yym,T1,k])))+2k Log[Abs[((2(xxp-F[u-σ1,T1,k])(u+σ1-yym))/((xxp-NF[yym,T1,k]) ϵ))(Sqrt[((Fprime[yym,T1,k])/(Fprime[u-σ1,T1,k]))])]])+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1,k]]))]]+((ϕ0+2ϕr π T0)/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.NSolveQESlatequench[u,σ1,T0,T1,k]]
tableSlatequench=Table[{U,NSgenlatequench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,5,31,(1/10)}];
ListPlot[{tableSlatequench}]
tableQESy1mlatequench=Table[{U,NSolveQESlatequench[U,vvσ1,vvT0,vvT1,vvk][[1,2]]},{U,5,31,(1/10)}];
tableQESx1platequench=Table[{U,NSolveQESlatequench[U,vvσ1,vvT0,vvT1,vvk][[2,2]]},{U,5,31,(1/10)}];
ListPlot[{tableQESy1mlatequench}];
ListPlot[{tableQESx1platequench}];

LateLate

SLateLate[x1p_, y1m_,x2p_,y2m_,σ2_ ,σ1_, u_,k_]:=(((2ϕ0+ϕ[x1p,y1m,k]+ϕ[x2p,y2m,k]))/(4 GN))+2k ((ϕr)/(4GN))Log[Abs[(((2^2) )/(ϵ^2))(((((u+σ1-y1m) (u+σ2-y2m) (x1p-ff[u-σ1,k]) (x2p-ff[u-σ2,k])))/(((x1p-ff[y1m,k]) (x2p-ff[y2m,k]))))) Sqrt[(((ffd[y1m,k] ffd[y2m,k]))/((ffd[u-σ1,k] ffd[u-σ2,k])))]]];

(*y1m and x1p are (SuperMinus[(Subscript[y, 1QL])]) and (SuperPlus[(Subscript[x, 1QL])]) respectively*)
(*y2m and x2p are (SuperMinus[(Subscript[y, 2QL])]) and (SuperPlus[(Subscript[x, 2QL])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, LL])*)

QESlatelate={(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),(((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k]))+((2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2))-(((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2))+((2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k])))+((k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/(NFprime[yqm,T1,k])))};
NSolveQESlatelate1[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlatelate[[1]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,yqm->yy1m},QESlatelate[[2]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,yqm->yy1m}},{yy1m,-((Log[((8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk))])/(2 (π TT1))),0,uu},{xx1p,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],(1/(π TT0))},WorkingPrecision->105,MaxIterations->200]]
NSolveQESlatelate2[UU_?NumberQ,σσ2_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlatelate[[1]]/.{T1->TT1,k->kk,u->uu,σ1->σσ2,xqp->xx2p,yqm->yy2m},QESlatelate[[2]]/.{T1->TT1,k->kk,u->uu,σ1->σσ2,xqp->xx2p,yqm->yy2m}},{yy2m,-((Log[((8 (E^(-2 π TT1 (uu-σσ2))) π TT1)/(3 kk))])/(2 (π TT1))),0,uu},{xx2p,+(1/3) (-NF[uu-σσ2,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],(1/(π TT0))},WorkingPrecision->105,MaxIterations->200]]
NSgenlatelate[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},2((ϕ0)/(4GN))+((ϕr)/(4GN))( (((Fpprime[yy1m,T1,k])/(Fprime[yy1m,T1,k]))+2 ((Fprime[yy1m,T1,k])/(xx1p- F[yy1m,T1,k])))+2k Log[Abs[((2(xx1p-F[u-σ1,T1,k])(u+σ1-yy1m))/((xx1p-F[yy1m,T1,k]) ϵ))(Sqrt[((Fprime[yy1m,T1,k])/(Fprime[u-σ1,T1,k]))])]])+((ϕr)/(4GN))( (((Fpprime[yy2m,T1,k])/(Fprime[yy2m,T1,k]))+2 ((Fprime[yy2m,T1,k])/(xx2p- F[yy2m,T1,k])))+2k Log[Abs[((2(xx2p-F[u-σ2,T1,k])(u+σ2-yy2m))/((xx2p-F[yy2m,T1,k]) ϵ))(Sqrt[((Fprime[yy2m,T1,k])/(Fprime[u-σ2,T1,k]))])]])/.NSolveQESlatelate1[u,σ1,T0,T1,k]/.NSolveQESlatelate2[u,σ2,T0,T1,k]]
tableQESy1mlatelate=Table[{U,NSolveQESlatelate2[U,vvσ1,vvT0,vvT1,vvk][[1,2]]},{U,15,31,(1/10)}];
tableQESy2mlatelate=Table[{U,NSolveQESlatelate2[U,vvσ2,vvT0,vvT1,vvk][[1,2]]},{U,15,31,(1/10)}];
tableQESx1platelate=Table[{U,NSolveQESlatelate2[U,vvσ1,vvT0,vvT1,vvk][[2,2]]},{U,15,31,(1/10)}];
tableQESx2platelate=Table[{U,NSolveQESlatelate2[U,vvσ2,vvT0,vvT1,vvk][[2,2]]},{U,15,31,(1/10)}];tableSlatelate=Table[{U,NSgenlatelate[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,15,31,(1/10)}];
ListPlot[{tableSlatelate}]
ListPlot[{tableQESy1mlatelate,tableQESy2mlatelate}];
ListPlot[{tableQESx1platelate,tableQESx2platelate}];

Combined plot of nontrivial QES saddles

ListPlot[{tableSscramblingtrivial4,tableSlatetrivial,tableSscramblingquench4,tableSscramblingscrambling4,tableSlatescrambling4,tableSlatequench,tableSlatelate},PlotLegends->{"ScramblingTrivial Saddle","LateTrivial Saddle","ScramblingQuench Saddle","ScramblingScrambling Saddle","LateScrambling Saddle","LateQuench Saddle","LateLate Saddle"}]

Combined plot of trivial QES saddles

ListPlot[{tableSCrossDisconnected,tableSCrossConnected,tableSFutureConnected},PlotLegends-> {"CrossDisconnected Saddle","CrossConnected Saddle","FutureConnected Saddle"},PlotRange->All]

Combined plot of trivial QES and nontrivial QES saddles

ListPlot[{tableSCrossDisconnected,tableSCrossConnected,tableSFutureConnected,tableSscramblingtrivial4,tableSlatetrivial,tableSscramblingquench4,tableSscramblingscrambling4,tableSlatescrambling4,tableSlatequench,tableSlatelate},PlotLegends->{"CrossDisconnected Saddle","CrossConnected Saddle","FutureConnected Saddle","ScramblingTrivial Saddle","LateTrivial Saddle","ScramblingQuench Saddle","ScramblingScrambling Saddle","LateScrambling Saddle","LateQuench Saddle","LateLate Saddle"},PlotRange->All]

Entropy for [σ1=1,σ2=30] υ (Subscript[QM, L]) (scenario 1)

(*From here on, we change the values of the parameters in order to have a better plots *)

(*Let us take T1=(1/π);T0=(459/(512 π)); as reference and we vary T1 from [(449/(512 π)),(459/(512 π))]*)

vvϕ0=0;
vvσ1=1;
vvσ2=30;
vvT0=(449/(512 π));
vvk=Nkfirstbound[(459/(512π)),(1/π),1,30,1];
vvT1=(1/π);vvϵ=1;vvϕr=(1/(1024^2));vvGN=(3/(1024^4));vvc=((3vvk vvϕr)/vvGN); vvEs=((vvc π)/(12vvk))((vvT1^2)-(vvT0^2));vvm=12 π (vvEs/(vvc vvϵ));

(*Notice that we use a specific value of k defined via a function. This function is found analytically and gives us a lower bound on the dimensionless parameter ((Subscript[E, S])/(c(T1+T0)))=(π/12)((T1-T0)/k) in order for temporary blindness to occur*)

Nkfirstbound[(459/(512π)),(1/π),1,30,1]

Trivial QES saddles

SEarlyConnected[σ1_, σ2_,ϕr_,ϵ_,GN_,k_]:= 4 k ((ϕr)/(4GN))Log[((σ2-σ1)/ϵ)]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SEarlyDisconnected[σ1_, σ2_,ϕr_,ϵ_,GN_,k_]:= 2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]+2 k ((ϕr)/(4GN))Log[2((σ1)/ϵ)]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SCrossDisconnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, m_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[m(u+σ1)((F[u-σ1,vvT1,k])/(Sqrt[Fprime[u-σ1,vvT1,k]]))]]+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SCrossConnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, m_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(m/ϵ) (u-σ2)(((σ1-σ2)F[u-σ1,vvT1, k])/(Sqrt[Fprime[u-σ1,vvT1, k]]))]]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SFutureConnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(F[u-σ1,vvT1, k]-F[u-σ2,vvT1, k])(((σ2-σ1))/((ϵ^2)(Sqrt[Fprime[u-σ1,vvT1, k]Fprime[u-σ2,vvT1, k]])))]]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;

tableSCrossDisconnected=Table[{U,N[SCrossDisconnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN,vvm, vvk]/.T1-> vvT1,105]},{U,1,30,(1/10)}];
ListPlot[{tableSCrossDisconnected}]

tableSCrossConnected=Table[{U,N[SCrossConnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN,vvm, vvk]/.T1-> vvT1,105]},{U,2,29.9,(1/10)}];
ListPlot[{tableSCrossConnected}]

(*Segment in the future of the shock*)

tableSFutureConnected=Table[{U,N[SFutureConnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN, vvk]/.T1-> vvT1,105]},{U,30,40,(1/10)}];
ListPlot[{tableSFutureConnected}]

Nontrivial QES saddles: intermediate times σ1<u<σ2

ScramblingTrivial Saddle

Sscramblingtrivial[xqm_, xqp_, σ1_,σ2_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[-2m xqm(u+σ1)((( xqp-ff[u-σ1,k]))/((xqp-xqm)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]

(*xqm and xqp are (SuperMinus[(Subscript[x, QS])]) and (SuperPlus[(Subscript[x, QS])]) respectively*)

(*Extemizing the generalised entropy defined above, gives the following QES equations*)

eq1=(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2)));
eq2= (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp)) ff[u-σ1,k]);

(*Numerical computation of the QES location and the (Subscript[S, ST])*)

QESscramblingtrivial={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscramblingtrivial[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscramblingtrivial[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm},QESscramblingtrivial[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm}},{xxm,xxp},WorkingPrecision->105]]
NSgenscramblingtrivial3[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[-2m xxm(u+σ1)((( xxp-F[u-σ1,T1,k]))/((xxp-xxm)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]/.NSolveQESscramblingtrivial[u,σ1,T0,T1,k][[3]]]
NSgenscramblingtrivial4[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[-2m xxm(u+σ1)((( xxp-F[u-σ1,T1,k]))/((xxp-xxm)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]/.NSolveQESscramblingtrivial[u,σ1,T0,T1,k][[4]]]
tableSscramblingtrivial3=Table[{U,NSgenscramblingtrivial3[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,1,30,(1/10)}];
tableSscramblingtrivial4=Table[{U,NSgenscramblingtrivial4[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,1,30,(1/10)}];
ListPlot[{tableSscramblingtrivial3,tableSscramblingtrivial4}]
tableQESx1mscramblingtrivial=Table[{U,NSolveQESscramblingtrivial[U,vvσ1,vvT0,vvT1,vvk][[4,1,2]]},{U,1,30,(1/10)}];
tableQESx1pscramblingtrivial=Table[{U,NSolveQESscramblingtrivial[U,vvσ1,vvT0,vvT1,vvk][[4,2,2]]},{U,1,30,(1/10)}];
ListPlot[{tableQESx1mscramblingtrivial}];
ListPlot[{tableQESx1pscramblingtrivial}];

LateTrivial Saddle

Slatetrivial[yqm_,xqp_, σ1_,σ2_, u_, k_]:=(((ϕ0+ϕ[xqp,yqm,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)(((((u+σ1-yqm) (xqp-ff[u-σ1,k]) ))/(((xqp-ff[yqm,k]) )))) Sqrt[(((ffd[yqm,k] ))/((ffd[u-σ1,k] )))]]]+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]

(*xqp and yqm are (SuperPlus[(Subscript[x, QL])]) and (SuperMinus[(Subscript[y, QL])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, LT])*)

QESlatetrivial={(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k])+(2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2)-((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2)+(2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k]))+(k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/NFprime[yqm,T1,k])};
NSolveQESlatetrivial[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_,ϕϕr_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlatetrivial[[1]]/.{T1->TT1,k->kk,ϕr-> ϕϕr,u->uu,σ1->σσ1,xqp->xxp,yqm->yym},QESlatetrivial[[2]]/.{T1->TT1,k->kk,ϕr-> ϕϕr,u->uu,σ1->σσ1,xqp->xxp,yqm->yym}},{yym,-(Log[(8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk)]/(2 (π TT1))),0,uu},{xxp,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],1/(π TT0)},WorkingPrecision->105,MaxIterations->200]]
NSgenlatetrivial[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},(ϕr/(4 GN)) ( (Fpprime[yym,T1,k]/Fprime[yym,T1,k]+2 (Fprime[yym,T1,k]/(xxp- F[yym,T1,k])))+2 k  Log[Abs[((2(xxp-F[u-σ1,T1,k])(u+σ1-yym))/(ϵ (xxp-F[yym,T1,k]) ))√(Fprime[yym,T1,k]/Fprime[u-σ1,T1,k])]])+2 k (ϕr/(4 GN)) Log[Abs[2(σ2/ϵ)]]/.NSolveQESlatetrivial[u,σ1,T0,T1,k,ϕr]]
tableSlatetrivial=Table[{U,NSgenlatetrivial[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN]},{U,5,30,1/10}];
ListPlot[{tableSlatetrivial}]
tableQESy1mlatetrivial=Table[{U,NSolveQESlatetrivial[U,vvσ1,vvT0,vvT1,vvk,vvϕr][[1,2]]},{U,5,30,1/10}];
tableQESx1platetrivial=Table[{U,NSolveQESlatetrivial[U,vvσ1,vvT0,vvT1,vvk,vvϕr][[2,2]]},{U,5,30,1/10}];
ListPlot[{tableQESy1mlatetrivial}];
ListPlot[{tableQESx1platetrivial}];

Nontrivial QES saddles: late times σ1<σ2<u

ScramblingQuench Saddle

Sscramblingquench[xqp_, xqm_,σ2_ ,σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[2m xqm(u+σ1)((( ff[u -σ1,k]-xqp))/((xqp-xqm)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[m(u+σ2)((ff[u-σ2,k])/(Sqrt[ffd[u-σ2,k]]))]

(*xqm and xqp are (SuperMinus[(Subscript[x, QS])]) and (SuperPlus[(Subscript[x, QS])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, SQ])*)

QESscramblingquench={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscramblingquench[UU_?NumberQ,σσ1_?NumberQ,σσ2_?NumberQ,TT0_,TT1_,kk_,ϕϕr_,GGN_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscramblingquench[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,xqm->xx1m},QESscramblingquench[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,xqm->xx1m}},{xx1m,xx1p},WorkingPrecision->105]]
NSgenscramblingquench3[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[2m xx1m(u+σ1)((( F[u -σ1,T1,k]-xx1p))/((xx1p-xx1m)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx1p xx1m))/(xx1p-xx1m)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1,k]]))]]/.NSolveQESscramblingquench[u,σ1,σ2,T0,T1,k,ϕr,GN][[3]]]
NSgenscramblingquench4[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[2m xx1m(u+σ1)((( F[u -σ1,T1,k]-xx1p))/((xx1p-xx1m)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx1p xx1m))/(xx1p-xx1m)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1, k]]))]]/.NSolveQESscramblingquench[u,σ1,σ2,T0,T1,k,ϕr,GN][[4]]]
tableSscramblingquench3=Table[{U,NSgenscramblingquench3[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,30,41,(1/10)}];
tableSscramblingquench4=Table[{U,NSgenscramblingquench4[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,30,41,(1/10)}];
ListPlot[{tableSscramblingquench3,tableSscramblingquench4}]
tableQESx1mscramblingquench=Table[{U,NSolveQESscramblingquench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,1,2]]},{U,30,41,(1/10)}];
tableQESx1pscramblingquench=Table[{U,NSolveQESscramblingquench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,2,2]]},{U,30,41,(1/10)}];
ListPlot[{tableQESx1mscramblingquench}];
ListPlot[{tableQESx1pscramblingquench}];

LateQuench Saddle

Slatequench[xqp_, yqm_,σ2_ ,σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)( (u+σ1)-yqm)(((xqp-ff[u-σ1,k])/(-ff[yqm,k]+xqp)))((Sqrt[ffd[yqm,k]])/(Sqrt[ffd[u-σ1,k]]))]]+(((ϕ0+ϕ[xqp,yqm,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((ff[u-σ2,k])/(Sqrt[ffd[u-σ2,k]]))]]

(*yqm and xqp are (SuperMinus[(Subscript[y, QL])]) and (SuperPlus[(Subscript[x, QL])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, LQ])*)

(*Now let us do some numerics*)
QESlatequench={(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),(((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k]))+((2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2))-(((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2))+((2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k])))+((k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/(NFprime[yqm,T1,k])))};
NSolveQESlatequench[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlatequench[[1]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym},QESlatequench[[2]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym}},{yym,-((Log[((8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk))])/(2 (π TT1))),0,uu},{xxp,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],(1/(π TT0))},WorkingPrecision->105,MaxIterations->200]]
NSgenlatequench[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},((ϕr)/(4GN))( (((Fpprime[yym,T1,k])/(Fprime[yym,T1,k]))+2 ((Fprime[yym,T1,k])/(xxp- F[yym,T1,k])))+2k Log[Abs[((2(xxp-F[u-σ1,T1,k])(u+σ1-yym))/((xxp-NF[yym,T1,k]) ϵ))(Sqrt[((Fprime[yym,T1,k])/(Fprime[u-σ1,T1,k]))])]])+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1,k]]))]]/.NSolveQESlatequench[u,σ1,T0,T1,k]]
tableSlatequench=Table[{U,NSgenlatequench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,30,41,(1/10)}];
ListPlot[{tableSlatequench}]
tableQESy1mlatequench=Table[{U,NSolveQESlatequench[U,vvσ1,vvT0,vvT1,vvk][[1,2]]},{U,30,41,(1/10)}];
tableQESx1platequench=Table[{U,NSolveQESlatequench[U,vvσ1,vvT0,vvT1,vvk][[2,2]]},{U,30,41,(1/10)}];
ListPlot[{tableQESy1mlatequench}];
ListPlot[{tableQESx1platequench}];

Combined plot of nontrivial QES saddles

ListPlot[{tableSscramblingtrivial4,tableSlatetrivial,tableSscramblingquench4,tableSlatequench},PlotLegends->{"ScramblingTrivial Saddle","LateTrivial Saddle","ScramblingQuench Saddle","LateQuench Saddle"}]

Combined plot of trivial QES saddles

ListPlot[{tableSCrossDisconnected,tableSCrossConnected,tableSFutureConnected},PlotLegends-> {"CrossDisconnected Saddle","CrossConnected Saddle","FutureConnected Saddle"},PlotRange->All]

Combined plot of trivial and nontrivial QES saddles

ListPlot[{tableSCrossDisconnected,tableSCrossConnected,tableSFutureConnected,tableSscramblingtrivial4,tableSlatetrivial,tableSscramblingquench4,tableSlatequench},PlotLegends-> {"CrossDisconnected Saddle","CrossConnected Saddle","FutureConnected Saddle","ScramblingTrivial Saddle","LateTrivial Saddle","ScramblingQuench Saddle","LateQuench Saddle"}]

Size of the entanglement wedge and subregion complexity

tableQESx1mlatetrivial=Table[F[tableQESy1mlatetrivial[[i,2]],vvT1,vvk],{i,1,Length[tableQESy1mlatetrivial]}];
tableQESx1mlatequench=Table[F[tableQESy1mlatequench[[i,2]],vvT1,vvk],{i,1,Length[tableQESy1mlatequench]}];

tableQES2sscramblingtrivial=Table[{tableQESx1pscramblingtrivial[[i,1]],tableQESx1pscramblingtrivial[[i,2]]-tableQESx1mscramblingtrivial[[i,2]]},{i,1,Length[tableQESx1pscramblingtrivial]}];
tableQES2slatetrivial=Table[{tableQESx1platetrivial[[i,1]],tableQESx1platetrivial[[i,2]]-tableQESx1mlatetrivial[[i]]},{i,1,Length[tableQESx1platetrivial]}];
tableQES2sscramblingquench=Table[{tableQESx1pscramblingquench[[i,1]],tableQESx1pscramblingquench[[i,2]]-tableQESx1mscramblingquench[[i,2]]},{i,1,Length[tableQESx1pscramblingquench]}];
tableQES2slatequench=Table[{tableQESx1platequench[[i,1]],tableQESx1platequench[[i,2]]-tableQESx1mlatequench[[i]]},{i,1,Length[tableQESx1platequench]}];

(*In order to compute the increase in subregion complexity, we use equation 4.22*)

tableQES2sscramblingtrivialevol=Table[{tableQESx1pscramblingtrivial[[i,1]],Log[(2/(π vvT0(tableQESx1pscramblingtrivial[[i,2]]-tableQESx1mscramblingtrivial[[i,2]])))]},{i,9,253}];

ListPlot[{tableQES2sscramblingtrivialevol},PlotLegends->{"Δsubregion complexity ScramblingTrivial"},PlotRange->All,PlotStyle->{Red}]

ListPlot[{tableQES2sscramblingtrivialevol},PlotLegends->{"Δsubregion complexity ScramblingTrivial"},PlotRange->{{0,50},{0,30}},PlotStyle->{Red}]

Entropy for [σ1=1,σ2=30] υ (Subscript[QM, L]) (scenario 3)

(*From here on, we change the values of the parameters in order to have a better plots *)

(*Let us take T1=(1/π);T0=(459/(512 π)); as reference and we vary T1 from [(449/(512 π)),(459/(512 π))]*)

vvϕ0=0;
vvσ1=1;
vvσ2=30;
vvT0=((453+(1/2))/(512 π));
vvk=Nkfirstbound[(459/(512π)),(1/π),1,30,1];
vvT1=(1/π);vvϵ=1;vvϕr=(1/(1024^2));vvGN=(3/(1024^4));vvc=((3vvk vvϕr)/vvGN); vvEs=((vvc π)/(12vvk))((vvT1^2)-(vvT0^2));vvm=12 π (vvEs/(vvc vvϵ));

(*Notice that we use a specific value of k defined via a function. This function is found analytically and gives us a lower bound on the dimensionless parameter ((Subscript[E, S])/(c(T1+T0)))=(π/12)((T1-T0)/k) in order for temporary blindness to occur*)

Nkfirstbound[(459/(512π)),(1/π),1,30,1]

Trivial QES saddles

SEarlyConnected[σ1_, σ2_,ϕr_,ϵ_,GN_,k_]:= 4 k ((ϕr)/(4GN))Log[((σ2-σ1)/ϵ)]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SEarlyDisconnected[σ1_, σ2_,ϕr_,ϵ_,GN_,k_]:= 2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]+2 k ((ϕr)/(4GN))Log[2((σ1)/ϵ)]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SCrossDisconnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, m_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[m(u+σ1)((F[u-σ1,vvT1,k])/(Sqrt[Fprime[u-σ1,vvT1,k]]))]]+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SCrossConnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, m_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(m/ϵ) (u-σ2)(((σ1-σ2)F[u-σ1,vvT1, k])/(Sqrt[Fprime[u-σ1,vvT1, k]]))]]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SFutureConnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(F[u-σ1,vvT1, k]-F[u-σ2,vvT1, k])(((σ2-σ1))/((ϵ^2)(Sqrt[Fprime[u-σ1,vvT1, k]Fprime[u-σ2,vvT1, k]])))]]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;

tableSCrossDisconnected=Table[{U,N[SCrossDisconnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN,vvm, vvk]/.T1-> vvT1,105]},{U,1,30,(1/10)}];
ListPlot[{tableSCrossDisconnected}]

tableSCrossConnected=Table[{U,N[SCrossConnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN,vvm, vvk]/.T1-> vvT1,105]},{U,2,29.9,(1/10)}];
ListPlot[{tableSCrossConnected}]

(*Segment in the future of the shock*)

tableSFutureConnected=Table[{U,N[SFutureConnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN, vvk]/.T1-> vvT1,105]},{U,30,40,(1/10)}];
ListPlot[{tableSFutureConnected}]

Nontrivial QES saddles: intermediate times σ1<u<σ2

ScramblingTrivial Saddle

Sscramblingtrivial[xqm_, xqp_, σ1_,σ2_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[-2m xqm(u+σ1)((( xqp-ff[u-σ1,k]))/((xqp-xqm)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]

(*xqm and xqp are (SuperMinus[(Subscript[x, QS])]) and (SuperPlus[(Subscript[x, QS])]) respectively*)

(*Extemizing the generalised entropy defined above, gives the following QES equations*)

eq1=(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2)));
eq2= (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp)) ff[u-σ1,k]);

(*Numerical computation of the QES location and the (Subscript[S, ST])*)

QESscramblingtrivial={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscramblingtrivial[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscramblingtrivial[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm},QESscramblingtrivial[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm}},{xxm,xxp},WorkingPrecision->105]]
NSgenscramblingtrivial3[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[-2m xxm(u+σ1)((( xxp-F[u-σ1,T1,k]))/((xxp-xxm)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]/.NSolveQESscramblingtrivial[u,σ1,T0,T1,k][[3]]]
NSgenscramblingtrivial4[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[-2m xxm(u+σ1)((( xxp-F[u-σ1,T1,k]))/((xxp-xxm)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]/.NSolveQESscramblingtrivial[u,σ1,T0,T1,k][[4]]]
tableSscramblingtrivial3=Table[{U,NSgenscramblingtrivial3[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,1,30,(1/10)}];
tableSscramblingtrivial4=Table[{U,NSgenscramblingtrivial4[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,1,30,(1/10)}];
ListPlot[{tableSscramblingtrivial3,tableSscramblingtrivial4}]
tableQESx1mscramblingtrivial=Table[{U,NSolveQESscramblingtrivial[U,vvσ1,vvT0,vvT1,vvk][[4,1,2]]},{U,1,30,(1/10)}];
tableQESx1pscramblingtrivial=Table[{U,NSolveQESscramblingtrivial[U,vvσ1,vvT0,vvT1,vvk][[4,2,2]]},{U,1,30,(1/10)}];
ListPlot[{tableQESx1mscramblingtrivial}];
ListPlot[{tableQESx1pscramblingtrivial}];

LateTrivial Saddle

Slatetrivial[yqm_,xqp_, σ1_,σ2_, u_, k_]:=(((ϕ0+ϕ[xqp,yqm,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)(((((u+σ1-yqm) (xqp-ff[u-σ1,k]) ))/(((xqp-ff[yqm,k]) )))) Sqrt[(((ffd[yqm,k] ))/((ffd[u-σ1,k] )))]]]+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]

(*xqp and yqm are (SuperPlus[(Subscript[x, QL])]) and (SuperMinus[(Subscript[y, QL])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, LT])*)

QESlatetrivial={(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k])+(2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2)-((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2)+(2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k]))+(k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/NFprime[yqm,T1,k])};
NSolveQESlatetrivial[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_,ϕϕr_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlatetrivial[[1]]/.{T1->TT1,k->kk,ϕr-> ϕϕr,u->uu,σ1->σσ1,xqp->xxp,yqm->yym},QESlatetrivial[[2]]/.{T1->TT1,k->kk,ϕr-> ϕϕr,u->uu,σ1->σσ1,xqp->xxp,yqm->yym}},{yym,-(Log[(8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk)]/(2 (π TT1))),0,uu},{xxp,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],1/(π TT0)},WorkingPrecision->105,MaxIterations->200]]
NSgenlatetrivial[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},(ϕr/(4 GN)) ( (Fpprime[yym,T1,k]/Fprime[yym,T1,k]+2 (Fprime[yym,T1,k]/(xxp- F[yym,T1,k])))+2 k  Log[Abs[((2(xxp-F[u-σ1,T1,k])(u+σ1-yym))/(ϵ (xxp-F[yym,T1,k]) ))√(Fprime[yym,T1,k]/Fprime[u-σ1,T1,k])]])+2 k (ϕr/(4 GN)) Log[Abs[2(σ2/ϵ)]]/.NSolveQESlatetrivial[u,σ1,T0,T1,k,ϕr]]
tableSlatetrivial=Table[{U,NSgenlatetrivial[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN]},{U,5,30,1/10}];
ListPlot[{tableSlatetrivial}]
tableQESy1mlatetrivial=Table[{U,NSolveQESlatetrivial[U,vvσ1,vvT0,vvT1,vvk,vvϕr][[1,2]]},{U,5,30,1/10}];
tableQESx1platetrivial=Table[{U,NSolveQESlatetrivial[U,vvσ1,vvT0,vvT1,vvk,vvϕr][[2,2]]},{U,5,30,1/10}];
ListPlot[{tableQESy1mlatetrivial}];
ListPlot[{tableQESx1platetrivial}];

Nontrivial QES saddles: late times σ1<σ2<u

ScramblingQuench Saddle

Sscramblingquench[xqp_, xqm_,σ2_ ,σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[2m xqm(u+σ1)((( ff[u -σ1,k]-xqp))/((xqp-xqm)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[m(u+σ2)((ff[u-σ2,k])/(Sqrt[ffd[u-σ2,k]]))]

(*xqm and xqp are (SuperMinus[(Subscript[x, QS])]) and (SuperPlus[(Subscript[x, QS])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, SQ])*)

QESscramblingquench={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscramblingquench[UU_?NumberQ,σσ1_?NumberQ,σσ2_?NumberQ,TT0_,TT1_,kk_,ϕϕr_,GGN_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscramblingquench[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,xqm->xx1m},QESscramblingquench[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,xqm->xx1m}},{xx1m,xx1p},WorkingPrecision->105]]
NSgenscramblingquench3[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[2m xx1m(u+σ1)((( F[u -σ1,T1,k]-xx1p))/((xx1p-xx1m)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx1p xx1m))/(xx1p-xx1m)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1,k]]))]]/.NSolveQESscramblingquench[u,σ1,σ2,T0,T1,k,ϕr,GN][[3]]]
NSgenscramblingquench4[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[2m xx1m(u+σ1)((( F[u -σ1,T1,k]-xx1p))/((xx1p-xx1m)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx1p xx1m))/(xx1p-xx1m)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1, k]]))]]/.NSolveQESscramblingquench[u,σ1,σ2,T0,T1,k,ϕr,GN][[4]]]
tableSscramblingquench3=Table[{U,NSgenscramblingquench3[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,30,41,(1/10)}];
tableSscramblingquench4=Table[{U,NSgenscramblingquench4[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,30,41,(1/10)}];
ListPlot[{tableSscramblingquench3,tableSscramblingquench4}]
tableQESx1mscramblingquench=Table[{U,NSolveQESscramblingquench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,1,2]]},{U,30,41,(1/10)}];
tableQESx1pscramblingquench=Table[{U,NSolveQESscramblingquench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,2,2]]},{U,30,41,(1/10)}];
ListPlot[{tableQESx1mscramblingquench}];
ListPlot[{tableQESx1pscramblingquench}];

LateQuench Saddle

Slatequench[xqp_, yqm_,σ2_ ,σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)( (u+σ1)-yqm)(((xqp-ff[u-σ1,k])/(-ff[yqm,k]+xqp)))((Sqrt[ffd[yqm,k]])/(Sqrt[ffd[u-σ1,k]]))]]+(((ϕ0+ϕ[xqp,yqm,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((ff[u-σ2,k])/(Sqrt[ffd[u-σ2,k]]))]]

(*yqm and xqp are (SuperMinus[(Subscript[y, QL])]) and (SuperPlus[(Subscript[x, QL])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, LQ])*)

(*Now let us do some numerics*)
QESlatequench={(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),(((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k]))+((2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2))-(((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2))+((2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k])))+((k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/(NFprime[yqm,T1,k])))};
NSolveQESlatequench[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlatequench[[1]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym},QESlatequench[[2]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym}},{yym,-((Log[((8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk))])/(2 (π TT1))),0,uu},{xxp,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],(1/(π TT0))},WorkingPrecision->105,MaxIterations->200]]
NSgenlatequench[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},((ϕr)/(4GN))( (((Fpprime[yym,T1,k])/(Fprime[yym,T1,k]))+2 ((Fprime[yym,T1,k])/(xxp- F[yym,T1,k])))+2k Log[Abs[((2(xxp-F[u-σ1,T1,k])(u+σ1-yym))/((xxp-NF[yym,T1,k]) ϵ))(Sqrt[((Fprime[yym,T1,k])/(Fprime[u-σ1,T1,k]))])]])+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1,k]]))]]/.NSolveQESlatequench[u,σ1,T0,T1,k]]
tableSlatequench=Table[{U,NSgenlatequench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,30,41,(1/10)}];
ListPlot[{tableSlatequench}]
tableQESy1mlatequench=Table[{U,NSolveQESlatequench[U,vvσ1,vvT0,vvT1,vvk][[1,2]]},{U,30,41,(1/10)}];
tableQESx1platequench=Table[{U,NSolveQESlatequench[U,vvσ1,vvT0,vvT1,vvk][[2,2]]},{U,30,41,(1/10)}];
ListPlot[{tableQESy1mlatequench}];
ListPlot[{tableQESx1platequench}];

Combined plot of nontrivial QES saddles

ListPlot[{tableSscramblingtrivial4,tableSlatetrivial,tableSscramblingquench4,tableSlatequench},PlotLegends->{"ScramblingTrivial Saddle","LateTrivial Saddle","ScramblingQuench Saddle","LateQuench Saddle"}]

Combined plot of trivial QES saddles

ListPlot[{tableSCrossDisconnected,tableSCrossConnected,tableSFutureConnected},PlotLegends-> {"CrossDisconnected Saddle","CrossConnected Saddle","FutureConnected Saddle"},PlotRange->All]

Combined plot of trivial and nontrivial QES saddles

ListPlot[{tableSCrossDisconnected,tableSCrossConnected,tableSFutureConnected,tableSscramblingtrivial4,tableSlatetrivial,tableSscramblingquench4,tableSlatequench},PlotLegends-> {"CrossDisconnected Saddle","CrossConnected Saddle","FutureConnected Saddle","ScramblingTrivial Saddle","LateTrivial Saddle","ScramblingQuench Saddle","LateQuench Saddle"}]

Size of the entanglement wedge and subregion complexity

tableQESx1mlatetrivial=Table[F[tableQESy1mlatetrivial[[i,2]],vvT1,vvk],{i,1,Length[tableQESy1mlatetrivial]}];
tableQESx1mlatequench=Table[F[tableQESy1mlatequench[[i,2]],vvT1,vvk],{i,1,Length[tableQESy1mlatequench]}];

tableQES2sscramblingtrivial=Table[{tableQESx1pscramblingtrivial[[i,1]],tableQESx1pscramblingtrivial[[i,2]]-tableQESx1mscramblingtrivial[[i,2]]},{i,1,Length[tableQESx1pscramblingtrivial]}];
tableQES2slatetrivial=Table[{tableQESx1platetrivial[[i,1]],tableQESx1platetrivial[[i,2]]-tableQESx1mlatetrivial[[i]]},{i,1,Length[tableQESx1platetrivial]}];
tableQES2sscramblingquench=Table[{tableQESx1pscramblingquench[[i,1]],tableQESx1pscramblingquench[[i,2]]-tableQESx1mscramblingquench[[i,2]]},{i,1,Length[tableQESx1pscramblingquench]}];
tableQES2slatequench=Table[{tableQESx1platequench[[i,1]],tableQESx1platequench[[i,2]]-tableQESx1mlatequench[[i]]},{i,1,Length[tableQESx1platequench]}];

(*In order to compute the increase in subregion complexity, we use equation 4.22*)

tableQES2sscramblingtrivialevol=Table[{tableQESx1pscramblingtrivial[[i,1]],Log[(2/(π vvT0(tableQESx1pscramblingtrivial[[i,2]]-tableQESx1mscramblingtrivial[[i,2]])))]},{i,9,255}];

tableQES2slatetrivialevol=Table[{tableQESx1platetrivial[[i,1]],Log[(2/(π vvT0(tableQESx1platetrivial[[i,2]]-tableQESx1mlatetrivial[[i]])))]},{i,231,248}];

ListPlot[{tableQES2sscramblingtrivialevol,tableQES2slatetrivialevol},PlotLegends->{"Δsubregion complexity ScramblingTrivial","Δ subregion complexity LateTrivial"},PlotRange->All,PlotStyle->{Red,Purple}]

Phase diagram and contour plots

tableScrossConn3D=Table[N[SCrossConnected[vvσ1,s2, U,vvϕr,vvϵ,vvGN,vvm, vvk]/.T1-> vvT1, 105], {s2, 28,32, 1/10},{U,26,29.9,1/20}];

ListContourPlot[tableScrossConn3D, DataRange-> {{26,29.9},{28,32}}]

tableSfuture=Table[{U,N[SFutureConnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN, vvk]/.T1-> vvT1,105]},{U,30,40,1/10}];
tableSfuture3D=Table[N[SFutureConnected[vvσ1,s2, U,vvϕr,vvϵ,vvGN, vvk]/.T1-> vvT1,105], {s2, 28, 32, 1/10},{U,30,40,1/10}];
ListPlot[{tableSfuture}]
ListContourPlot[tableSfuture3D, DataRange->{{30, 40}, {28, 32}}]

tableSscramblingtrivial43D=Table[NSgenscramblingtrivial4[U,vvσ1,s2,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN], {s2, 28, 32, 1/10},{U,26, 29.9,(1/20)}];
ListContourPlot[tableSscramblingtrivial43D, DataRange->{{26, 29.9}, {28,32}}]

tableSlatetrivial3D=Table[NSgenlatetrivial[U,vvσ1,s2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN],{s2, 28, 32, 1/10},{U,26,29.9,1/20}];
ListContourPlot[tableSlatetrivial3D, DataRange->{{26, 29.9}, {28,32}}]

tableSall3D=Table[Min[N[SCrossConnected[vvσ1,s2, U,vvϕr,vvϵ,vvGN,vvm, vvk]/.T1-> vvT1, 105], NSgenscramblingtrivial4[U,vvσ1,s2,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN], NSgenlatetrivial[U,vvσ1,s2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN](*,N[Sfuture[vvσ1,s2, U,vvϕr,vvϵ,vvGN, vvk]/.T1-> vvT1,105]*)] ,{s2, 28, 32, 1/10},{U,20,29.9,1/20}];

ListContourPlot[tableSall3D, DataRange->{{20, 29.9}, {28,32}}, RegionFunction->Function[{x,y,z}, x<y]]

(*The phase diagram*)

ListPlot3D[{-tableScrossConn3D,-tableSscramblingtrivial43D,-tableSlatetrivial3D}, DataRange->{{26, 29.9}, {28,32}}, RegionFunction->Function[{x,y,z}, x<y], ViewPoint->{0,0,Infinity},Mesh->None,Lighting->{{"Ambient",White}},PlotRangePadding->0,Axes->{True,True,False}, ImageSize->Large]

Entropy for [σ1=1,σ2=30] υ (Subscript[QM, L]) (scenario 2)

(*From here on, we change the values of the parameters in order to have a better plots *)

(*Let us take T1=(1/π);T0=(459/(512 π)); as reference and we vary T1 from [(449/(512 π)),(459/(512 π))]*)

vvϕ0=0;
vvσ1=1;
vvσ2=30;
vvT0=(459/(512 π));
vvk=Nkfirstbound[(459/(512π)),(1/π),1,30,1];
vvT1=(1/π);vvϵ=1;vvϕr=(1/(1024^2));vvGN=(3/(1024^4));vvc=((3vvk vvϕr)/vvGN); vvEs=((vvc π)/(12vvk))((vvT1^2)-(vvT0^2));vvm=12 π (vvEs/(vvc vvϵ));

(*Notice that we use a specific value of k defined via a function. This function is found analytically and gives us a lower bound on the dimensionless parameter ((Subscript[E, S])/(c(T1+T0)))=(π/12)((T1-T0)/k) in order for temporary blindness to occur*)

Nkfirstbound[(459/(512π)),(1/π),1,30,1]

Trivial QES saddles

SEarlyConnected[σ1_, σ2_,ϕr_,ϵ_,GN_,k_]:= 4 k ((ϕr)/(4GN))Log[((σ2-σ1)/ϵ)]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SEarlyDisconnected[σ1_, σ2_,ϕr_,ϵ_,GN_,k_]:= 2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]+2 k ((ϕr)/(4GN))Log[2((σ1)/ϵ)]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SCrossDisconnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, m_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[m(u+σ1)((F[u-σ1,vvT1,k])/(Sqrt[Fprime[u-σ1,vvT1,k]]))]]+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SCrossConnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, m_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(m/ϵ) (u-σ2)(((σ1-σ2)F[u-σ1,vvT1, k])/(Sqrt[Fprime[u-σ1,vvT1, k]]))]]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;
SFutureConnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(F[u-σ1,vvT1, k]-F[u-σ2,vvT1, k])(((σ2-σ1))/((ϵ^2)(Sqrt[Fprime[u-σ1,vvT1, k]Fprime[u-σ2,vvT1, k]])))]]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;

tableSCrossDisconnected=Table[{U,N[SCrossDisconnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN,vvm, vvk]/.T1-> vvT1,105]},{U,1,30,(1/10)}];
ListPlot[{tableSCrossDisconnected}]

tableSCrossConnected=Table[{U,N[SCrossConnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN,vvm, vvk]/.T1-> vvT1,105]},{U,2,29.9,(1/10)}];
ListPlot[{tableSCrossConnected}]

(*Segment in the future of the shock*)

tableSFutureConnected=Table[{U,N[SFutureConnected[vvσ1,vvσ2, U,vvϕr,vvϵ,vvGN, vvk]/.T1-> vvT1,105]},{U,30,40,(1/10)}];
ListPlot[{tableSFutureConnected}]

Nontrivial QES saddles: intermediate times σ1<u<σ2

ScramblingTrivial Saddle

Sscramblingtrivial[xqm_, xqp_, σ1_,σ2_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[-2m xqm(u+σ1)((( xqp-ff[u-σ1,k]))/((xqp-xqm)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]

(*xqm and xqp are (SuperMinus[(Subscript[x, QS])]) and (SuperPlus[(Subscript[x, QS])]) respectively*)

(*Extemizing the generalised entropy defined above, gives the following QES equations*)

eq1=(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2)));
eq2= (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp)) ff[u-σ1,k]);

(*Numerical computation of the QES location and the (Subscript[S, ST])*)

QESscramblingtrivial={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscramblingtrivial[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscramblingtrivial[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm},QESscramblingtrivial[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,xqm->xxm}},{xxm,xxp},WorkingPrecision->105]]
NSgenscramblingtrivial3[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[-2m xxm(u+σ1)((( xxp-F[u-σ1,T1,k]))/((xxp-xxm)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]/.NSolveQESscramblingtrivial[u,σ1,T0,T1,k][[3]]]
NSgenscramblingtrivial4[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,m_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[-2m xxm(u+σ1)((( xxp-F[u-σ1,T1,k]))/((xxp-xxm)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xxp xxm))/(xxp-xxm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]/.NSolveQESscramblingtrivial[u,σ1,T0,T1,k][[4]]]
tableSscramblingtrivial3=Table[{U,NSgenscramblingtrivial3[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,1,30,(1/10)}];
tableSscramblingtrivial4=Table[{U,NSgenscramblingtrivial4[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvm,vvϕ0,vvϕr,vvGN]},{U,1,30,(1/10)}];
ListPlot[{tableSscramblingtrivial3,tableSscramblingtrivial4}]
tableQESx1mscramblingtrivial=Table[{U,NSolveQESscramblingtrivial[U,vvσ1,vvT0,vvT1,vvk][[4,1,2]]},{U,1,30,(1/10)}];
tableQESx1pscramblingtrivial=Table[{U,NSolveQESscramblingtrivial[U,vvσ1,vvT0,vvT1,vvk][[4,2,2]]},{U,1,30,(1/10)}];
ListPlot[{tableQESx1mscramblingtrivial}];
ListPlot[{tableQESx1pscramblingtrivial}];

LateTrivial Saddle

Slatetrivial[yqm_,xqp_, σ1_,σ2_, u_, k_]:=(((ϕ0+ϕ[xqp,yqm,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)(((((u+σ1-yqm) (xqp-ff[u-σ1,k]) ))/(((xqp-ff[yqm,k]) )))) Sqrt[(((ffd[yqm,k] ))/((ffd[u-σ1,k] )))]]]+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]

(*xqp and yqm are (SuperPlus[(Subscript[x, QL])]) and (SuperMinus[(Subscript[y, QL])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, LT])*)

QESlatetrivial={(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k])+(2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2)-((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2)+(2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k]))+(k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/NFprime[yqm,T1,k])};
NSolveQESlatetrivial[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_,ϕϕr_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlatetrivial[[1]]/.{T1->TT1,k->kk,ϕr-> ϕϕr,u->uu,σ1->σσ1,xqp->xxp,yqm->yym},QESlatetrivial[[2]]/.{T1->TT1,k->kk,ϕr-> ϕϕr,u->uu,σ1->σσ1,xqp->xxp,yqm->yym}},{yym,-(Log[(8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk)]/(2 (π TT1))),0,uu},{xxp,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],1/(π TT0)},WorkingPrecision->105,MaxIterations->200]]
NSgenlatetrivial[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_]:=With[{u=SetPrecision[UU,150]},(ϕr/(4 GN)) ( (Fpprime[yym,T1,k]/Fprime[yym,T1,k]+2 (Fprime[yym,T1,k]/(xxp- F[yym,T1,k])))+2 k  Log[Abs[((2(xxp-F[u-σ1,T1,k])(u+σ1-yym))/(ϵ (xxp-F[yym,T1,k]) ))√(Fprime[yym,T1,k]/Fprime[u-σ1,T1,k])]])+2 k (ϕr/(4 GN)) Log[Abs[2(σ2/ϵ)]]/.NSolveQESlatetrivial[u,σ1,T0,T1,k,ϕr]]
tableSlatetrivial=Table[{U,NSgenlatetrivial[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN]},{U,6,30,1/10}];
ListPlot[{tableSlatetrivial}]
tableQESy1mlatetrivial=Table[{U,NSolveQESlatetrivial[U,vvσ1,vvT0,vvT1,vvk,vvϕr][[1,2]]},{U,6,30,1/10}];
tableQESx1platetrivial=Table[{U,NSolveQESlatetrivial[U,vvσ1,vvT0,vvT1,vvk,vvϕr][[2,2]]},{U,6,30,1/10}];
ListPlot[{tableQESy1mlatetrivial}];
ListPlot[{tableQESx1platetrivial}];

Nontrivial QES saddles: late times σ1<σ2<u

ScramblingQuench Saddle

Sscramblingquench[xqp_, xqm_,σ2_ ,σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[2m xqm(u+σ1)((( ff[u -σ1,k]-xqp))/((xqp-xqm)(Sqrt[ffd[u-σ1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[m(u+σ2)((ff[u-σ2,k])/(Sqrt[ffd[u-σ2,k]]))]

(*xqm and xqp are (SuperMinus[(Subscript[x, QS])]) and (SuperPlus[(Subscript[x, QS])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, SQ])*)

QESscramblingquench={(k (xqp^2)-xqm (-1+k xqp+(π^2) (T0^2) (xqp^2))),
 (k xqm (xqm-xqp)+(-1+(π^2) (T0^2) (xqm^2)) xqp+(1-(π^2) (T0^2) (xqm^2)+k (-xqm+xqp))NF[u-σ1,T1,k])};
NSolveQESscramblingquench[UU_?NumberQ,σσ1_?NumberQ,σσ2_?NumberQ,TT0_,TT1_,kk_,ϕϕr_,GGN_]:=With[{uu=SetPrecision[UU,150]},NSolve[{QESscramblingquench[[1]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,xqm->xx1m},QESscramblingquench[[2]]/.{T0->TT0,T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xx1p,xqm->xx1m}},{xx1m,xx1p},WorkingPrecision->105]]
NSgenscramblingquench3[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[2m xx1m(u+σ1)((( F[u -σ1,T1,k]-xx1p))/((xx1p-xx1m)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx1p xx1m))/(xx1p-xx1m)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1,k]]))]]/.NSolveQESscramblingquench[u,σ1,σ2,T0,T1,k,ϕr,GN][[3]]]
NSgenscramblingquench4[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},2 k ((ϕr)/(4GN))Log[Abs[2m xx1m(u+σ1)((( F[u -σ1,T1,k]-xx1p))/((xx1p-xx1m)(Sqrt[Fprime[u-σ1,T1,k]])))]]+((ϕ0+2ϕr(((1-(((π T0))^2) xx1p xx1m))/(xx1p-xx1m)))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1, k]]))]]/.NSolveQESscramblingquench[u,σ1,σ2,T0,T1,k,ϕr,GN][[4]]]
tableSscramblingquench3=Table[{U,NSgenscramblingquench3[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,30,41,(1/10)}];
tableSscramblingquench4=Table[{U,NSgenscramblingquench4[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,30,41,(1/10)}];
ListPlot[{tableSscramblingquench3,tableSscramblingquench4}]
tableQESx1mscramblingquench=Table[{U,NSolveQESscramblingquench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,1,2]]},{U,30,41,(1/10)}];
tableQESx1pscramblingquench=Table[{U,NSolveQESscramblingquench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϕr,vvGN][[4,2,2]]},{U,30,41,(1/10)}];
ListPlot[{tableQESx1mscramblingquench}];
ListPlot[{tableQESx1pscramblingquench}];

LateQuench Saddle

Slatequench[xqp_, yqm_,σ2_ ,σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)( (u+σ1)-yqm)(((xqp-ff[u-σ1,k])/(-ff[yqm,k]+xqp)))((Sqrt[ffd[yqm,k]])/(Sqrt[ffd[u-σ1,k]]))]]+(((ϕ0+ϕ[xqp,yqm,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((ff[u-σ2,k])/(Sqrt[ffd[u-σ2,k]]))]]

(*yqm and xqp are (SuperMinus[(Subscript[y, QL])]) and (SuperPlus[(Subscript[x, QL])]) respectively*)

(*Numerical computation of the QES location and the (Subscript[S, LQ])*)

(*Now let us do some numerics*)
QESlatequench={(-k  ((NF[yqm,T1,k])^2)+k NF[yqm,T1,k] (xqp+ NF[u-σ1,T1,k])+xqp NFprime[yqm,T1,k]- NF[u-σ1,T1,k] (k xqp+NFprime[yqm,T1,k])),(((2 k NFprime[yqm,T1,k])/(xqp-NF[yqm,T1,k]))+((2((NFprime[yqm,T1,k])^2))/(((xqp-NF[yqm,T1,k]))^2))-(((NFpprime[yqm,T1,k])^2)/((NFprime[yqm,T1,k])^2))+((2 (-k xqp+k NF[yqm,T1,k]+(u-yqm+σ1) NFpprime[yqm,T1,k]))/((u-yqm+σ1) (xqp-NF[yqm,T1,k])))+((k NFpprime[yqm,T1,k]+NFppprime[yqm,T1,k])/(NFprime[yqm,T1,k])))};
NSolveQESlatequench[UU_?NumberQ,σσ1_?NumberQ,TT0_,TT1_,kk_]:=With[{uu=SetPrecision[UU,150]},FindRoot[{QESlatequench[[1]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym},QESlatequench[[2]]/.{T1->TT1,k->kk,u->uu,σ1->σσ1,xqp->xxp,yqm->yym}},{yym,-((Log[((8 (E^(-2 π TT1 (uu-σσ1))) π TT1)/(3 kk))])/(2 (π TT1))),0,uu},{xxp,+(1/3) (-NF[uu-σσ1,TT1,kk]+4Ntinfinity[TT1,kk]),Ntinfinity[TT1,kk],(1/(π TT0))},WorkingPrecision->105,MaxIterations->200]]
NSgenlatequench[UU_?NumberQ,σ1_?NumberQ,σ2_?NumberQ,T0_,T1_,k_,ϵ_,ϕ0_,ϕr_,GN_,m_]:=With[{u=SetPrecision[UU,150]},((ϕr)/(4GN))( (((Fpprime[yym,T1,k])/(Fprime[yym,T1,k]))+2 ((Fprime[yym,T1,k])/(xxp- F[yym,T1,k])))+2k Log[Abs[((2(xxp-F[u-σ1,T1,k])(u+σ1-yym))/((xxp-NF[yym,T1,k]) ϵ))(Sqrt[((Fprime[yym,T1,k])/(Fprime[u-σ1,T1,k]))])]])+2 k ((ϕr)/(4GN))Log[Abs[m(u+σ2)((F[u-σ2,T1,k])/(Sqrt[Fprime[u-σ2,T1,k]]))]]/.NSolveQESlatequench[u,σ1,T0,T1,k]]
tableSlatequench=Table[{U,NSgenlatequench[U,vvσ1,vvσ2,vvT0,vvT1,vvk,vvϵ,vvϕ0,vvϕr,vvGN,vvm]},{U,30,41,(1/10)}];
ListPlot[{tableSlatequench}]
tableQESy1mlatequench=Table[{U,NSolveQESlatequench[U,vvσ1,vvT0,vvT1,vvk][[1,2]]},{U,30,41,(1/10)}];
tableQESx1platequench=Table[{U,NSolveQESlatequench[U,vvσ1,vvT0,vvT1,vvk][[2,2]]},{U,30,41,(1/10)}];
ListPlot[{tableQESy1mlatequench}];
ListPlot[{tableQESx1platequench}];

Combined plot of nontrivial QES saddles

ListPlot[{tableSscramblingtrivial4,tableSlatetrivial,tableSscramblingquench4,tableSlatequench},PlotLegends->{"ScramblingTrivial Saddle","LateTrivial Saddle","ScramblingQuench Saddle","LateQuench Saddle"}]

Combined plot of trivial QES saddles

ListPlot[{tableSCrossDisconnected,tableSCrossConnected,tableSFutureConnected},PlotLegends-> {"CrossDisconnected Saddle","CrossConnected Saddle","FutureConnected Saddle"},PlotRange->All]

Combined plot of trivial and nontrivial QES saddles

ListPlot[{tableSCrossDisconnected,tableSCrossConnected,tableSFutureConnected,tableSscramblingtrivial4,tableSlatetrivial,tableSscramblingquench4,tableSlatequench},PlotLegends-> {"CrossDisconnected Saddle","CrossConnected Saddle","FutureConnected Saddle","ScramblingTrivial Saddle","LateTrivial Saddle","ScramblingQuench Saddle","LateQuench Saddle"}]

Size of the entanglement wedge and subregion complexity

tableQESx1mlatetrivial=Table[F[tableQESy1mlatetrivial[[i,2]],vvT1,vvk],{i,1,Length[tableQESy1mlatetrivial]}];
tableQESx1mlatequench=Table[F[tableQESy1mlatequench[[i,2]],vvT1,vvk],{i,1,Length[tableQESy1mlatequench]}];

tableQES2sscramblingtrivial=Table[{tableQESx1pscramblingtrivial[[i,1]],tableQESx1pscramblingtrivial[[i,2]]-tableQESx1mscramblingtrivial[[i,2]]},{i,1,Length[tableQESx1pscramblingtrivial]}];
tableQES2slatetrivial=Table[{tableQESx1platetrivial[[i,1]],tableQESx1platetrivial[[i,2]]-tableQESx1mlatetrivial[[i]]},{i,1,Length[tableQESx1platetrivial]}];
tableQES2sscramblingquench=Table[{tableQESx1pscramblingquench[[i,1]],tableQESx1pscramblingquench[[i,2]]-tableQESx1mscramblingquench[[i,2]]},{i,1,Length[tableQESx1pscramblingquench]}];
tableQES2slatequench=Table[{tableQESx1platequench[[i,1]],tableQESx1platequench[[i,2]]-tableQESx1mlatequench[[i]]},{i,1,Length[tableQESx1platequench]}];

(*In order to compute the increase in subregion complexity, we use equation 4.22*)

tableQES2sscramblingtrivialevol=Table[{tableQESx1pscramblingtrivial[[i,1]],Log[2/(π vvT0(tableQESx1pscramblingtrivial[[i,2]]-tableQESx1mscramblingtrivial[[i,2]]))]},{i,9,243}];

tableQES2slatetrivialevol=Table[{tableQESx1platetrivial[[i,1]],Log[2/(π vvT0(tableQESx1platetrivial[[i,2]]-tableQESx1mlatetrivial[[i]]))]},{i,193,241}];

ListPlot[{tableQES2sscramblingtrivialevol,tableQES2slatetrivialevol},PlotLegends->{"Δsubregion complexity ScramblingTrivial","Δ subregion complexity LateTrivial"},PlotRange->All,PlotStyle->{Red,Purple}]

Analytical derivations

von Neumann entropy of the radiation

12 π (Es/c)=m

Zmin[y_]:=-I y
Zplus[y_]:=(((12 π (Es/c)))^(-2))(I/(f[y]))
Zbarplus[y_]:=-(((12 π (Es/c)))^(-2))(I/(f[y]))
Zbarmin[y_]:=I y

Wmin[x_]:=I g[-x]
Wplus[x_]:=(((12 π (Es/c)))^(-2))(I/x)
Wbarplus[x_]:=-(((12 π (Es/c)))^(-2))(I/x)
Wbarmin[x_]:=-I g[-x]

(*First we look at semi-infinte intervals*)

(* endpoint in AdS, above the shock*)

Assuming[{Es>0, c>0,xp>0,xm>0,f[ym]>0,f'[ym]>0},Series[-I (((w[x]-wbar[xbar]))/(δ Ω1))/.Ω1-> ((x+xbar)/2)(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.w-> Wmin/.wbar->Wbarplus/.g[-x]->- y/.g'[-x]->-(((f'[y]))^(-1))/.x->-xm/.xbar->xp/.y->-ym/.f'[-ym]->-f'[ym]//Simplify,{Es,∞,0}]]//Simplify

(* endpoint in Bath, above the shock*)

Assuming[{Es>0, c>0,xm>0,xp>0,f[ym]>0,f[yp]>0},Series[-I (((z[y]-zbar[ybar]))/(δ Ω1))/.Ω1-> ϵ (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.z-> Zmin/.zbar->Zbarplus/.y->-ym/.ybar->yp//Simplify,{Es,∞,0}]]//Simplify

(*endpoint in AdS, below the shock*)

Assuming[{Es>0, c>0,xm<0,xp>0,f[ym]<0,f[yp]>0},Series[-I (((w[x]-wbar[xbar]))/(δ Ω1))/.Ω1-> ((x+xbar)/2)(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.w-> Wplus/.wbar->Wbarplus/.g[-x]->- y/.g'[-x]->-(((f'[y]))^(-1))/.x->-xm/.xbar->xp/.y->-ym/.f'[-ym]->-f'[ym]//Simplify,{Es,∞,0}]]//Simplify

(*endpoint in Bath, below shock*)

Assuming[{Es>0, c>0,xm>0,xp<0,f[ym]>0,f[yp]<0},Series[-I (((z[y]-zbar[ybar]))/(δ Ω1))/.Ω1-> ϵ (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.z-> Zmin/.zbar->Zbarmin/.y->-ym/.ybar->yp//Simplify,{Es,∞,0}]]//Simplify

(*Now let us look at finite intervals, they have two endpoints referred to as "1" and "2" *)
(*{Disconnected η<(1/2),Connected η>(1/2),η}*)

(*1 in bath below the shock, 2 in ads above the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1<0,f[ym1]>0,f[yp1]<0,f'[ym2]>0,xp2>0,xm2>0,f[ym2]>0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ϵ (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->((x+xbar)/2)(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Zmin/.zbar->Zbarmin/.y->-ym1/.ybar->yp1/.w-> Wmin/.wbar->Wbarplus/.g[-x]->- y2/.g'[-x]->-(((f'[y2]))^(-1))/.x->-xm2/.xbar->xp2/.y2->-ym2/.f'[-ym2]->-f'[ym2]//Simplify,{Es,∞,0}]]//Simplify

(*1 in bath below the shock, 2 in bath above the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1<0,f[ym1]>0,f[yp1]<0,f'[ym2]>0,xp2>0,xm2>0,f[ym2]>0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ϵ (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->ϵ(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Zmin/.zbar->Zbarmin/.y->-ym1/.ybar->yp1/.w-> Zmin/.wbar->Zbarplus/.x->-ym2/.xbar->yp2//Simplify,{Es,∞,0}]]//Simplify

(*1 in bath below the shock, 2 in ads below the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1<0,f[ym1]>0,f[yp1]<0,f'[ym2]<0,xp2>0,xm2<0,f[ym2]<0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ϵ (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->((x+xbar)/2)(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Zmin/.zbar->Zbarmin/.y->-ym1/.ybar->yp1/.w-> Wplus/.wbar->Wbarplus/.g[-x]->- y2/.g'[-x]->-(((f'[y2]))^(-1))/.x->-xm2/.xbar->xp2/.y2->-ym2/.f'[-ym2]->-f'[ym2]//Simplify,{Es,∞,0}]]//Simplify

(*η=0 so always in disconnected case*)

(*1 in bath below the shock, 2 in bathbelow the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1<0,f[ym1]>0,f[yp1]<0,f'[ym2]>0,xp2<0,xm2>0,f[ym2]>0,f[yp2]<0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ϵ (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->ϵ(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Zmin/.zbar->Zbarmin/.y->-ym1/.ybar->yp1/.w-> Zmin/.wbar->Zbarmin/.x->-ym2/.xbar->yp2//Simplify,{Es,∞,0}]]//Simplify

(*1 in bath above the shock, 2 in ads above the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1>0,f[ym1]>0,f[yp1]>0,f'[ym2]>0,xp2>0,xm2>0,f[ym2]>0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ϵ (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->((x+xbar)/2)(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Zmin/.zbar->Zbarplus/.y->-ym1/.ybar->yp1/.w-> Wmin/.wbar->Wbarplus/.g[-x]->- y2/.g'[-x]->-(((f'[y2]))^(-1))/.x->-xm2/.xbar->xp2/.y2->-ym2/.f'[-ym2]->-f'[ym2]//Simplify,{Es,∞,0}]]//Simplify

(*1 in bath above the shock, 2 in bath above the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1>0,f[ym1]>0,f[yp1]>0,f'[ym2]>0,xp2>0,xm2>0,f[ym2]>0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ϵ (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->ϵ(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Zmin/.zbar->Zbarplus/.y->-ym1/.ybar->yp1/.w-> Zmin/.wbar->Zbarplus/.x->-ym2/.xbar->yp2//Simplify,{Es,∞,0}]]//Simplify

(*1 in bath above the shock, 2 in ads below the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1>0,f[ym1]>0,f[yp1]>0,f'[ym2]<0,xp2>0,xm2<0,f[ym2]<0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ϵ (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->((x+xbar)/2)(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Zmin/.zbar->Zbarplus/.y->-ym1/.ybar->yp1/.w-> Wplus/.wbar->Wbarplus/.g[-x]->- y2/.g'[-x]->-(((f'[y2]))^(-1))/.x->-xm2/.xbar->xp2/.y2->-ym2/.f'[-ym2]->-f'[ym2]//Simplify,{Es,∞,0}]]//Simplify

(*1 in bath above the shock, 2 in bath below the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1>0,f[ym1]>0,f[yp1]>0,f'[ym2]>0,xp2<0,xm2>0,f[ym2]>0,f[yp2]<0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ϵ (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->ϵ(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Zmin/.zbar->Zbarplus/.y->-ym1/.ybar->yp1/.w-> Zmin/.wbar->Zbarmin/.x->-ym2/.xbar->yp2//Simplify,{Es,∞,0}]]//Simplify

(*1 in AdS below the shock, 2 in ads above the shock*)

Assuming[{Es>0, c>0,xm1<0,xp1>0,f[ym1]<0,f[yp1]>0,f'[ym2]>0,xp2>0,xm2>0,f[ym2]>0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ((y+ybar)/2) (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->((x+xbar)/2)(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Wplus/.zbar->Wbarplus/.g[-y]->- y1/.g'[-y]->-(((f'[y1]))^(-1))/.y->-xm1/.ybar->xp1/.y1->-ym1/.f'[-ym1]->-f'[ym1]/.w-> Wmin/.wbar->Wbarplus/.g[-x]->- y2/.g'[-x]->-(((f'[y2]))^(-1))/.x->-xm2/.xbar->xp2/.y2->-ym2/.f'[-ym2]->-f'[ym2]//Simplify,{Es,∞,0}]]//Simplify

(*1 in AdS below the shock, 2 in bath above the shock*)

Assuming[{Es>0, c>0,xm1<0,xp1>0,f[ym1]<0,f[yp1]>0,f'[ym2]>0,xp2>0,xm2>0,f[ym2]>0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ((y+ybar)/2) (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->ϵ(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))//.z-> Wplus/.zbar->Wbarplus/.g[-y]->- y1/.g'[-y]->-(((f'[y1]))^(-1))/.y->-xm1/.ybar->xp1/.y1->-ym1/.f'[-ym1]->-f'[ym1]/.w-> Zmin/.wbar->Zbarplus/.x->-ym2/.xbar->yp2//Simplify,{Es,∞,0}]]//Simplify

(*1 in AdS below the shock, 2 in ads below the shock*)

Assuming[{Es>0, c>0,xm1<0,xp1>0,f[ym1]<0,f[yp1]>0,f'[ym2]<0,xp2>0,xm2<0,f[ym2]<0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ((y+ybar)/2)(((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->((x+xbar)/2)(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Wplus/.zbar->Wbarplus/.g[-y]->- y1/.g'[-y]->-(((f'[y1]))^(-1))/.y->-xm1/.ybar->xp1/.y1->-ym1/.f'[-ym1]->-f'[ym1]/.w-> Wplus/.wbar->Wbarplus/.g[-x]->- y2/.g'[-x]->-(((f'[y2]))^(-1))/.x->-xm2/.xbar->xp2/.y2->-ym2/.f'[-ym2]->-f'[ym2]//Simplify,{Es,∞,0}]]//Simplify

(*1 in AdS below the shock, 2 in bathbelow the shock*)

Assuming[{Es>0, c>0,xm1<0,xp1>0,f[ym1]<0,f[yp1]>0,f'[ym2]>0,xp2<0,xm2>0,f[ym2]>0,f[yp2]<0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ((y+ybar)/2) (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->ϵ(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Wplus/.zbar->Wbarplus/.g[-y]->- y1/.g'[-y]->-(((f'[y1]))^(-1))/.y->-xm1/.ybar->xp1/.y1->-ym1/.f'[-ym1]->-f'[ym1]/.w-> Zmin/.wbar->Zbarmin/.x->-ym2/.xbar->yp2//Simplify,{Es,∞,0}]]//Simplify

(*1 in aDs above the shock, 2 in ads above the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1>0,f[ym1]>0,f[yp1]>0,f'[ym1]>0,f'[ym2]>0,xp2>0,xm2>0,f[ym2]>0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ((y+ybar)/2) (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->((x+xbar)/2)(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Wmin/.zbar->Wbarplus/.g[-y]->- y1/.g'[-y]->-(((f'[y1]))^(-1))/.y->-xm1/.ybar->xp1/.y1->-ym1/.f'[-ym1]->-f'[ym1]/.w-> Wmin/.wbar->Wbarplus/.g[-x]->- y2/.g'[-x]->-(((f'[y2]))^(-1))/.x->-xm2/.xbar->xp2/.y2->-ym2/.f'[-ym2]->-f'[ym2]//Simplify,{Es,∞,0}]]//Simplify

(*1 in AdS above the shock, 2 in bath above the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1>0,f[ym1]>0,f[yp1]>0,f'[ym1]>0,f'[ym2]>0,xp2>0,xm2>0,f[ym2]>0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ((y+ybar)/2) (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->ϵ(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Wmin/.zbar->Wbarplus/.g[-y]->- y1/.g'[-y]->-(((f'[y1]))^(-1))/.y->-xm1/.ybar->xp1/.y1->-ym1/.f'[-ym1]->-f'[ym1]/.w-> Zmin/.wbar->Zbarplus/.x->-ym2/.xbar->yp2//Simplify,{Es,∞,0}]]//Simplify

(*1 in bath above the shock, 2 in ads below the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1>0,f[ym1]>0,f[yp1]>0,f'[ym1]>0,f'[ym2]<0,xp2>0,xm2<0,f[ym2]<0,f[yp2]>0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ((y+ybar)/2) (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->((x+xbar)/2)(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Wmin/.zbar->Wbarplus/.g[-y]->- y1/.g'[-y]->-(((f'[y1]))^(-1))/.y->-xm1/.ybar->xp1/.y1->-ym1/.f'[-ym1]->-f'[ym1]/.w-> Wplus/.wbar->Wbarplus/.g[-x]->- y2/.g'[-x]->-(((f'[y2]))^(-1))/.x->-xm2/.xbar->xp2/.y2->-ym2/.f'[-ym2]->-f'[ym2]//Simplify,{Es,∞,0}]]//Simplify

(*1 in bath above the shock, 2 in ads above the shock*)

Assuming[{Es>0, c>0,xm1>0,xp1>0,f[ym1]>0,f[yp1]>0,f'[ym1]>0,f'[ym2]>0,xp2<0,xm2>0,f[ym2]>0,f[yp2]<0},Series[{(((z[y]-zbar[ybar]))/(δ Ω1))(((w[x]-wbar[xbar]))/(δ Ω2)),(((z[y]-w[x]))/(δ Ω1))(((zbar[ybar]-wbar[xbar]))/(δ Ω2)),(((z[y]-zbar[ybar]))/((z[y]-wbar[xbar])))(((w[x]-wbar[xbar]))/((w[x]-zbar[ybar])))}/.Ω1-> ((y+ybar)/2) (((D[z[y],y]D[zbar[ybar],ybar]))^(1/2))/.Ω2->ϵ(((D[w[x],x]D[wbar[xbar],xbar]))^(1/2))/.z-> Wmin/.zbar->Wbarplus/.g[-y]->- y1/.g'[-y]->-(((f'[y1]))^(-1))/.y->-xm1/.ybar->xp1/.y1->-ym1/.f'[-ym1]->-f'[ym1]/.w-> Zmin/.wbar->Zbarmin/.x->-ym2/.xbar->yp2//Simplify,{Es,∞,0}]]//Simplify

Approximate analytic solutions for (Subscript[x, QL])

Quit[];

Function f and substitutions

(*No need to run f to do approximations once subs are defined*)

f[x_,k_]:=(1/(π T1))(1/(BesselI[1, ((2π T1)/k)] BesselK[0, 2π (T1/k)(E^(-k (x/2)))]+BesselK[1, 2π (T1/k)] BesselI[0,2π (T1/k)(E^(-k (x/2)))]))(BesselI[0, 2π (T1/k)] BesselK[0, 2π (T1/k)(E^(-k (x/2)))]- BesselK[0, 2π (T1/k)] BesselI[0, 2π (T1/k)(E^(-k (x/2)))])
Df[x_,k_]:=((k^2)/(4 (π^2) (T1^2) (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^2)))
DDf[x_,k_]:=(((E^(-((k x)/2))) (k^2) (BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]-BesselI[1,((2 π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)]))/(2 π T1 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^3)))
DDDf[x_,k_]:=-(((E^(-k x)) (k^2) ((((BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)])^2)-3 ((BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)])^2)) ((BesselK[1,((2 π T1)/k)])^2)+2 BesselI[1,((2 π T1)/k)] BesselK[1,((2 π T1)/k)] (BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+3 BesselI[1,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)])+((BesselI[1,((2 π T1)/k)])^2) (((BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)])^2)-3 ((BesselK[1,((2 (E^(-((k x)/2))) π T1)/k)])^2))))/(2 (((BesselI[1,((2 π T1)/k)] BesselK[0,((2 (E^(-((k x)/2))) π T1)/k)]+BesselI[0,((2 (E^(-((k x)/2))) π T1)/k)] BesselK[1,((2 π T1)/k)]))^4)))

(*Now lets define the substitutions we will need next*)

subs={BesselK[0,(α/k)]-> (((((π k)/(2α))))^(1/2))ϵ K0[α,k],BesselI[0,(α/k)]-> ((((( k)/(2π α))))^(1/2))(1/ϵ)I0[α,k],BesselK[0,(β/k)]-> (((((π k)/(2β))))^(1/2))ξ K0[β,k],BesselI[0,(β/k)]-> ((((( k)/(2π β))))^(1/2))(1/ξ)I0[β,k],BesselK[0,(γ/k)]-> (((((π k)/(2γ))))^(1/2))δ K0[γ,k],BesselI[0,(γ/k)]-> ((((( k)/(2π γ))))^(1/2))(1/δ)I0[γ,k],BesselK[1,(β/k)]-> (((((π k)/(2β))))^(1/2))ξ K1[β,k],BesselI[1,(β/k)]-> ((((( k)/(2π β))))^(1/2))(1/ξ)I1[β,k],BesselK[1,(α/k)]-> (((((π k)/(2α))))^(1/2))ϵ(K1[α,k]),BesselI[1,(α/k)]-> ((((( k)/(2π α))))^(1/2))(1/ϵ)I1[α,k],BesselK[1,(γ/k)]-> (((((π k)/(2γ))))^(1/2))δ K1[γ,k],BesselI[1,(γ/k)]-> ((((( k)/(2π γ))))^(1/2))(1/δ)I1[γ,k]};
KK0[α_,k_]:=(1-(k/(8α))+((9(k^2))/(2 (8^2)(α^2)))(*-25*((9 (k^3))/(6*(8^3)(α^3)))+((3675  (k^4) )/(32768 (α^4)))-((59535 (k^5))/(262144 (α^5)))*))
KK1[γ_,k_]:=(1+(((4-1)k)/(8γ))+(((4-1)(4-9)(k^2))/(2 (8^2)(γ^2)))(*+(((4-1)(4-9)(4-25) (k^3))/(6*(8^3)(γ^3)))-((4725 (k^4))/(32768 (γ^4)))+((72765 (k^5))/(262144 (γ^5)))*))
II1[α_,k_]:=(1-(((4-1)k)/(8α))+(((4-1)(4-9)(k^2))/(2 (8^2)(α^2)))(*-(((4-1)(4-9)(4-25) (k^3))/(6*(8^3)(α^3)))-((4725 (k^4))/(32768 (α^4)))-((72765 (k^5))/(262144 (α^5)))*))
II0[α_,k_]:=(1+(k/(8α))+((9(k^2))/(2 (8^2)(α^2)))(*+25*((9 (k^3))/(6*(8^3)(α^3)))+((3675  (k^4) )/(32768 (α^4)))+((59535 (k^5))/(262144 (α^5)))*))

subsexp={K0->KK0,K1->KK1 ,I0->II0,I1->II1};
subsλ={ϵ->ξ (λ^(1/2))};
subs2={ϵ->(E^(-(α/k))),ξ-> (E^(-(β/k))),δ-> (E^(-(γ/k)))};
subs2new={ξ-> (E^(-(β/k))),δ-> (E^(-(γ/k)))};
subs3={α->2π T1,β->2 (E^(-(1/2) U)) π T1,γ->2 T1 (E^(-Y1m/2)) π};

(*U=k u, Y1m= k y1m*)

EOM for (Subscript[x, QL])

(*Now let us look at the eom*)
eqQESx1p[x1p_,y1m_,S1_,u_,k_]:=((Df[y1m,k] (-x1p+ f[-S1+u,k])+k (x1p- f[y1m,k]) ( f[-S1+u,k]- f[y1m,k]))/(k (x1p- f[-S1+u,k]) (((x1p- f[y1m,k]))^2)))
eqQESy1m[x1p_,y1m_,S1_,u_,k_]:=(1/(2 ((Df[y1m,k])^2) (((x1p- f[y1m,k]))^2)))((1/k)(Df[y1m,k] (((x1p- f[y1m,k]))^2) DDDf[y1m,k]+2 ((Df[y1m,k])^2) (x1p- f[y1m,k]) DDf[y1m,k]-DDf[y1m,k] (((x1p- f[y1m,k]))^2) DDf[y1m,k]+2 ((Df[y1m,k])^3) Df[y1m,k])+(1/(S1+u-y1m))Df[y1m,k] (x1p- f[y1m,k]) ((S1+u-y1m) (x1p- f[y1m,k]) DDf[y1m,k]-2 Df[y1m,k] (x1p- f[y1m,k]-(S1+u-y1m) Df[y1m,k])))

(*y1m and x1p are (SuperMinus[(Subscript[y, QS])]) and (SuperPlus[(Subscript[x, QS])]) respectively*)

(*S1 is used instead of σ1*)

t∞-t expansion and the new small parameter λ

(*We also define (Subscript[t, ∞]) and t*)
t∞=(1/(π T1))((BesselI[0,(α/k)])/(BesselI[1,(α/k)]));
t=(1/(π T1))(((BesselI[0, (α/k)] BesselK[0, (β/k)]- BesselK[0,(α/k)] BesselI[0, (β/k)]))/((BesselI[1, (α/k)] BesselK[0,(β/k)]+BesselK[1, (α/k)] BesselI[0,(β/k)])));

(((t∞-t))/(2t∞))//Simplify

((BesselI[0,(β/k)] (BesselI[1,(α/k)] BesselK[0,(α/k)]+BesselI[0,(α/k)] BesselK[1,(α/k)]))/(2 BesselI[0,(α/k)] (BesselI[1,(α/k)] BesselK[0,(β/k)]+BesselI[0,(β/k)] BesselK[1,(α/k)])))

(((t∞-t))/(2t∞))/.subs//Simplify

(((ϵ^2) I0[β,k] (I1[α,k] K0[α,k]+I0[α,k] K1[α,k]))/(2 I0[α,k] ((ξ^2) I1[α,k] K0[β,k]+(ϵ^2) I0[β,k] K1[α,k])))

(*Some intermediate thoughts*)

Series[(((t∞-t))/(2t∞))/.subs/.subsexp/.subs3,{k,0,1}]//Simplify

((ϵ^2)/((ϵ^2)+(ξ^2)))+(((-2 (ϵ^4)+(1+(e^(U/2))) (ϵ^2) (ξ^2)) k)/(8 π T1 ((((ϵ^2)+(ξ^2)))^2)))+((O[k])^(3/2))

(*and if one performed a series and assumed that (ϵ^2)(ξ^(-2)) is small*)

((ϵ^2)/((ϵ^2)+(ξ^2)))=(((ξ^2)+(ϵ^2))/((ξ^2)+(ϵ^2)))-((ξ^2)/((ξ^2)+(ϵ^2)))=1-((ξ^2)/((ξ^2)+(ϵ^2)))=1-(1/(1+((ϵ^2)/(ξ^2))))=1-(1-((ϵ^2)/(ξ^2))+(((((ϵ^2)/(ξ^2))))^2)-(((((ϵ^2)/(ξ^2))))^3))=(ϵ^2)(ξ^(-2))+O[((((ϵ^2)(ξ^(-2))))^2)]
(*and *)

(ϵ^2)(ξ^(-2))/.subs2/.subs3//FullSimplify

(e^((4 (-1+(e^(-U/2))) π T1)/k))

(*So at last we define*)
λ=((ϵ^2)/(ξ^2))

(*Such that*)
Series[(((t∞-t))/(2t∞))/.subs/.subs3/.subsλ,{λ,0,1}]//Simplify

((I0[2 (e^(-U/2)) π T1,k] (I1[2 π T1,k] K0[2 π T1,k]+I0[2 π T1,k] K1[2 π T1,k]) λ)/(2 I0[2 π T1,k] I1[2 π T1,k] K0[2 (e^(-U/2)) π T1,k]))+((O[λ])^(3/2))

((tinfty-f[u,k])/(2 tinfty))=( λ(1+# k+# (k^2))+O[(λ^2)])

Series[((I0[2 (e^(-U/2)) π T1,k] (I1[2 π T1,k] K0[2 π T1,k]+I0[2 π T1,k] K1[2 π T1,k]) )/(2 I0[2 π T1,k] I1[2 π T1,k] K0[2 (e^(-U/2)) π T1,k]))/.subsexp,{k,0,2}]

1+(((1+(e^(U/2))) k)/(8 π T1))+(((5+2 (e^(U/2))+(e^U)) (k^2))/(128 (π^2) (T1^2)))+((O[k])^3)

x1m-t expansion and the new small parameter τ which is BIGGER than λ but still smaller than k

(*We also define tinfinity and what we also can call t*)
t∞=(1/(π T1))((BesselI[0,(α/k)])/(BesselI[1,(α/k)]));
x1m=(1/(π T1))(((BesselI[0, (α/k)] BesselK[0, (γ/k)]- BesselK[0,(α/k)] BesselI[0, (γ/k)]))/((BesselI[1, (α/k)] BesselK[0,(γ/k)]+BesselK[1, (α/k)] BesselI[0,(γ/k)])));

(((t∞-x1m))/(2t∞))//Simplify

((BesselI[0,(γ/k)] (BesselI[1,(α/k)] BesselK[0,(α/k)]+BesselI[0,(α/k)] BesselK[1,(α/k)]))/(2 BesselI[0,(α/k)] (BesselI[1,(α/k)] BesselK[0,(γ/k)]+BesselI[0,(γ/k)] BesselK[1,(α/k)])))

(((t∞-x1m))/(2t∞))/.subs//Simplify

(((ϵ^2) I0[γ,k] (I1[α,k] K0[α,k]+I0[α,k] K1[α,k]))/(2 I0[α,k] ((δ^2) I1[α,k] K0[γ,k]+(ϵ^2) I0[γ,k] K1[α,k])))

(*In order to get to arrive at a usefull expression we performe a series and assume that (ϵ^2)(δ^(-2)) is small such that*)

((ϵ^2)/((ϵ^2)+(δ^2)))=(((δ^2)+(ϵ^2))/((δ^2)+(ϵ^2)))-((δ^2)/((δ^2)+(ϵ^2)))=1-((δ^2)/((δ^2)+(ϵ^2)))=1-(1/(1+((ϵ^2)/(δ^2))))=1-(1-((ϵ^2)/(δ^2))+(((((ϵ^2)/(δ^2))))^2)-(((((ϵ^2)/(δ^2))))^3))=(ϵ^2)(δ^(-2))+O[((((ϵ^2)(δ^(-2))))^2)]

(ϵ^2)(δ^(-2))/.subs2/.subs3//FullSimplify

(e^((4 (-1+(e^(-Y1m/2))) π T1)/k))

(*So at last we define*)
τ=((ϵ^2)/(δ^2))

(((ϵ^2) I0[γ,k] (I1[α,k] K0[α,k]+I0[α,k] K1[α,k]))/(2 I0[α,k] ((δ^2) I1[α,k] K0[γ,k]+(ϵ^2) I0[γ,k] K1[α,k])))/.ϵ->δ √τ//Simplify

((τ I0[γ,k] (I1[α,k] K0[α,k]+I0[α,k] K1[α,k]))/(2 I0[α,k] (I1[α,k] K0[γ,k]+τ I0[γ,k] K1[α,k])))

Series[((τ I0[γ,k] (I1[α,k] K0[α,k]+I0[α,k] K1[α,k]))/(2 I0[α,k] (I1[α,k] K0[γ,k]+τ I0[γ,k] K1[α,k]))),{τ,0,1}]

((I0[γ,k] (I1[α,k] K0[α,k]+I0[α,k] K1[α,k]) τ)/(2 I0[α,k] I1[α,k] K0[γ,k]))+((O[τ])^2)

(*Such that*)
((tinfty-f[y1m,k])/(2 tinfty))=(((I0[γ,k] (I1[α,k] K0[α,k]+I0[α,k] K1[α,k]) τ)/(2 I0[α,k] I1[α,k] K0[γ,k]))+O[(τ^2)])=( τ(1+# k+# (k^2))+O[(τ^2)])

Additional subs

(*Let us now use that*)
((tinfty-f[u,k])/(2 tinfty))=( λ(1+(((1+(e^(U/2))) k)/(8 π T1))+# (k^2))+O[(λ^2)])
((tinfty-f[y1m,k])/(2 tinfty))=( τ(1+(((1+(e^(Y1m/2))) k)/(8 π T1))+# (k^2))+O[(τ^2)])
τ=(e^((4 (-1+(e^(-Y1m/2))) π T1)/k))
λ=(e^((4 (-1+(e^(-U/2))) π T1)/k))

(*Now we can use that*)

Series[Log[λ(1+(((1+(e^(k u/2))) k)/(8 π T1))+(((5+2 (e^(U/2))+(e^U)))/(128 (π^2) (T1^2))) (k^2)+γ (k^3))+δ[k] (λ^2)],{λ,0,1},{k,0,2}]

(Log[λ]+(k/(4 π T1))+(((1+2 (e^(U/2))+(e^U)+8 π T1 u) (k^2))/(128 (π^2) (T1^2)))+((O[k])^3))+(δ[0]+(((-δ[0]+4 π T1 δ'[0]) k)/(4 π T1))+(((3 δ[0]-2 (e^(U/2)) δ[0]-(e^U) δ[0]-8 π T1 u δ[0]-32 π T1 δ'[0]+64 (π^2) (T1^2) δ''[0]) (k^2))/(128 (π^2) (T1^2)))+((O[k])^3)) λ+((O[λ])^2)

(*Let us write it out*)
Log[((tinfty-f[u,k])/(2 tinfty))]=-4π (T1/k)(1-(e^((-ku)/2)))+(k/(4 π T1))+(((1+2 (e^(k u/2))+(e^(k u))+8 π T1 u) (k^2))/(128 (π^2) (T1^2)))+((O[k])^3)
(*So this means that*)
f[u,k]=tinfty-2 tinfty (e^(-4π (T1/k)(1-(e^((-ku)/2)))+(k/(4 π T1))+((O[k])^2)))
(*at least approximately we can say that*)
f[u,k]=tinfty-2 tinfty (e^(-4π (T1/k)(1-(e^((-ku)/2)))+(k/(4 π T1))+((O[k])^2)))

(*We write it as*)

f[u_]:=t∞(1-2(e^(((-4π T1)/k)(1-(e^(-k u/2)))+(k/(4 π T1))+((O[k])^2))))
2(e^(((-4π T1)/k)(1-(e^(-k u/2)))+(k/(4 π T1))+((O[k])^2)))=t∞-f[u]

(*Also we can use that*)

D[tinfty(1-2(e^(((-4π T1)/k)(1-(e^(-k u/2)))+(k/(4 π T1))+(((1+2 (e^(k u/2))+(e^(k u))+8 π T1 u) (k^2))/(128 (π^2) (T1^2)))+α[u,k](k^3)))),u]//Simplify

-2 (e^((k/(4 π T1))+((4 (-1+(e^(-((k u)/2)))) π T1)/k)+(((k^2) (1+2 (e^((k u)/2))+(e^(k u))+8 π T1 u))/(128 (π^2) (T1^2)))+(k^3) α[u,k])) tinfty (-2 (e^(-((k u)/2))) π T1+(((k^2) ((e^((k u)/2)) k+(e^(k u)) k+8 π T1))/(128 (π^2) (T1^2)))+(k^3) Derivative[1,0][α][u,k])

(*Which we can rewrite as*)

Df[u,k]=2π T1 (e^(-k u/2))(t∞-f[u])-(t∞-f[u])((k^2)/(16 π T1))+O[(k^3)]

Df[u_]:=2π T1 (e^(-k u/2))(t∞-f[u])

(*Let us see what happens for higher derivatives, we can use that*)

D[-2 (e^((k/(4 π T1))+((4 (-1+(e^(-((k u)/2)))) π T1)/k)+(((k^2) (1+2 (e^((k u)/2))+(e^(k u))+8 π T1 u))/(128 (π^2) (T1^2)))+(k^3) α[u,k])) tinfty (-2 (e^(-((k u)/2))) π T1+(((k^2) (8 π T1))/(128 (π^2) (T1^2)))+(k^3)( Derivative[1,0][α][u,k]+((((e^((k u)/2)) +(e^(k u)) ))/(128 (π^2) (T1^2))))),u]//Simplify

2 (e^((k/(4 π T1))+((4 (-1+(e^(-((k u)/2)))) π T1)/k)+(((k^2) (1+2 (e^((k u)/2))+(e^(k u))+8 π T1 u))/(128 (π^2) (T1^2)))+(k^3) α[u,k])) tinfty (-(e^(-((k u)/2))) k π T1-(((e^(-k u)) ((((e^(k u)) (k^3)+(e^((3 k u)/2)) (k^3)+8 (e^((k u)/2)) (k^2) π T1-256 (π^3) (T1^3)+128 (e^((k u)/2)) (k^3) (π^2) (T1^2) Derivative[1,0][α][u,k]))^2))/(16384 (π^4) (T1^4)))-(k^3) (((((e^((k u)/2))+2 (e^(k u))) k)/(256 (π^2) (T1^2)))+Derivative[2,0][α][u,k]))

Collect[(-(e^(-((k u)/2))) k π T1-(((e^(-k u)) ((((e^(k u)) (k^3)+(e^((3 k u)/2)) (k^3)+8 (e^((k u)/2)) (k^2) π T1-256 (π^3) (T1^3)+128 (e^((k u)/2)) (k^3) (π^2) (T1^2) Derivative[1,0][α][u,k]))^2))/(16384 (π^4) (T1^4)))-(k^3) (((((e^((k u)/2))+2 (e^(k u))) k)/(256 (π^2) (T1^2)))+Derivative[2,0][α][u,k])),{k,(k^2),(k^3),(k^4)},Simplify]

(1/4) (e^(-((k u)/2))) (k^2)-(((1+(e^((k u)/2))+2 (e^(k u))) (k^4))/(256 (π^2) (T1^2)))-(e^(-((k u)/2))) k π T1-4 (e^(-k u)) (π^2) (T1^2)-(((k^5) ((e^((k u)/2))+(e^(k u))+128 (π^2) (T1^2) Derivative[1,0][α][u,k]))/(1024 (π^3) (T1^3)))-(((k^6) ((((e^((k u)/2))+(e^(k u))+128 (π^2) (T1^2) Derivative[1,0][α][u,k]))^2))/(16384 (π^4) (T1^4)))+(((k^3) (1+(e^((k u)/2))+128 (e^(-((k u)/2))) (π^2) (T1^2) Derivative[1,0][α][u,k]-32 π T1 Derivative[2,0][α][u,k]))/(32 π T1))

D[2 (e^((k/(4 π T1))+((4 (-1+(e^(-((k u)/2)))) π T1)/k)+(((k^2) (1+2 (e^((k u)/2))+(e^(k u))+8 π T1 u))/(128 (π^2) (T1^2)))+(k^3) α[u,k])) tinfty (-(((e^((k u)/2)) (k^4)+2 (e^(k u)) (k^4))/(256 (π^2) (T1^2)))-(e^(-((k u)/2))) k π T1-(((-2 (e^(-((k u)/2))) π T1+(((k^2) ((e^((k u)/2)) k+(e^(k u)) k+8 π T1))/(128 (π^2) (T1^2)))+(k^3) Derivative[1,0][α][u,k]))^2)-(k^3) Derivative[2,0][α][u,k]),u]//Simplify

2 (e^((k/(4 π T1))+((4 (-1+(e^(-((k u)/2)))) π T1)/k)+(((k^2) (1+2 (e^((k u)/2))+(e^(k u))+8 π T1 u))/(128 (π^2) (T1^2)))+(k^3) α[u,k])) tinfty (-(((e^((k u)/2)) (k^5)+4 (e^(k u)) (k^5))/(512 (π^2) (T1^2)))+(1/2) (e^(-((k u)/2))) (k^2) π T1+(-2 (e^(-((k u)/2))) π T1+(((k^2) ((e^((k u)/2)) k+(e^(k u)) k+8 π T1))/(128 (π^2) (T1^2)))+(k^3) Derivative[1,0][α][u,k]) (-(((e^((k u)/2)) (k^4)+2 (e^(k u)) (k^4))/(256 (π^2) (T1^2)))-(e^(-((k u)/2))) k π T1-(((-2 (e^(-((k u)/2))) π T1+(((k^2) ((e^((k u)/2)) k+(e^(k u)) k+8 π T1))/(128 (π^2) (T1^2)))+(k^3) Derivative[1,0][α][u,k]))^2)-(k^3) Derivative[2,0][α][u,k])-2 (-2 (e^(-((k u)/2))) π T1+(((k^2) ((e^((k u)/2)) k+(e^(k u)) k+8 π T1))/(128 (π^2) (T1^2)))+(k^3) Derivative[1,0][α][u,k]) ((((e^((k u)/2)) (k^4)+2 (e^(k u)) (k^4))/(256 (π^2) (T1^2)))+(e^(-((k u)/2))) k π T1+(k^3) Derivative[2,0][α][u,k])-(k^3) Derivative[3,0][α][u,k])

Collect[ (-(((e^((k u)/2)) (k^5)+4 (e^(k u)) (k^5))/(512 (π^2) (T1^2)))+(1/2) (e^(-((k u)/2))) (k^2) π T1+(-2 (e^(-((k u)/2))) π T1+(((k^2) ((e^((k u)/2)) k+(e^(k u)) k+8 π T1))/(128 (π^2) (T1^2)))+(k^3) Derivative[1,0][α][u,k]) (-(((e^((k u)/2)) (k^4)+2 (e^(k u)) (k^4))/(256 (π^2) (T1^2)))-(e^(-((k u)/2))) k π T1-(((-2 (e^(-((k u)/2))) π T1+(((k^2) ((e^((k u)/2)) k+(e^(k u)) k+8 π T1))/(128 (π^2) (T1^2)))+(k^3) Derivative[1,0][α][u,k]))^2)-(k^3) Derivative[2,0][α][u,k])-2 (-2 (e^(-((k u)/2))) π T1+(((k^2) ((e^((k u)/2)) k+(e^(k u)) k+8 π T1))/(128 (π^2) (T1^2)))+(k^3) Derivative[1,0][α][u,k]) ((((e^((k u)/2)) (k^4)+2 (e^(k u)) (k^4))/(256 (π^2) (T1^2)))+(e^(-((k u)/2))) k π T1+(k^3) Derivative[2,0][α][u,k])-(k^3) Derivative[3,0][α][u,k]),{k,(k^2),(k^3),(k^4)},Simplify]

(1/4) (e^(-k u)) (-3+2 (e^((k u)/2))) (k^2) π T1+6 (e^(-k u)) k (π^2) (T1^2)+8 (e^(-((3 k u)/2))) (π^3) (T1^3)+((3 (e^(-((k u)/2))) (k^4) (1+(e^(k u))-128 (π^2) (T1^2) Derivative[1,0][α][u,k]))/(128 π T1))-((3 (1+(e^((k u)/2))+2 (e^(k u))) (k^7) ((e^((k u)/2))+(e^(k u))+128 (π^2) (T1^2) Derivative[1,0][α][u,k]))/(32768 (π^4) (T1^4)))-((3 (k^8) ((((e^((k u)/2))+(e^(k u))+128 (π^2) (T1^2) Derivative[1,0][α][u,k]))^2))/(262144 (π^5) (T1^5)))-(((k^9) ((((e^((k u)/2))+(e^(k u))+128 (π^2) (T1^2) Derivative[1,0][α][u,k]))^3))/(2097152 (π^6) (T1^6)))+(k^5) ((3/4) (e^(-((k u)/2))) Derivative[1,0][α][u,k]+((3+2 (e^((k u)/2))-4 (e^(k u))-96 π T1 Derivative[2,0][α][u,k])/(512 (π^2) (T1^2))))+(1/(8192 (π^3) (T1^3)))(e^(-((k u)/2))) (k^6) (-2 (e^((k u)/2))-3 (e^(k u))-6 (e^((3 k u)/2))+3 (e^(2 k u))+49152 (π^4) (T1^4) ((Derivative[1,0][α][u,k])^2)-192 (e^(k u)) (1+(e^((k u)/2))) π T1 Derivative[2,0][α][u,k]+768 (e^((k u)/2)) (π^2) (T1^2) Derivative[1,0][α][u,k] (1+(e^((k u)/2))-32 π T1 Derivative[2,0][α][u,k]))+(k^3) (-(3/32)-(9/32) (e^(-((k u)/2)))-12 (e^(-k u)) (π^2) (T1^2) Derivative[1,0][α][u,k]+6 (e^(-((k u)/2))) π T1 Derivative[2,0][α][u,k]-Derivative[3,0][α][u,k])

DDf[u,k]=(t∞-f[u])((1/4) (e^(-((k u)/2))) (k^2)-(((1+(e^((k u)/2))+2 (e^(k u))) (k^4))/(256 (π^2) (T1^2)))-(e^(-((k u)/2))) k π T1-4 (e^(-k u)) (π^2) (T1^2)+((O[k])^3))=-4 (e^(-k u)) (π^2) (T1^2)(t∞-f[u])(1+  (((e^((k u)/2)) k)/(4  π T1))- (e^((k u)/2)) ((k^2)/(16  (π^2) (T1^2)))+((O[k])^3))
DDDf[u,k]=(t∞-f[u])((1/4) (e^(-k u)) (-3+2 (e^((k u)/2))) (k^2) π T1+6 (e^(-k u)) k (π^2) (T1^2)+8 (e^(-((3 k u)/2))) (π^3) (T1^3)+((O[k])^3))=8 (e^(-((3 k u)/2))) (π^3) (T1^3)(t∞-f[u])((((e^((k u)/2)) (-3+2 (e^((k u)/2))) (k^2))/(32 (π^2) (T1^2)))+((3 (e^((k u)/2)) k)/(4 π T1))+1+((O[k])^3))

(*So we define the approximation as*)
DDf[u_]:=- 4 (π^2) (T1^2) (e^(-k u))(t∞-f[u])

DDDf[u_]:=(e^(-((3 k u)/2))) 8 (π^3) (T1^3)(t∞-f[u])

Summary ( RUN FROM HERE )

subsfλ={f[u,k]-> t∞(1-2μ ν λ),Df[u,k]-> 2π T1 (E^(-((k u)/2)))2t∞ μ ν λ(1+Dfu2  ((k^2)/2)+Dfun[k](k^3)),DDf[u,k]->- 4 (π^2) (T1^2)(E^(-k u))2t∞ μ ν λ(1+(((E^((k u)/2)) k)/(4  π T1))+DDfu2  ((k^2)/2)+DDfun[k](k^3)),DDDf[u,k]->(E^(-((3k u)/2))) 8 (π^3) (T1^3) 2t∞ μ ν λ(1+((3 (E^((k u)/2)) k)/(4 π T1))k+DDDfu2  ((k^2)/2)+DDDfun[k](k^3))};
subsfτ={f[y1m,k]-> t∞(1-2μ  τ),Df[y1m,k]-> 2π T1 (E^(-(Y1m/2)))2t∞ μ τ(1+Dfy2  ((k^2)/2)+Dfyn[k](k^3)),DDf[y1m,k]->- 4 (π^2) (T1^2)(E^(-Y1m))2t∞ μ τ(1+(((E^(Y1m/2)) k)/(4  π T1))+DDfy2  ((k^2)/2)+DDfyn[k](k^3)),DDDf[y1m,k]->(E^(-((3Y1m)/2))) 8 (π^3) (T1^3) 2t∞ μ τ(1+((3 (E^(Y1m/2)) k)/(4 π T1))+DDDfy2  ((k^2)/2)+DDDfyn[k](k^3))};

(*At last we define the series for the variables*)

subsx={x1p->t∞(1 +x1pdev)};
subsxdevλ={x1pdev-> x1pdevl0+x1pdevl1 ν μ λ(*+x1pdevl(((ν μ))^2)2 ((λ^2)/2)+x1pdevl3 (((ν μ))^3) ((λ^3)/6)+x1pdevl4 (((ν μ ))^4)((λ^4)/24)+x1pdevl5 (((ν μ))^5) ((λ^5)/(5*24))*)};
subsxdevτ={x1pdevl0-> x1pl0devt0+x1pl0devt1 ( μ)τ(*+x1pl0devt2 ((( μ))^2)((τ^2)/2)+x1pl0devt3 (((μ))^3) ((τ^3)/6)+x1pl0devt4 ((( μ))^4) ((τ^4)/24)+x1pl0devt5 ((( μ))^5) ((τ^5)/(5*24))*),x1pdevl1-> x1pl1devt0+x1pl1devt1 μ τ(*+x1pl1devt2 (μ^2)((τ^2)/2)+x1pl1devt3 (μ^3) ((τ^3)/6)+x1pl1devt4 (μ^4)((τ^4)/24)+x1pl1devt5 (μ^5) ((τ^5)/(5*24))*),x1pdevl2-> x1pl2devt0+x1pl2devt1 μ τ(*+x1pl2devt2 (μ^2)((τ^2)/2)+x1pl2devt3 (μ^3)((τ^3)/6)+x1pl2devt4 (μ^4) ((τ^4)/24)+x1pl2devt5 (μ^5) ((τ^5)/(5*24))*),x1pdevl3-> x1pl3devt0+x1pl3devt1 μ τ(*+x1pl3devt2 (μ^2)((τ^2)/2)+x1pl3devt3 (μ^3) ((τ^3)/6)+x1pl3devt4 (μ^4)((τ^4)/24)+x1pl3devt5 (μ^5) ((τ^5)/(5*24))*),
x1pdevl4-> x1pl4devt0+x1pl4devt1 μ τ(*+x1pl4devt2 (μ^2)((τ^2)/2)+x1pl4devt3 (μ^3) ((τ^3)/6)+x1pl4devt4 (μ^4) ((τ^4)/24)+x1pl4devt5 (μ^5) ((τ^5)/(5*24))*),x1pdevl5-> x1pl5devt0+x1pl5devt1 μ τ(*+x1pl5devt2 (μ^2)((τ^2)/2)+x1pl5devt3 (μ^3) ((τ^3)/6)+x1pl5devt4 (μ^4) ((τ^4)/24)+x1pl5devt5 (μ^5)((τ^5)/(5*24))*)};
subsxdevk={x1pl0devt0-> x1pl0t0devk0+x1pl0t0devk1 ν k+x1pl0t0devk2 (ν^2)((k^2)/2)(*+x1pl0t0devk3 (ν^3) ((k^3)/6)+x1pl0t0devk4 (ν^4) ((k^4)/24)+x1pl0t0devk5 (ν^5)((k^5)/(5*24)) *),x1pl0devt1-> x1pl0t1devk0+x1pl0t1devk1 ν k+x1pl0t1devk2 (ν^2) ((k^2)/2)(*+x1pl0t1devk3 (ν^3) ((k^3)/6)+x1pl0t1devk4 (ν^4) ((k^4)/24)+x1pl0t1devk5 (ν^5) ((k^5)/(5*24))*),x1pl0devt2-> x1pl0t2devk0+x1pl0t2devk1 ν k(*+x1pl0t2devk2 (ν^2)((k^2)/2)+x1pl0t2devk3 (ν^3) ((k^3)/6)+x1pl0t2devk4 (ν^4)((k^4)/24)+x1pl0t2devk5 (ν^5)((k^5)/(5*24))*),x1pl0devt3-> x1pl0t3devk0+x1pl0t3devk1 ν k(*+x1pl0t3devk2 (ν^2)((k^2)/2)+x1pl0t3devk3 (ν^3)((k^3)/6)+x1pl0t3devk4 (ν^4)((k^4)/24)+x1pl0t3devk5 (ν^5)((k^5)/(5*24))*),x1pl0devt4-> x1pl0t4devk0+x1pl0t4devk1 ν k+x1pl0t4devk2 (ν^2)((k^2)/2)(*+x1pl0t4devk3 (ν^3)((k^3)/6)+x1pl0t4devk4 (ν^4)((k^4)/24)+x1pl0t4devk5 (ν^5)((k^5)/(5*24))*),x1pl0devt5-> x1pl0t5devk0+x1pl0t5devk1 ν k+x1pl0t5devk2 (ν^2)((k^2)/2)(*+x1pl0t5devk3 (ν^3)((k^3)/6)+x1pl0t5devk4 (ν^4)((k^4)/24)+x1pl0t5devk5 (ν^5)((k^5)/(5*24))*),x1pl1devt0-> x1pl1t0devk0+x1pl1t0devk1 ν k+x1pl1t0devk2 (ν^2)((k^2)/2)(*+x1pl1t0devk3 (ν^3)((k^3)/6)+x1pl1t0devk4 (ν^4)((k^4)/24)+x1pl1t0devk5 (ν^5)((k^5)/(5*24))*),x1pl1devt1-> x1pl1t1devk0+x1pl1t1devk1 ν k+x1pl1t1devk2 (ν^2)((k^2)/2)(*+x1pl1t1devk3 (ν^3)((k^3)/6)+x1pl1t1devk4 (ν^4)((k^4)/24)+x1pl1t1devk5 (ν^5)((k^5)/(5*24))*),x1pl1devt2-> x1pl1t2devk0+x1pl1t2devk1 ν k+x1pl1t2devk2 (ν^2)((k^2)/2)(*+x1pl1t2devk3 (ν^3)((k^3)/6)+x1pl1t2devk4 (ν^4)((k^4)/24)+x1pl1t2devk5 (ν^5)((k^5)/(5*24))*),x1pl1devt3-> x1pl1t3devk0+x1pl1t3devk1 ν k+x1pl1t3devk2 (ν^2)((k^2)/2)(*+x1pl1t3devk3 (ν^3)((k^3)/6)+x1pl1t3devk4 (ν^4)((k^4)/24)+x1pl1t3devk5 (ν^5)((k^5)/(5*24))*),x1pl1devt4-> x1pl1t4devk0+x1pl1t4devk1 ν k+x1pl1t4devk2 (ν^2)((k^2)/2)(*+x1pl1t4devk3 (ν^3)((k^3)/6)+x1pl1t4devk4 (ν^4)((k^4)/24)+x1pl1t4devk5 (ν^5)((k^5)/(5*24))*),x1pl1devt5-> x1pl1t5devk0+x1pl1t5devk1 ν k+x1pl1t5devk2 (ν^2)((k^2)/2)(*+x1pl1t5devk3 (ν^3)((k^3)/6)+x1pl1t5devk4 (ν^4)((k^4)/24)+x1pl1t5devk5 (ν^5)((k^5)/(5*24))*),x1pl2devt0-> x1pl2t0devk0+x1pl2t0devk1 ν k+x1pl2t0devk2 (ν^2)((k^2)/2)(*x1pl2t0devk3 (ν^3)((k^3)/6)+x1pl2t0devk4 (ν^4)((k^4)/24)+x1pl2t0devk5 (ν^5)((k^5)/(5*24))*),x1pl2devt1-> x1pl2t1devk0+x1pl2t1devk1 ν k+x1pl2t1devk2 (ν^2)((k^2)/2)(*+x1pl2t1devk3 (ν^3)((k^3)/6)+x1pl2t1devk4 (ν^4)((k^4)/24)+x1pl2t1devk5 (ν^5)((k^5)/(5*24))*),x1pl2devt2-> x1pl2t2devk0+x1pl2t2devk1 ν k+x1pl2t2devk2 (ν^2)((k^2)/2)(*+x1pl2t2devk3 (ν^3)((k^3)/6)+x1pl2t2devk4 (ν^4)((k^4)/24)+x1pl2t2devk5 (ν^5)((k^5)/(5*24))*),x1pl2devl3-> x1pl2t3devk0+x1pl2t3devk1 ν k+x1pl2t3devk2 (ν^2)((k^2)/2)(*+x1pl2t3devk3 (ν^3)((k^3)/6)+x1pl2t3devk4 (ν^4)((k^4)/24)+x1pl2t3devk5 (ν^5)((k^5)/(5*24))*),x1pl2devl4-> x1pl2t4devk0+x1pl2t4devk1 ν k+x1pl2t4devk2 (ν^2)((k^2)/2)(*+x1pl2t4devk3 (ν^3)((k^3)/6)+x1pl2t4devk4 (ν^4)((k^4)/24)+x1pl2t4devk5 (ν^5)((k^5)/(5*24))*),x1pl2devl5-> x1pl2t5devk0+x1pl2t5devk1 ν k+x1pl2t5devk2 (ν^2)((k^2)/2)(*+x1pl2t5devk3 (ν^3)((k^3)/6)+x1pl2t5devk4 (ν^4)((k^4)/24)+x1pl2t5devk5 (ν^5)((k^5)/(5*24))*),x1pl3devt0-> x1pl3t0devk0+x1pl3t0devk1 ν k+x1pl3t0devk2 (ν^2)((k^2)/2)(*+x1pl3t0devk3 (ν^3)((k^3)/6)+x1pl3t0devk4 (ν^4)((k^4)/24)+x1pl3t0devk5 (ν^5)((k^5)/(5*24))*),x1pl3devt1-> x1pl3t1devk0+x1pl3t1devk1 ν k+x1pl3t1devk2 (ν^2)((k^2)/2)(*+x1pl3t1devk3 (ν^3)((k^3)/6)+x1pl3t1devk4 (ν^4)((k^4)/24)+x1pl3t1devk5 (ν^5)((k^5)/(5*24))*),x1pl3devt2-> x1pl3t2devk0+x1pl3t2devk1 ν k+x1pl3t2devk2 (ν^2)((k^2)/2)(*+x1pl3t2devk3 (ν^3)((k^3)/6)+x1pl3t2devk4 (ν^4)((k^4)/24)+x1pl3t2devk5 (ν^5)((k^5)/(5*24))*),x1pl3devt3-> x1pl3t3devk0+x1pl3t3devk1 ν k+x1pl3t3devk2 (ν^2)((k^2)/2)(*+x1pl3t3devk3 (ν^3)((k^3)/6)+x1pl3t3devk4 (ν^4)((k^4)/24)+x1pl3t3devk5 (ν^5)((k^5)/(5*24))*),x1pl3devt4-> x1pl3t4devk0+x1pl3t4devk1 ν k+x1pl3t4devk2 (ν^2)((k^2)/2)(*+x1pl3t4devk3 (ν^3)((k^3)/6)+x1pl3t4devk4 (ν^4)((k^4)/24)+x1pl3t4devk5 (ν^5)((k^5)/(5*24))*),x1pl3devt5-> x1pl3t5devk0+x1pl3t5devk1 ν k+x1pl3t5devk2 (ν^2)((k^2)/2)(*+x1pl3t5devk3 (ν^3)((k^3)/6)+x1pl3t5devk4 (ν^4)((k^4)/24)+x1pl3t5devk5 (ν^5)((k^5)/(5*24))*),x1pl4devt0-> x1pl4t0devk0+x1pl4t0devk1 ν k+x1pl4t0devk2 (ν^2)((k^2)/2)(*+x1pl4t0devk3 (ν^3)((k^3)/6)+x1pl4t0devk4 (ν^4)((k^4)/24)+x1pl4t0devk5 (ν^5)((k^5)/(5*24))*),x1pl4devt1-> x1pl4t1devk0+x1pl4t1devk1 ν k+x1pl4t1devk2 (ν^2)((k^2)/2)(*+x1pl4t1devk3 (ν^3)((k^3)/6)+x1pl4t1devk4 (ν^4)((k^4)/24)+x1pl4t1devk5 (ν^5)((k^5)/(5*24))*),x1pl4devt2-> x1pl4t2devk0+x1pl4t2devk1 ν k+x1pl4t2devk2 (ν^2)((k^2)/2)(*+x1pl4t2devk3 (ν^3)((k^3)/6)+x1pl4t2devk4 (ν^4)((k^4)/24)+x1pl4t2devk5 (ν^5)((k^5)/(5*24))*),x1pl4devt3-> x1pl4t3devk0+x1pl4t3devk1 ν k+x1pl4t3devk2 (ν^2)((k^2)/2)(*+x1pl4t3devk3 (ν^3)((k^3)/6)+x1pl4t3devk4 (ν^4)((k^4)/24)+x1pl4t3devk5 (ν^5)((k^5)/(5*24))*),x1pl4devt4-> x1pl4t4devk0+x1pl4t4devk1 ν k+x1pl4t4devk2 (ν^2)((k^2)/2)(*+x1pl4t4devk3 (ν^3)((k^3)/6)+x1pl4t4devk4 (ν^4)((k^4)/24)+x1pl4t4devk5 (ν^5)((k^5)/(5*24))*),x1pl4devt5-> x1pl4t5devk0+x1pl4t5devk1 ν k+x1pl4t5devk2 (ν^2)((k^2)/2)(*+x1pl4t5devk3 (ν^3)((k^3)/6)+x1pl4t5devk4 (ν^4)((k^4)/24)+x1pl4t5devk5 (ν^5)((k^5)/(5*24))*)};
subsy={y1m->yp-y1mdev };
(*also, we allow for one very bigg parameter B*)
subsydevλ={y1mdev-> y1mdevl0+y1mdevl1 μ ν  λ(*+y1mdevl2 ((((λ))^2)/2)+y1mdevl3((((λ))^3)/6)*)};
subsydevτ={y1mdevl0-> y1ml0devt0+y1ml0devt1 μ  τ(*+y1ml0devt2 ((((τ))^2)/2)+y1ml0devt3 ((((τ))^3)/6)*),y1mdevl1-> y1ml1devt0+y1ml1devt1 μ τ(*+y1ml1devt2 ((((τ))^2)/2)+Y1ml1devt3 ((((τ))^3)/6)*),y1mdevl2->y1ml2devt0+y1ml2devt1 μ τ(*+y1ml2devt2 ((((τ))^2)/2)+y1ml2devt3 ((((τ))^3)/6)*),y1mdevl3-> y1ml3devt0+y1ml3devt1 μ (τ)(*+y1ml3devt2 ((((τ))^2)/2)+y1ml3devt3 ((((τ))^3)/6)*)};
subsydevk={y1ml0devt0-> y1ml0t0devk0+y1ml0t0devk1 ν k+y1ml0t0devk2 (ν^2) ((k^2)/2)(*+y1ml0t0devk3 ((k^3)/6)+y1ml0t0devk4 ((k^4)/24)+y1ml0t0devk5 ((k^5)/(5*24))*) ,y1ml0devt1-> y1ml0t1devk0+y1ml0t1devk1 ν k+y1ml0t1devk2 (ν^2) ((k^2)/2)(*+y1ml0t1devk3 ((k^3)/6)+y1ml0t1devk4 ((k^4)/24)+y1ml0t1devk5 ((k^5)/(5*24))*),y1ml0devt2-> y1ml0t2devk0+y1ml0t2devk1 ν k+y1ml0t2devk2 (ν^2) ((k^2)/2)(*+y1ml0t2devk3 ((k^3)/6)+y1ml0t2devk4 ((k^4)/24)+y1ml0t2devk5 ((k^5)/(5*24))*),y1ml0devt3-> y1ml0t3devk0+y1ml0t3devk1 ν k+y1ml0t3devk2 (ν^2)((k^2)/2)(*+y1ml0t3devk3 ((k^3)/6)+y1ml0t3devk4 ((k^4)/24)+y1ml0t3devk5 ((k^5)/(5*24))*),y1ml1devt0-> y1ml1t0devk0+y1ml1t0devk1 ν k+y1ml1t0devk2 (ν^2) ((k^2)/2)(*+y1ml1t0devk3 ((k^3)/6)+y1ml1t0devk4 ((k^4)/24)+y1ml1t0devk5 ((k^5)/(5*24))*),y1ml1devt1-> y1ml1t1devk0+y1ml1t1devk1 ν k+y1ml1t1devk2 (ν^2) ((k^2)/2)(*+y1ml1t1devk3 ((k^3)/6)+y1ml1t1devk4 ((k^4)/24)+y1ml1t1devk5 ((k^5)/(5*24))*),y1ml1devt2-> y1ml1t2devk0+y1ml1t2devk1 ν k+y1ml1t2devk2 (ν^2) ((k^2)/2)(*+y1ml1t2devk3 ((k^3)/6)+y1ml1t2devk4 ((k^4)/24)+y1ml1t2devk5 ((k^5)/(5*24))*),y1ml1devt3-> y1ml1t3devk0+y1ml1t3devk1 k+y1ml1t3devk2 ((k^2)/2)(*+y1ml1t3devk3 ((k^3)/6)+y1ml1t3devk4 ((k^4)/24)+y1ml1t3devk5 ((k^5)/(5*24))*),y1ml2devt0-> y1ml2t0devk0+y1ml2t0devk1 ν k+y1ml2t0devk2 (ν^2) ((k^2)/2)(*+y1ml2t0devk3 ((k^3)/6)+y1ml2t0devk4 ((k^4)/24)+y1ml2t0devk5 ((k^5)/(5*24))*),y1ml2devt1-> y1ml2t1devk0+y1ml2t1devk1 ν k+y1ml2t1devk2 (ν^2) ((k^2)/2)(*+y1ml2t1devk3 ((k^3)/6)+y1ml2t1devk4 ((k^4)/24)+y1ml2t1devk5 ((k^5)/(5*24))*),y1ml2devt2-> y1ml2t2devk0+y1ml2t2devk1 k+y1ml2t2devk2 ((k^2)/2)(*+y1ml2t2devk3 ((k^3)/6)+y1ml2t2devk4 ((k^4)/24)+y1ml2t2devk5 ((k^5)/(5*24))*),y1ml2devl3-> y1ml2t3devk0+y1ml2t3devk1 ν k+y1ml2t3devk2 (ν^2) ((k^2)/2)(*+y1ml2t3devk3 ((k^3)/6)+y1ml2t3devk4 ((k^4)/24)+y1ml2t3devk5 ((k^5)/(5*24))*),y1ml3devt0-> y1ml3t0devk0+y1ml3t0devk1 ν k+y1ml3t0devk2 (ν^2)((k^2)/2)(*+y1ml3t0devk3 ((k^3)/6)+y1ml3t0devk4 ((k^4)/24)+y1ml3t0devk5 ((k^5)/(5*24))*),y1ml3devt1-> y1ml3t1devk0+y1ml3t1devk1 ν k+y1ml3t1devk2 (ν^2) ((k^2)/2)(*+y1ml3t1devk3 ((k^3)/6)+y1ml3t1devk4 ((k^4)/24)+y1ml3t1devk5 ((k^5)/(5*24))*),y1ml3devt2-> y1ml3t2devk0+y1ml3t2devk1 ν k+y1ml3t2devk2 (ν^2) ((k^2)/2)(*+y1ml3t2devk3 ((k^3)/6)+y1ml3t2devk4 ((k^4)/24)+y1ml3t2devk5 ((k^5)/(5*24))*),y1ml3devt3-> y1ml3t3devk0+y1ml3t3devk1 ν k+y1ml3t3devk2 (ν^2)((k^2)/2)(*+y1ml3t3devk3 ((k^3)/6)+y1ml3t3devk4 ((k^4)/24)+y1ml3t3devk5 ((k^5)/(5*24))*)};
subsydevB={y1ml0t0devk0->y1ml0t0devk0Bp1 B +y1ml0t0devk0B0+y1ml0t0devk0B1 (1/B)+y1ml0t0devk0B2 (1/((B^2) ((Log[k])^2))),y1ml1t0devk0->y1ml1t0devk0Bp1 B +y1ml1t0devk0B0+y1ml1t0devk0B1 (1/(B ))+y1ml1t0devk0B2 (1/((B^2) )),y1ml0t1devk0->y1ml0t1devk0Bp1 B +y1ml0t1devk0B0+y1ml0t1devk0B1 (1/(B ))+y1ml0t1devk0B2 (1/((B^2) )),y1ml1t1devk0->y1ml1t1devk0Bp1 B +y1ml1t1devk0B0+y1ml1t1devk0B1 (1/(B L))+y1ml1t1devk0B2 (1/(B^2)),y1ml0t0devk1->y1ml0t0devk1Bp2 (B^2) +y1ml0t0devk1Bp1 B+y1ml0t0devk1B0+y1ml0t0devk1B1 (1/(B ))+y1ml0t0devk1B2 (1/((B^2) )),y1ml1t0devk1->y1ml1t0devk2Bp2 (B^2) +y1ml1t0devk1Bp1 B +y1ml1t0devk1B0+y1ml1t0devk1B1 (1/(B ))+y1ml1t0devk1B2 (1/((B^2) )),y1ml0t1devk1->y1ml0t1devk2Bp2 (B^2) +y1ml0t1devk1Bp1 B +y1ml0t1devk1B0+y1ml0t1devk1B1 (1/B)+y1ml0t1devk1B2 (1/((B^2) )),y1ml1t1devk1->y1ml1t1devk2Bp2 (B^2) +y1ml1t1devk1Bp1 B +y1ml1t1devk1B0+y1ml1t1devk1B1 (1/(B ))+y1ml1t1devk1B2 (1/((B^2) )),y1ml0t0devk2->y1ml0t0devk2Bp3 (B^3) +y1ml0t0devk2Bp2 (B^2) +y1ml0t0devk2Bp1 B +y1ml0t0devk2B0+y1ml0t0devk2B1 (1/(B ))+y1ml0t0devk2B2 (1/((B^2) )),y1ml1t0devk2->y1ml1t0devk2Bp3 (B^3) +y1ml1t0devk2Bp2 (B^2) +y1ml1t0devk2Bp1 B +y1ml1t0devk2B0+y1ml1t0devk2B1 (1/(B ))+y1ml1t0devk2B2 (1/((B^2) )),y1ml0t1devk2->y1ml0t1devk2Bp3 (B^3) +y1ml0t1devk2Bp2 (B^2) +y1ml0t1devk2Bp1 B +y1ml0t1devk2B0+y1ml0t1devk2B1 (1/(B ))+y1ml0t1devk2B2 (1/((B^2) )),y1ml1t1devk2->y1ml1t1devk2Bp3 (B^3) +y1ml1t1devk2Bp2 (B^2) +y1ml1t1devk2Bp1 B +y1ml1t1devk2B0+y1ml1t1devk2B1 (1/(B ))+y1ml1t1devk2B2 (1/((B^2) ))};

Approximate equations and solutions

First EOM and location of (SuperPlus[(Subscript[x, QL])])

(*Now let us look at the equation*)
Df[y1m,k] (-x1p+ f[-S1+u,k])+k (x1p- f[y1m,k]) ( f[-S1+u,k]- f[y1m,k])

(*we redefine -S1+u->u and introduce a new small parameter ν for each k*)

(*Now we also insert the series of the solution and we make a series in the smallest parameter*)
Series[Df[y1m,k] (-x1p+ f[u,k])+ k (x1p- f[y1m,k]) ( f[u,k]- f[y1m,k])/.subsfλ/.subsfτ/.k->ν k/.subsx/.subsxdevλ/.subsxdevτ,{μ,0,1}]//Simplify

(*So this tells us that x1pl0devt0  =0*)

Series[Df[y1m,k] (-x1p+ f[u,k])+ k (x1p- f[y1m,k]) ( f[u,k]- f[y1m,k])/.subsfλ/.subsfτ/.k->ν k/.subsx/.subsxdevλ/.subsxdevτ/.x1pl0devt0  -> 0,{μ,0,2}]//Simplify

(*Now we need to do the expansion in ν*)

Series[Df[y1m,k] (-x1p+ f[u,k])+ k (x1p- f[y1m,k]) ( f[u,k]- f[y1m,k])/.subsfλ/.subsfτ/.k->ν k/.subsx/.subsxdevλ/.subsxdevτ/.x1pl0devt0  -> 0/.subsxdevk,{μ,0,2},{ν,0,1}]//Simplify

(*From here we extract that x1pl0t1devk0=0*)
Series[Df[y1m,k] (-x1p+ f[u,k])+ k (x1p- f[y1m,k]) ( f[u,k]- f[y1m,k])/.subsfλ/.subsfτ/.k->ν k/.subsx/.subsxdevλ/.subsxdevτ/.x1pl0devt0  -> 0/.subsxdevk/.x1pl0t1devk0->0,{μ,0,2},{ν,0,1}]//Simplify

Solve[k τ-π T1(E^(-Y1m/2))  ((2+x1pl1t0devk0) λ+k x1pl0t1devk1 τ)==0,x1pl1t0devk0 ]//Simplify

(*In the linear regime where Y1m(=k y1m) is small, (E^(-Y1m/2)) can be approximated by 1*)

Series[Df[y1m,k] (-x1p+ f[u,k])+ k (x1p- f[y1m,k]) ( f[u,k]- f[y1m,k])/.subsfλ/.subsfτ/.k->ν k/.subsx/.subsxdevλ/.subsxdevτ/.x1pl0devt0  -> 0/.subsxdevk/.x1pl0t1devk0->0/.x1pl1t0devk0->-2+(((E^(Y1m/2)) k τ)/(π T1 λ))-((k x1pl0t1devk1 τ)/λ),{μ,0,2},{ν,0,1}]//Simplify

(*Now we know to what order the equation is solved!*)

Series[t∞ (1+x1pdev)/.subsxdevλ/.subsxdevτ/.x1pl0devt0  -> 0/.subsxdevk/.x1pl0t1devk0->0/.x1pl1t0devk0->-2+(((E^(Y1m/2)) k τ)/(π T1 λ))-((k x1pl0t1devk1 τ)/λ)/.μ->1/.ν->1,{λ,0,1},{τ ,0,1},{k,0,1}]//Simplify

(*We find that*)

t∞+(((( t∞ k)/(π T1))) τ)+((-2 t∞)) λ+O[k (λ^2)]

(*Notice that t∞ itself is (1/(π T1))+O[ν] so this approximation can only be trusted to the order we just found!*)

Series[t∞ (1+x1pdev)/.subsxdevλ/.subsxdevτ/.x1pl0devt0  -> 0/.subsxdevk/.x1pl0t1devk0->0/.x1pl1t0devk0->-2+(((E^(Y1m/2)) k τ)/(π T1 λ))-((k x1pl0t1devk1 τ)/λ),{μ,0,1},{ν ,0,1}]//Simplify

t∞ (-2 λ+((k τ)/(π T1))) ν+((O[ν])^2)/.t∞-> (1/(π T1))+α ν//Simplify

(*Hence to be truly correct we must say that*)

x1p=t∞+((((-2 π T1 λ+(E^(Y1m/2))k τ) ν)/((π^2) (T1^2)))+((O[ν])^2))μ+((O[μ])^2)

Second EOM and location of (SuperMinus[(Subscript[y, QL])])

(*Now let us look at the equation*)
((1/k)(Df[y1m,k] (((x1p- f[y1m,k]))^2) DDDf[y1m,k]+2 ((Df[y1m,k])^2) (x1p- f[y1m,k]) DDf[y1m,k]-DDf[y1m,k] (((x1p- f[y1m,k]))^2) DDf[y1m,k]+2 ((Df[y1m,k])^3) Df[y1m,k])+(1/(S1+u-y1m))Df[y1m,k] (x1p- f[y1m,k]) ((S1+u-y1m) (x1p- f[y1m,k]) DDf[y1m,k]-2 Df[y1m,k] (x1p- f[y1m,k]-(S1+u-y1m) Df[y1m,k])))

(*we redefine -S1+u->u and introduce a ν for each k and redefine S1+u==y1m*)

(*Now we also insert the series of the solution and we make a series in the smallest parameters*)
Series[(y1m-ym)k((1/k)(Df[y1m,k] (((x1p- f[y1m,k]))^2) DDDf[y1m,k]+2 ((Df[y1m,k])^2) (x1p- f[y1m,k]) DDf[y1m,k]-DDf[y1m,k] (((x1p- f[y1m,k]))^2) DDf[y1m,k]+2 ((Df[y1m,k])^3) Df[y1m,k])+(1/( ym-y1m))Df[y1m,k] (x1p- f[y1m,k]) (( ym-y1m) (x1p- f[y1m,k]) DDf[y1m,k]-2 Df[y1m,k] (x1p- f[y1m,k]-( ym-y1m) Df[y1m,k])))/.subsfλ/.subsfτ/.subsy/.subsx/.k->ν k/.subsxdevλ/.subsxdevτ/.x1pl0devt0  -> 0/.subsxdevk/.x1pl0t1devk0->0/.x1pl1t0devk0->-2+(((E^(Y1m/2)) k τ)/(π T1 λ))-((k x1pl0t1devk1 τ)/λ)/.subsydevλ/.subsydevτ/.subsydevk,{μ,0,4},{ν,0,1}]//Simplify

Series[(y1m-ym)k((1/k)(Df[y1m,k] (((x1p- f[y1m,k]))^2) DDDf[y1m,k]+2 ((Df[y1m,k])^2) (x1p- f[y1m,k]) DDf[y1m,k]-DDf[y1m,k] (((x1p- f[y1m,k]))^2) DDf[y1m,k]+2 ((Df[y1m,k])^3) Df[y1m,k])+(1/(ym-y1m))Df[y1m,k] (x1p- f[y1m,k]) ((ym-y1m) (x1p- f[y1m,k]) DDf[y1m,k]-2 Df[y1m,k] (x1p- f[y1m,k]-(ym-y1m) Df[y1m,k])))/.subsfλ/.subsfτ/.k->ν k/.subsx/.subsxdevλ/.subsxdevτ/.x1pl0devt0  -> 0/.subsxdevk/.x1pl0t1devk0->0/.x1pl1t0devk0->-2+(((E^(Y1m/2)) k τ)/(π T1 λ))-((k x1pl0t1devk1 τ)/λ)/.subsy/.subsydevλ/.subsydevτ/.subsydevk/.subsydevB,{μ,0,4},{ν,0,1},{B,∞,0}]//Simplify

(*Let us clarify this equation*)
(*The lowest order still tells you that y1ml0t0devk0Bp1=0!!!! *)
(*In order words, we do not find the coefficient in front of the log, but there is now another way to solve this equation since τ itself depends on y1m! *)

Solve[(-8 π T1 λ+3 (E^(Y1m/2)) k τ)==0,λ]

λ=(E^(-4π (T1/k)(1-(E^((-k yp)/2)))+O[k]))
τ=(E^(-4π (T1/k)(1-(E^((-k y1m)/2)))+O[k]))

Solve[(E^(-4π (T1/k)(1-(E^((-k yp)/2)))))==(((3(E^(k y1m/2))) k )/(8 π T1))((E^(-4π (T1/k)(1-(E^((-k y1m)/2)))))),y1m]//Simplify

Solve[(E^(-4π (T1/k)(1-(E^((-k yp)/2)))))==((3 (E^(-((4 (1-(E^(-((k y1m)/2)))) π T1)/k))) k)/(8 π T1)),y1m]//Simplify

Series[((2 (Log[((4 π T1)/(4 π T1-k Log[((3 (E^((4 (1-(E^(-((k yp)/2)))) π T1)/k)) k)/(8 π T1))]))]))/k),{k,0,1}]//Simplify

(*But actually there are corrections*)

Solve[(E^(-4π (T1/k)(1-(E^((-k yp)/2)))+α k))==((3 k )/(8 π T1))((E^(-4π (T1/k)(1-(E^((-k y1m)/2)))+β k))),y1m]//Simplify

Series[((2 (Log[((4 π T1)/(4 π T1-(k^2) β+k Log[((8 (E^(((4 (-1+(E^(-((k yp)/2)))) π T1)/k)+k α)) π T1)/(3 k))]))]))/k),{k,0,1}]//Simplify

(*What about x1p?*)

t∞+(((( t∞ k)/(π T1))) τ)+((-2 t∞)) λ/.τ->(((((3 k )/(8 π T1))))^(-1))λ//Simplify

t∞+(((( t∞ k)/(π T1))) τ)+((-2 t∞)) λ/.τ->(((((3 k )/(8 π T1))))^(-1))λ/.λ->((t∞-t)/(2 t∞))//Simplify

Island transition for semi-finite intervals

SQuench[m_,u_,σ1_,k_,T1_,ϕr_,GN_]:=2 k ((ϕr)/(4GN))Log[m (u+σ1)((( ff[u-σ1,k]))/(Sqrt[ffd[u-σ1,k]]))];

SLate[x1p_, y1m_, σ1_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[(2/ϵ)( (u+σ1)-y1m)(((x1p-ff[u-σ1,k])/(-ff[y1m,k]+x1p)))((Sqrt[ffd[y1m,k]])/(Sqrt[ffd[u-σ1,k]]))]+(((ϕ0+ϕ[x1p,y1m,k]))/(4 GN))+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2];

(*We write SQuench-SLate as (c/6)Log[X]*)

X= k (π^(2 ))((T1^2)-(T0^2))ym f[yp](((xqp-xqm))/(4(ym-yqm)(xqp-f[yp])(((f'[yqm]))^(1/2))))(E^((-2ϕ0-ϕr (2π T0 + f'' [yqm,k]/f'[yqm,k]+2 f'[yqm,k]/(xqp-f[yqm,k])))/(2k ϕr)))

(*Let us expand using*)

subsfλ={f[u,k]-> t∞(1-2μ ν λ),Df[u,k]-> 2π T1 (E^(-((k u)/2)))2t∞ μ ν λ(1+Dfu2  ((k^2)/2)),DDf[u,k]->- 4 (π^2) (T1^2)(E^(-k u))2t∞ μ ν λ(1+(((E^((k u)/2)) k)/(4  π T1))+DDfu2  ((k^2)/2)),DDDf[u,k]->(E^(-((3k u)/2))) 8 (π^3) (T1^3) 2t∞ μ ν λ(1+((3 (E^((k u)/2)) k)/(4 π T1))k+DDDfu2  ((k^2)/2))};
subsfτ={f[y1m,k]-> t∞(1-2μ  τ),Df[y1m,k]-> 2π T1 (E^(-(Y1m/2)))2t∞ μ τ(1+Dfy2  ((k^2)/2)),DDf[y1m,k]->- 4 (π^2) (T1^2)(E^(-Y1m))2t∞ μ τ(1+(((E^(Y1m/2)) k)/(4  π T1))+DDfy2  ((k^2)/2)),DDDf[y1m,k]->(E^(-((3Y1m)/2))) 8 (π^3) (T1^3) 2t∞ μ τ(1+((3 (E^(Y1m/2)) k)/(4 π T1))+DDDfy2  ((k^2)/2))};

(*The QES location is approximately given by*)
/.xqp->+(1/3) (-t∞(1-2μ ν λ)+4t∞)
/.yqm->(u-σ)-(Log[(8  π T1)/(3 ν k)]/(2 (π T1)))

(*We also know that*)
/.Dfu2-> -(1/(16 π T1 π T1 (E^(-((k u)/2)))))/.DDfu2-> ((-(E^((k u)/2)))/(8π T1 π T1))

-((ν k (π^3) T1 (-(T0^2)+(T1^2)) t∞  (-1+2 λ μ ν) (u+σ) (λ ν+3 τ))/(8 λ ν (4 π T1 σ+Log[((8 π T1)/(3 k ))])(((2π T1 (E^(-(Yqm/2)))2t∞ μ τ(1+Dfy2 (ν^2) ((k^2)/2))))^(1/2))))(E^((-2ϕ0-ϕr (2π T0- 4 (π^2) (T1^2)(E^(-Y1m))2t∞ μ τ(1+(((E^(Yqm/2)) ν k)/(4  π T1))+DDfy2 (ν^2) ((k^2)/2))/2π T1 (E^(-(Yqm/2)))2t∞ μ τ(1+Dfy2 (ν^2)  ((k^2)/2))+2 2π T1 (E^(-(Yqm/2)))2t∞ μ τ(1+Dfy2 (ν^2)  ((k^2)/2))/(2/3) t∞ μ (λ ν+3 τ)))/(2 ν k ϕr)))

Series[-((ν k (π^3) T1 (-(T0^2)+(T1^2)) t∞  (-1+2 λ μ ν) (Ym/(k ν)) (λ ν+3 τ))/(8 λ ν (4 π T1 σ+Log[((8 π T1)/(3 k  ))])(((2π T1 (E^(-(Yqm/2)))2t∞ μ τ(1+Dfy2 (ν^2) ((k^2)/2))))^(1/2))))(E^((-2ϕ0-ϕr (2π T0- 4 (π^2) (T1^2)(E^(-Yqm))2t∞ μ τ(1+(((E^(Yqm/2)) ν k)/(4  π T1))+DDfy2 (ν^2) ((k^2)/2))/(2π T1 (E^(-(Yqm/2)))2t∞ μ τ(1+Dfy2 (ν^2)  ((k^2)/2)))+2 2π T1 (E^(-(Yqm/2)))2t∞ μ τ(1+Dfy2 (ν^2)  ((k^2)/2))/((2/3) t∞ μ (λ ν+3 τ))))/(2 ν k ϕr))),{μ,0,0}]

(((E^((-2 ϕ0-(2 π T0-((2 (E^(-Yqm/2)) π T1 (1+(((E^(Yqm/2)) k ν)/(4 π T1))+(1/2) DDfy2 (k^2) (ν^2)))/(1+(1/2) Dfy2 (k^2) (ν^2)))+((12 (E^(-Yqm/2)) π T1 (1+(1/2) Dfy2 (k^2) (ν^2)) τ)/(λ ν+3 τ))) ϕr)/(2 k ν ϕr))) (π^(5/2)) T1 (-(T0^2)+(T1^2)) t∞ Ym (λ ν+3 τ))/(16 (λ ν (Sqrt[(E^(-Yqm/2)) T1 t∞ (1+(1/2) Dfy2 (k^2) (ν^2)) τ]) (4 π T1 σ+Log[((8 π T1)/(3 k))])) (Sqrt[μ])))/.τ->λ((((((3(E^(Yqm/2))) ν k )/(8 π T1))))^(-1))/.yqm->yp-Log[(8  π T1)/(3  k)]/(2 (π T1))/.Yqm->ν k yp-ν k Log[(8  π T1)/(3  k)]/(2 (π T1))/.λ->(1/(μ ν)) (E^(-4π (T1/(ν k))(1-(E^(-(Yp/2))))))/.Yp->ν k yp/.Ym->ν k ym/.ym->(U/(ν k))+σ/.yp->(U/(ν k))-σ//Simplify

Assuming[{t∞>0,yp>0,ym>0,σ>0,k>0,T0>0,T1>0,ν>0,μ>0,U>0},Series[(((Sqrt[3]) (E^(-((2 π T0+(((2^(4+((3 k ν)/(2 π T1)))) (π^(2+((k ν)/(2 π T1)))) (T1^2) ((((T1/k)))^((k ν)/(2 π T1))) (2+Dfy2 (k^2) (ν^2)))/((2^(3+((3 k ν)/(4 π T1)))) (3^((k ν)/(4 π T1))) (E^((1/2) (U-k ν σ))) (π^(1+((k ν)/(4 π T1)))) T1 ((((T1/k)))^((k ν)/(4 π T1)))+(3^((k ν)/(2 π T1))) (E^(U-k ν σ)) k (ν^2)))-((k ν+(2^(1+((3 k ν)/(4 π T1)))) (3^(-((k ν)/(4 π T1)))) (E^((1/2) (-U+k ν σ))) (π^(1+((k ν)/(4 π T1)))) T1 ((((T1/k)))^((k ν)/(4 π T1))) (2+DDfy2 (k^2) (ν^2)))/(2+Dfy2 (k^2) (ν^2)))-k ν (-U+k ν σ)+((2 ϕ0)/(ϕr)))/(2 k ν)))) (π^2) T1 (-(T0^2)+(T1^2)) t∞ ((2^(3+((3 k ν)/(4 π T1)))) (3^(-((k ν)/(4 π T1)))) (π^(1+((k ν)/(4 π T1)))) T1 ((((T1/k)))^((k ν)/(4 π T1)))+(E^((1/2) (U-k ν σ))) k (ν^2)) (U+k ν σ))/(32 k (Sqrt[μ]) (ν^2) (Sqrt[(((2^((3 k ν)/(2 π T1))) (E^(-U+((4 (-1+(E^((1/2) (-U+k ν σ)))) π T1)/(k ν))+k ν σ)) ((((π/3)))^((k ν)/(2 π T1))) T1 ((((T1/k)))^(1+((k ν)/(2 π T1)))) t∞ (2+Dfy2 (k^2) (ν^2)))/(μ (ν^2)))]) (4 π T1 σ+Log[((8 π T1)/(3 k))]))),{ν,0,-1}]]//Simplify

(E^(((-π T0+(2-3 (E^(-U/2))) π T1-((ϕ0)/(ϕr)))/(k ν))+(1/4) (E^(-U/2)) ((E^(U/2))-6 π T1 σ-Log[((8 π T1)/(3 k))]))) ((((Sqrt[(3/2)]) (π^3) (Sqrt[(T1/k)]) (-(T0^2)+(T1^2)) (Sqrt[T1 t∞]) U )/(4 (4 π T1 σ+Log[((8 π T1)/(3 k))]) ν)))

Solve[0==((-π T0+(2-3 (E^(-U/2))) π T1-((ϕ0)/(ϕr)))/(k ν)),U]//Simplify

{{U->(ConditionalExpression[2 (2 I π (C[1])+Log[-((3 π T1 ϕr)/(ϕ0+π (T0-2 T1) ϕr))]), (C[1])∈(Integers)])}}

Solve[ϕ0+π (T0-2 T1) ϕr==0]//Simplify

(*So as long as ((ϕ0)/(ϕr))<π (2 T1-T0) we can have a island transition*)

(*improving the calculation by extra corrections*)

Solve[Log[α]==((-π T0+(2-3 (E^(-U/2))) π T1-((ϕ0)/(ϕr)))/(k ν))+(1/4) (E^(-U/2)) ((E^(U/2))-6 π T1 σ-Log[((8 π T1)/(3 k))]),U]//Simplify

Series[2Log[((ϕr (6 π T1 (2+k ν σ)+k ν Log[((8 π T1)/(3 k))]))/(-4 ϕ0-4 π T0 ϕr+8 π T1 ϕr+k ν ϕr-4 k ν ϕr Log[α]))],{ν,0,1}]

Assuming[{t∞>0,yp>0,ym>0,σ>0,k>0,T0>0,T1>0,ν>0,μ>0,U>0},Solve[(α ((((Sqrt[(3/2)]) (π^3) (Sqrt[(T1/k)]) (-(T0^2)+(T1^2)) (Sqrt[T1 t∞]) U )/(4 (4 π T1 σ+Log[((8 π T1)/(3 k))]) ν)))/.U->2 Log[-((3 π T1 ϕr)/(ϕ0+π T0 ϕr-2 π T1 ϕr))])==1,α]]//Simplify

No island transition for finite intervals

SFutureConnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, k_]:=2 k ((ϕr)/(4GN))Log[Abs[(F[u-σ1,T1, k]-F[u-σ2,T1, k])](((σ2-σ1))/((ϵ^2)(Sqrt[Fprime[u-σ1,T1, k]Fprime[u-σ2,T1, k]])))]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))/.ϕ0->vvϕ0;

ϕ[xp_,ym_,k_]:= ϕr (ffdd[ym,k]/ffd[ym,k]+2 ffd[ym,k]/(xp-ff[ym,k]))

Slatelate[x1p_, y1m_,x2p_,y2m_,σ2_ ,σ1_, u_,k_]:=(((2ϕ0+ϕ[x1p,y1m,k]+ϕ[x2p,y2m,k]))/(4 GN))+2k ((ϕr)/(4GN))Log[(((2^2) )/(ϵ^2))(((((u+σ1-y1m) (u+σ2-y2m) (x1p-F[u-σ1,T1,k]) (x2p-F[u-σ2,T1,k])))/(((x1p-F[y1m,T1,k]) (x2p-F[y2m,T1,k]))))) Sqrt[(((Fprime[y1m,T1,k] Fprime[y2m,T1,k]))/((Fprime[u-σ1,T1,k] Fprime[u-σ2,T1,k])))]]+(((ϕ0+ϕr 2 π vvT0))/(4 GN))/.ϕ0->vvϕ0;

$Assumptions=T1>0&&δ[y1p]>0&&l>0&&k>0

(* The position of the QES*)

xpLate=t∞+(1/3)(t∞-xp);
xmLate=t∞-((8 π Tef)/(3k))(t∞-xp);
ymLate=yp-(1/(2π Tef))Log[((8π Tef)/(3k))];

(*Notice that we introduce the following notation for the effective temperture: Tef=T1 Exp[-((k(U-σ))/2)]*)

(*The approximation for f*)

fsub[u_]:=t∞-δ[u];
fpsub[u_]:=δ[u](2 π T1 Exp[-k u/2] + ((k^2)/(16π T1))Exp[k u/2]);
fppsub[u_]:=-2π T1 δ[u](2π T1 Exp[-k u]+(k/2)Exp[-k u/2]+((k^2)/(16π T1)));

(*Relating δ[yqm]=τ to δ[yp]=λ*)

δ[yqm]= 2t∞ Exp[-((4π T1)/k)(1-Exp[-k ymLate/2])]=2t∞ Exp[-((4π T1)/k)(1-Exp[-k yp/2+(k/(4π Tef))Log[((8π Tef)/(3k))]])]=2t∞ Exp[-((4π T1)/k)(1-Exp[-k yp/2](1+(k/(4π Tef))Log[((8π Tef)/(3k))]))]=2t∞ Exp[-((4π T1)/k)(1-Exp[-k yp/2])+((T1 Exp[-k yp/2])/Tef)Log[((8π Tef)/(3k))]]=δ[yp]Exp[Log[((8π Tef)/(3k))]]=δ[yp]((8π Tef)/(3k))

(*expression in the log for the bulk von Neumann entropy in the island phase*)

((((ym-yQm)(xQp-xp))/(xQp-xQm))(Sqrt[(fpyQm/fpyp)])/.{yQm->ymLate,xQp->xpLate,xQm->xmLate})/.{xp->fsub[yp],fpyQm->fpsub[yqm],fpyp->fpsub[yp]}//Simplify

%/.δ[yqm]-> ((8π Tef)/(3k))δ[yp]//Simplify

Series[%/.{ym->U/k+σ,yp->U/k-σ,yqm->YQm/k,Tef->T1 Exp[-((U-k σ)/2)]},{k,0,1}]//Simplify//Normal

Series[((%/.YQm->k ymLate)/.{yp->U/k-σ,Tef->T1 Exp[-((U-k σ)/2)]}),{k,0,1}]//Simplify//Normal

(*Exponentials of bulk entropies for both end points*)

expSbulk1=(((E^((3 U)/4)) (Sqrt[k]) (-4 (E^(-U/2)) π T1 σ1+Log[(8/(3k)) (E^(-U/2)) π T1]))/((Sqrt[6]) (((π T1))^(3/2)) ));
expSbulk2=(((E^((3 U)/4)) (Sqrt[k]) (-4 (E^(-U/2)) π T1 σ2+Log[(8/(3k)) (E^(-U/2)) π T1]))/((Sqrt[6]) (((π T1))^(3/2)) ));

(*bulk von Neumann entropy in the trivial phase*)

(((x1p-x2p)(y2m-y1m))/(Sqrt[fp1p fp2p]))/.{x1p->fsub[y1p],x2p->fsub[y2p],fp1p->fpsub[y1p],fp2p->fpsub[y2p]}//Simplify

(*Relating δ[y2p] to δ[y1p]*)

δ[y2p]= 2t∞ Exp[-((4π T1)/k)(1-Exp[-k y2p/2])]= 2t∞ Exp[-((4π T1)/k)(1-Exp[-k (y1p-l)/2])]= 2t∞ Exp[-((4π T1)/k)(1-Exp[-k y1p/2](1+ ((k l)/2)))]= 2t∞ Exp[-((4π T1)/k)(1-Exp[-k y1p/2])+2π T1 l Exp[-k y1p/2]]=t∞ Exp[-((4π T1)/k)(1-Exp[-k y1p/2])]Exp[2π T1 l Exp[- k y1p/2]] = δ[y1p]Exp[2π T1 l Exp[- k y1p/2]]

((16 π (y1m-y2m) (δ[y1p]-δ[y2p]))/(Sqrt[(((E^(-(1/2) k (y1p+y2p))) ((E^(k y1p)) (k^2)+32 (π^2) (T1^2)) ((E^(k y2p)) (k^2)+32 (π^2) (T1^2)) δ[y1p] δ[y2p])/(T1^2))]))/.δ[y2p]->δ[y1p]Exp[2π T1 l Exp[- k y1p/2]] //Simplify

%/.{y2p->y1p-l,y2m->y1m+l}//Simplify

%/.{y1p->U/k - σ1}//Simplify

Series[%,{k,0,0}]//Simplify//Normal

(*Exponential of bulk entropy in the connected phase*)

expScon=((((E^( (E^(-U/2)) l π T1))-(E^(- (E^(-U/2)) l π T1))) l (E^(U/2)))/(2  π T1));

(*Expanding in large time U = ku*)

expScon/.U->2Log[1/ϵ]

Series[%,{ϵ,0,2}]//Simplify//Normal

%/.ϵ->Exp[-U/2]//Simplify//Normal

(*Late time expression for the bulk entropy (exponentiated*)

expSconLate=(l^2)+(1/6) (E^(-U)) (l^4) (π^2) (T1^2);

Series[Log[(l^2)+(1/6) (l^4) (π^2) (T1^2) (ϵ^2)],{ϵ,0,2}]//Simplify//Normal

%/.ϵ->Exp[-U/2]//Simplify//Normal

(*dilaton contributions*)

((ϕr)/(4GN))(6/c)/.GN->((3 ϕr k)/c)//Simplify

(*the dilaton*)

((2fpym)/(xp-xm))+(fppym/fpym)/.{xp->xpLate,xm->xmLate,fpym->fpsub[yqm],fppym->fppsub[yqm]}//Simplify

%/.xp->fsub[yp]//Simplify

%/.δ[yqm]-> ((8π Tef)/(3k))δ[yp]//Simplify

(*Small k expansion, keeping k yqm fixed*)

Series[%/.yqm->Yqm/k,{k,0,1}]//Normal

%/.Yqm-> k ymLate/.Tef-> T1 Exp[-k yp/2]//Simplify

(*Small k expansion, keeping k u fixed*)

Series[%/.yp->U/k-σ1,{k,0,1}]//Normal

expDilaton1=Exp[((2 (E^(-U/2)) π T1)/(2k))+ ((1/4) Log[(8/(3k)) (E^(-U/2)) π T1]+(((E^(-U/2)) π T1 σ1-1)/2))];
expDilaton2=Exp[((2 (E^(-U/2)) π T1)/(2k))+ ((1/4) Log[(8/(3k)) (E^(-U/2)) π T1]+(((E^(-U/2)) π T1 σ2-1)/2))];

(*the (exponentiated) difference in the entropies is*)
expDilaton1 expDilaton1 expSbulk1 expSbulk2/expScon//Simplify

Series[%,{σ1,0,0},{σ2,0,0}]//Simplify//Normal

Series[%,{k,0,1}]//Simplify

(*Expanding in large U = k u*)

%/.U->2Log[(1/ϵ)]//Simplify

(*But recalling that (E^(-k u))= ϵ should still be >> k*)

Normal[%]/.k->ϵ kk//Simplify

Series[%,{ϵ,0,0}]//Simplify

Normal[%]/.kk->k/ϵ//Simplify

((2(E^(-2+((2 π T1 ϵ)/k))) (Sqrt[k]) ((((1/(π T1 ϵ))))^(5/2))  (((Log[((8 π T1 ϵ)/(3 k))]))^2))/(3 (Sqrt[6]) (l^2) ))

(*Since k<<ϵ, this is always very large, and so there is no island transition for finite intervals*)

Lower bound on ((Subscript[E, S])/(c(T1+T0)))=(π/12)((T1-T0)/k) for temporary blindness

(*The lower bound comes from the triple intersection between the CrossConnected, ScramblingTrivial and LateTrivial saddles*)

(*Let us first look where the CrossConnected and ScramblingTrivial saddles intersect*)

SCrossConnected[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, m_, k_]:=2 k ((ϕr)/(4GN))Log[(m/ϵ) (u-σ2)(((σ1-σ2)F[u-σ1,T1, k])/(Sqrt[Fprime[u-σ1,T1, k]]))]+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;

SScramblingTrivial[xqm_, xqp_, σ1_,σ2_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[-2m xqm(u+σ1)((( xqp-F[u-σ1,T1,k]))/((xqp-xqm)(Sqrt[Fprime[u-σ1,T1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]

(*This means that*)
Sscramblingtrivialminuscrossconnected=2 k ((ϕr)/(4GN))Log[-2m xqm(u+σ1)((( xqp-F[u-σ1,T1,k]))/((xqp-xqm)(Sqrt[Fprime[u-σ1,T1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]-(2 k ((ϕr)/(4GN))Log[(m/ϵ) (u-σ2)(((σ1-σ2)F[u-σ1,T1, k])/(Sqrt[Fprime[u-σ1,T1, k]]))]+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2])//Simplify

((ϕr (-π T0+((-1+(π^2) (T0^2) xqm xqp)/(xqm-xqp))+k Log[2]+k Log[((σ2)/ϵ)]+k Log[((m xqm (u+σ1) (xqp-F[u-σ1,T1,k]))/((xqm-xqp) (Sqrt[Fprime[u-σ1,T1,k]])))(((((m (u-σ2) (σ1-σ2) F[u-σ1,T1,k])/(ϵ (Sqrt[Fprime[u-σ1,T1,k]])))))^(-1))]))/(2 GN))

((ϕr (-π T0+((-1+(π^2) (T0^2) xqm xqp)/(xqm-xqp))+k Log[2σ2]+k Log[(( xqm (u+σ1) (xqp-F[u-σ1,T1,k]))/((xqm-xqp) ))((( (u-σ2) (σ1-σ2) F[u-σ1,T1,k]))^(-1))]))/(2 GN))

((ϕr (-π T0+((-1+(π^2) (T0^2) xqm xqp)/(xqm-xqp))+k Log[2σ2(( xqm (u+σ1) (xqp-F[u-σ1,T1,k]))/((xqm-xqp) ))((( (u-σ2) (σ1-σ2) F[u-σ1,T1,k]))^(-1))]))/(2 GN))

(*We use the following expansions*) subsfλ={f[u,k]-> t∞(1-2μ ν λ),Df[u,k]-> 2π T1 (E^(-((k u)/2)))2t∞ μ ν λ(1+Dfu2  ((k^2)/2)),DDf[u,k]->- 4 (π^2) (T1^2)(E^(-k u))2t∞ μ ν λ(1+(((E^((k u)/2)) k)/(4  π T1))+DDfu2  ((k^2)/2)),DDDf[u,k]->(E^(-((3k u)/2))) 8 (π^3) (T1^3) 2t∞ μ ν λ(1+((3 (E^((k u)/2)) k)/(4 π T1))k+DDDfu2  ((k^2)/2))};
(*And we also know that (Subscript[x, QS]) is located at*)
/.xqm->(1/(π T0))(-1+ν(k/(π T0))(((1+π T0 t∞(1-2 μ ν λ)))/(1-π T0 t∞(1-2μ ν λ))))
/.xqp->(1/(π T0))(1-((ν k)/(π T0)))

(*Let us look first at the dilaton contribution*)

Series[((ϕr (-π T0+((-1+(π^2) (T0^2) xqm xqp)/(xqm-xqp))))/(2 GN))/.xqm->(1/(π T0))(-1+ν(k/(π T0))(((1+π T0 t∞(1-2 μ ν λ)))/(1-π T0 t∞(1-2μ ν λ))))/.xqp->(1/(π T0))(1-((ν k)/(π T0))),{μ,0,1}]//Simplify

Series[-(((k^2) (1+π T0 t∞) (ν^2) ϕr)/(4 (GN (-π T0+(π^2) (T0^2) t∞+k ν)))),{ν,0,0}]//Simplify

(*We do the same for the logarithmic terms*)

ν k Log[2σ2(( xqm (u+σ1) (xqp-F[u-σ1,T1,k]))/((xqm-xqp) ))((( (u-σ2) (σ1-σ2) F[u-σ1,T1,k]))^(-1))]

Series[2σ2(( xqm (u+σ1) (xqp-t∞(1-2 μ ν λ)))/((xqm-xqp) ))((( (u-σ2) (σ1-σ2)t∞(1-2 μ ν λ)))^(-1))/.xqm->(1/(π T0))(-1+ν(k/(π T0))(((1+π T0 t∞(1-2 μ ν λ)))/(1-π T0 t∞(1-2μ ν λ))))/.xqp->(1/(π T0))(1-((ν k)/(π T0))),{μ,0,1}]//Simplify

Series[((((π^2) (T0^2) t∞+k ν+π T0 (-1+k t∞ ν)) (u+σ1) σ2)/((π^2) (T0^2) t∞ (u-σ2) (-σ1+σ2))),{ν,0,0}]//Simplify

(*The equation that we need to solve is Log[x]==0 -> x==1*)

Solve[(((-1+π T0 t∞) (u+σ1) σ2)/(π T0 t∞ (u-σ2) (-σ1+σ2)))==1,u]

(*Let us nowt look where the ScramblingTrivial LateTrivial saddles intersect. This is the page time.*)

Sscramblingtrivial[xqm_, xqp_, σ1_,σ2_, u_, m_, k_]:=2 k ((ϕr)/(4GN))Log[-2m xqm(u+σ1)((( xqp-F[u-σ1,T1,k]))/((xqp-xqm)(Sqrt[Fprime[u-σ1,T1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]
Slatetrivial[xqm_, xqp_, σ1_,σ2_, u_, m_, k_]:=(((ϕ0+ϕ[x1p,y1m,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)(((((u+σ1-y1m) (x1p-ff[u-S1,k]) ))/(((x1p-ff[y1m,k]) )))) Sqrt[(((ffd[y1m,k] ))/((ffd[u-σ1,k] )))]]]+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]

(*The difference is given by*)

2 k ((ϕr)/(4GN))Log[-2m xqm(u+σ1)((( xqp-F[u-σ1,T1,k]))/((xqp-xqm)(Sqrt[Fprime[u-σ1,T1,k]])))]+((ϕ0+2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm)))/(4 GN))-(((ϕ0+ϕ[x1p,y1m,k]))/(4 GN))-2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)(((((u+σ1-y1m) (x1p-ff[u-S1,k]) ))/(((x1p-ff[y1m,k]) )))) Sqrt[(((ffd[y1m,k] ))/((ffd[u-σ1,k] )))]]]
2 k ((ϕr)/(4GN))Log[-2m xqm(u+σ1)((( xqp-F[u-σ1,T1,k]))/((xqp-xqm)(Sqrt[Fprime[u-σ1,T1,k]])))((((2/ϵ)(((((u+σ1-y1m) (x1p-ff[u-S1,k]) ))/(((x1p-ff[y1m,k]) )))) Sqrt[(((ffd[y1m,k] ))/((ffd[u-σ1,k] )))]))^(-1))]+((2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm))-(ϕ[x1p,y1m,k]))/(4 GN))
2 k ((ϕr)/(4GN))Log[-2((π^2)/(ϵ k ))((T1^2)-(T0^2)) xqm(u+σ1)((( xqp-F[u-σ1,T1,k]))/((xqp-xqm)(Sqrt[Fprime[u-σ1,T1,k]])))((((2/ϵ)(((((u+σ1-y1m) (x1p-ff[u-S1,k]) ))/(((x1p-ff[y1m,k]) )))) Sqrt[(((ffd[y1m,k] ))/((ffd[u-σ1,k] )))]))^(-1))]+((2ϕr(((1-(((π T0))^2) xqp xqm))/(xqp-xqm))-(ϕ[x1p,y1m,k]))/(4 GN))
 ((ϕr)/(4GN))(2k Log[-((π^2)/(k ))((T1^2)-(T0^2)) xqm(u+σ1)((( xqp-F[u-σ1,T1,k]))/((xqp-xqm)))((((x1p-ff[y1m,k]))/((u+σ1-y1m) (x1p-ff[u-S1,k]))) Sqrt[(1/(ffd[y1m,k]))])]+2(((1-(((π T0))^2) xqp xqm))/(xqp-xqm))-(((Fpprime[y1m,T1,k])/(Fprime[y1m,T1,k]))+((2 Fprime[y1m,T1,k])/(x1p-F[y1m,T1,k]))))

(1/(μ )) (E^(-4π (T1/(ν k))(1-(E^(-ν k(y1m/2))))+ν (k/(4π T1))))

(*We use the following expansions*) subsfλ={f[u,k]-> t∞(1-2μ ν λ),Df[u,k]-> 2π T1 (E^(-((k u)/2)))2t∞ μ ν λ(1+Dfu2  ((k^2)/2)),DDf[u,k]->- 4 (π^2) (T1^2)(E^(-k u))2t∞ μ ν λ(1+(((E^((k u)/2)) k)/(4  π T1))+DDfu2  ((k^2)/2)),DDDf[u,k]->(E^(-((3k u)/2))) 8 (π^3) (T1^3) 2t∞ μ ν λ(1+((3 (E^((k u)/2)) k)/(4 π T1))k+DDDfu2  ((k^2)/2))};
subsfτ={f[y1m,k]-> t∞(1-2μ  τ),Df[y1m,k]-> 2π T1 (E^(-(Y1m/2)))2t∞ μ τ(1+Dfy2  ((k^2)/2)),DDf[y1m,k]->- 4 (π^2) (T1^2)(E^(-Y1m))2t∞ μ τ(1+(((E^(Y1m/2)) k)/(4  π T1))+DDfy2  ((k^2)/2)),DDDf[y1m,k]->(E^(-((3Y1m)/2))) 8 (π^3) (T1^3) 2t∞ μ τ(1+((3 (E^(Y1m/2)) k)/(4 π T1))+DDDfy2  ((k^2)/2))};
(*And we also know that the locations of (Subscript[x, QS]) and (Subscript[x, QL]) are*)
/.x1p->+(1/3) (-t∞(1-2μ ν λ)+4t∞)
/.y1m->(u-σ1)-(Log[(8  π T1)/(3 μ k)]/(2 (π T1)))
/.xqm->(1/(π T0))(-1+ν(k/(π T0))(((1+π T0 t∞(1-2 μ ν λ)))/(1-π T0 t∞(1-2μ ν λ))))
/.xqp->(1/(π T0))(1-((ν k)/(π T0)))

(*Let us first look at the dilaton terms*)

Series[2(((1-(((π T0))^2) xqp xqm))/(xqp-xqm))-(((- 4 (π^2) (T1^2)(E^(-Y1m))2t∞ μ τ(1+ν(((E^(Y1m/2)) k)/(4  π T1))+(ν^2)DDfy2  ((k^2)/2)))/( 2π T1 (E^(-(Y1m/2)))2t∞ μ τ(1+(ν^2)Dfy2  ((k^2)/2))))+((2  2π T1 (E^(-(Y1m/2)))2t∞ μ τ(1+(ν^2)Dfy2  ((k^2)/2)))/(x1p-t∞(1-2μ  τ))))/.xqm->(1/(π T0))(-1+ν(k/(π T0))(((1+π T0 t∞(1-2 μ ν λ)))/(1-π T0 t∞(1-2μ ν λ))))/.xqp->(1/(π T0))(1-((ν k)/(π T0)))/.x1p->+(1/3) (-t∞(1-2μ ν λ)+4t∞)/.y1m->(u-σ1)-(Log[(8  π T1)/(3 μ k)]/(2 (π T1))),{μ,0,0}]//Simplify

Series[(((-2 (π^2) (T0^2)+2 (π^3) (T0^3) t∞-(k^2) (ν^2)+k π T0 ν (2-k t∞ ν))/(-π T0+(π^2) (T0^2) t∞+k ν))+(((E^(-Y1m/2)) ((E^(Y1m/2)) k ν+2 π T1 (2+DDfy2 (k^2) (ν^2))))/(2+Dfy2 (k^2) (ν^2)))-((6 (E^(-Y1m/2)) π T1 (2+Dfy2 (k^2) (ν^2)) τ)/(λ ν+3 τ)))/.τ->((λ 8 π T1)/(k 3 (E^(Y1m/2)))),{ν,0,2}]//Simplify

(*Using the small k y1m approximation this becomes*)
Series[(E^(-Y1m/2))/.Y1m->ν k y1m/.y1m->(u-σ1)-(Log[(8  π T1)/(3 μ k)]/(2 (π T1))),{ν,0,2}]//Simplify

Series[(((-2 (π^2) (T0^2)+2 (π^3) (T0^3) t∞-(k^2) (ν^2)+k π T0 ν (2-k t∞ ν))/(-π T0+(π^2) (T0^2) t∞+k ν))+(((E^(-Y1m/2)) ((E^(Y1m/2)) k ν+2 π T1 (2+DDfy2 (k^2) (ν^2))))/(2+Dfy2 (k^2) (ν^2)))-((6 (E^(-Y1m/2)) π T1 (2+Dfy2 (k^2) (ν^2)) τ)/(λ ν+3 τ)))/.Dfy2->  -(1/(16 π T1 π T1 (E^(-((ν k y1m)/2)))))/.DDfy2-> ((-(E^((ν k y1m)/2)))/(8π T1 π T1))/.τ->((λ 8 π T1)/(k 3 (E^(Y1m/2))))/.Y1m->ν k y1m/.y1m->(u-σ1)-(Log[(8  π T1)/(3 ν k)]/(2 (π T1))),{ν,0,2}]//Simplify

(*Now let us look at the logarithmic terms*)

2k ν Log[-((π^2)/(k ))((T1^2)-(T0^2)) xqm(u+σ1)((( xqp-F[u-σ1,T1,k]))/((xqp-xqm)))((((x1p-ff[y1m,k]))/((u+σ1-y1m) (x1p-ff[u-S1,k]))) Sqrt[(1/(ffd[y1m,k]))])]

Assuming[{μ>0,ν>0,T1>0,T0>0,t∞>0,τ>0,λ>0},Series[-((π^2)/(ν k ))((T1^2)-(T0^2)) xqm(u+σ1)((( xqp-t∞(1-2μ ν λ)))/((xqp-xqm)))((((x1p-t∞(1-2μ τ )))/((u+σ1-y1m) (x1p-t∞(1-2μ ν λ)))) Sqrt[(1/( 2π T1 (E^(-(Y1m/2)))2t∞ μ τ(1+Dfy2 (ν^2) ((k^2)/2))))])/.xqm->(1/(π T0))(-1+ν(k/(π T0))(((1+π T0 t∞(1-2 μ ν λ)))/(1-π T0 t∞(1-2μ ν λ))))/.xqp->(1/(π T0))(1-((ν k)/(π T0)))/.x1p->+(1/3) (-t∞(1-2μ ν λ)+4t∞)/.y1m->(u-σ1)-(Log[(8  π T1)/(3 ν k)]/(2 (π T1))),{μ,0,1}]]//Simplify

Series[-(((Sqrt[π]) T1 (-(T0^2)+(T1^2)) (Sqrt[((E^(Y1m/2))/(4+2 Dfy2 (k^2) (ν^2)))]) ((π^2) (T0^2) t∞+k ν+π T0 (-1+k t∞ ν)) (u+σ1) (λ ν+3 τ))/(4 (k (T0^2) λ (ν^2) (Sqrt[T1 t∞ τ]) (4 π T1 σ1+Log[((8 π T1)/(3 k ν))])) (Sqrt[μ]))),{ν,0,-1}]//Simplify

2ν k Log[-((3 ((Sqrt[(E^(k ν y1m/2))]) (π^(3/2)) T1 (-(T0^2)+(T1^2)) (-1+π T0 t∞) (u+σ1) τ))/(8 (k T0 λ (Sqrt[μ]) (Sqrt[T1 t∞ τ]) (4 π T1 σ1+Log[((8 π T1)/(3 k ν))])) (ν^2)))]/.λ->(1/(μ ν )) (E^(-4π (T1/(ν k))(1-(E^(-ν k(((u-σ1))/2))))+ν (k/(4π T1))))/.τ->(1/(μ )) (E^(-4π (T1/(ν k))(1-(E^(-ν k(y1m/2))))+ν (k/(4π T1))))/.y1m->(u-σ1)-(Log[(8  π T1)/(3 ν k)]/(2 (π T1)))//Simplify

Series[- ν k(((4 (-1+(E^((k ν (2 π T1 (-u+σ1)+Log[A]))/(4 π T1)))) π T1)/(k ν))+((k ν)/(4 π T1)))+2k ν(1/4) ν k (u-σ1-((Log[A])/(2 π T1)))+2k ν (((4 (E^(-(1/2) k ν (u-σ1))) (-1+(E^((k ν Log[A])/(4 π T1)))) π T1)/(k ν)))+2 k ν Log[-((3   (π^(3/2)) T1 (-(T0^2)+(T1^2)) (-1+π T0 t∞) (u+σ1))/(8 k T0 (Sqrt[ T1 t∞]) ν (4 π T1 σ1+Log[A])))],{ν,0,1}]//Simplify

(*where A=8π T1/ 3 k *)

(*Hence we find that*)

Solve[2 π (T0-T1)+(1/2) (2 k+2 k π T1 u-2 k π T1 σ1-k Log[((8 π T1)/(3 k ν))]) ν-(1/(16 (π T0 T1 (-1+π T0 t∞))))((k^2) (16 T1+16 π T0 T1 t∞-4 (π^2) T0 (T1^2) (u^2)+4 (π^3) (T0^2) (T1^2) t∞ (u^2)+8 (π^2) T0 (T1^2) u σ1-8 (π^3) (T0^2) (T1^2) t∞ u σ1-4 (π^2) T0 (T1^2) ((σ1)^2)+4 (π^3) (T0^2) (T1^2) t∞ ((σ1)^2)+4 π T0 T1 u Log[((8 π T1)/(3 k ν))]-4 (π^2) (T0^2) T1 t∞ u Log[((8 π T1)/(3 k ν))]-4 π T0 T1 σ1 Log[((8 π T1)/(3 k ν))]+4 (π^2) (T0^2) T1 t∞ σ1 Log[((8 π T1)/(3 k ν))]-T0 ((Log[((8 π T1)/(3 k ν))])^2)+π (T0^2) t∞ ((Log[((8 π T1)/(3 k ν))])^2))) (ν^2)+             3 ν k (2 π T1 u-2 π T1 σ1+Log[((8 π T1)/(3 k ))]+2 Log[-((3 (π^(3/2)) T1 (-(T0^2)+(T1^2)) (-1+π T0 t∞) (u+σ1))/(8 k T0 (Sqrt[T1 t∞]) ν (4 π T1 σ1+Log[((8 π T1)/(3 k ))])))]) ν==0,u]//Expand

Solve[2 π (T0-T1)+(1/2) (2 k+2 k π T1 u-2 k π T1 σ1-k Log[((8 π T1)/(3 k ν))]) ν-(1/(16 (π T0 T1 (-1+π T0 t∞))))((k^2) (16 T1+16 π T0 T1 t∞-4 (π^2) T0 (T1^2) (u^2)+4 (π^3) (T0^2) (T1^2) t∞ (u^2)+8 (π^2) T0 (T1^2) u σ1-8 (π^3) (T0^2) (T1^2) t∞ u σ1-4 (π^2) T0 (T1^2) ((σ1)^2)+4 (π^3) (T0^2) (T1^2) t∞ ((σ1)^2)+4 π T0 T1 u Log[((8 π T1)/(3 k ν))]-4 (π^2) (T0^2) T1 t∞ u Log[((8 π T1)/(3 k ν))]-4 π T0 T1 σ1 Log[((8 π T1)/(3 k ν))]+4 (π^2) (T0^2) T1 t∞ σ1 Log[((8 π T1)/(3 k ν))]-T0 ((Log[((8 π T1)/(3 k ν))])^2)+π (T0^2) t∞ ((Log[((8 π T1)/(3 k ν))])^2))) (ν^2)+k (2 π T1 u-2 π T1 σ1+Log[((8 π T1)/(3 k ))](*+2 Log[-((3 (π^(3/2)) T1 (-(T0^2)+(T1^2)) (-1+π T0 t∞) (u+σ1))/(8 k T0 (Sqrt[T1 t∞]) ν (4 π T1 σ1+Log[((8 π T1)/(3 k ))])))]*)) ν==0,u]//Expand

Series[-(1/(3 π T1))+(2/(3 k ν))-((2 T0)/(3 k T1 ν))+σ1-((Log[((8 π T1)/(3 k))])/(6 π T1)),{ν,0,1}]//Simplify

upagezero=-((2 (T0-T1))/(3 (k T1) ν))-((2-6 π T1 σ1+Log[((8 π T1)/(3 k))])/(6 (π T1)));

Solve[2 π (T0-T1)+(1/2) (2 k+2 k π T1 u-2 k π T1 σ1-k Log[((8 π T1)/(3 k ))]) ν+k (2 π T1 u-2 π T1 σ1+Log[((8 π T1)/(3 k ))]+2 Log[-((3 (π^(3/2)) T1 (-(T0^2)+(T1^2)) (-1+π T0 t∞) (upagezero+σ1))/(8 k T0 (Sqrt[T1 t∞]) ν (4 π T1 σ1+Log[((8 π T1)/(3 k ))])))]) ν==0,u]//Simplify

(*Now we equate the two intersection times to find the lower bound*)

Assuming[{k>0,T1>0,t∞>0,σ2>0},Solve[-((4 π T0-4 π T1+2 k ν-6 k π T1 ν σ1(*+k ν Log[((8 π T1)/(3 k))]+4 k ν Log[-((3 (π^(3/2)) T1 (-(T0^2)+(T1^2)) (-1+π T0 t∞) (upagezero+σ1))/(8 k T0 (Sqrt[T1 t∞]) ν (4 π T1 σ1+Log[((8 π T1)/(3 k))])))]*))/(6 k π T1 ν))==-((σ2 (-σ1+π T0 t∞ σ2))/(π T0 t∞ σ1-σ2)),k]]//Simplify

kfirstzero=((2 π (T0-T1) (π T0 t∞ σ1-σ2))/(ν (σ2-π σ1 (T0 t∞+6 T1 σ2)+3 (π^2) T0 T1 t∞ (((σ1)^2)+((σ2)^2)))));

Assuming[{k>0,T1>0,t∞>0,σ2>0},Solve[-((4 π T0-4 π T1+2 k ν-6 k π T1 ν σ1+k ν Log[((8 π T1)/(3 kfirstzero))]+4 k ν Log[-((3 (π^(3/2)) T1 (-(T0^2)+(T1^2)) (-1+π T0 t∞) (upagezero+σ1))/(8 kfirstzero T0 (Sqrt[T1 t∞]) ν (4 π T1 σ1+Log[((8 π T1)/(3 kfirstzero))])))])/(6 k π T1 ν))==-((σ2 (-σ1+π T0 t∞ σ2))/(π T0 t∞ σ1-σ2))-(1/(π T1))-σ1,k]]//Simplify

(*Lower bound on (Es/(c(T1+T0)))*)
(( π)/(12k))(T1-T0)/.k->(4 π (T0-T1) (π T0 t∞ σ1-σ2))/(ν (-4 σ2+2 π σ1 (2 T0 t∞-9 T1 σ2)+6 (π^2) T0 T1 t∞ (2 ((σ1)^2)+((σ2)^2))+(-π T0 t∞ σ1+σ2) Log[((8 π T1)/(3 kfirstzero))]-4 (π T0 t∞ σ1-σ2) Log[-((3 (π^(3/2)) T1 (-(T0^2)+(T1^2)) (-1+π T0 t∞) (upagezero+σ1))/(8 kfirstzero T0 (Sqrt[T1 t∞]) ν (4 π T1 σ1+Log[((8 π T1)/(3 kfirstzero))])))]))//Simplify

Upper bound on ((Subscript[E, S])/(c(T1+T0)))=(π/12)((T1-T0)/k) for temporary blindness

(*For the upper bound, we use that the entropies and the time derivatives of the CrossConneected and LateTrivial saddles are be equal*)

(*Let us first look at the entropies themselves and determine where they cross*)

ScrossConn[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, m_, k_]:=2 k ((ϕr)/(4GN))Log[(m/ϵ) (u-σ2)(((σ1-σ2)F[u-σ1,T1, k])/(Sqrt[Fprime[u-σ1,T1, k]]))]+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;

Slatetrivial[xqm_, xqp_, σ1_,σ2_, u_, m_, k_]:=(((ϕ0+ϕ[x1p,y1m,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)(((((u+σ1-y1m) (x1p-ff[u-S1,k]) ))/(((x1p-ff[y1m,k]) )))) Sqrt[(((ffd[y1m,k] ))/((ffd[u-σ1,k] )))]]]+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]

(*This means that the difference becomes zero somewhere*)
Slatetrivialminuscrossconnected=(((ϕ0+ ϕr (Fpprime[y1m,T1,k]/Fprime[y1m,T1,k]+2 Fprime[y1m,T1,k]/(x1p-F[y1m,T1,k]))))/(4 GN))+2 k ((ϕr)/(4GN))Log[(2/ϵ)(((((u+σ1-y1m) (x1p-F[u-S1,T1,k]) ))/(((x1p-F[y1m,T1,k]) )))) Sqrt[(((Fprime[y1m,T1,k] ))/((Fprime[u-σ1,T1,k] )))]]+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]-(2 k ((ϕr)/(4GN))Log[(m/ϵ) (u-σ2)(((σ1-σ2)F[u-σ1,T1, k])/(Sqrt[Fprime[u-σ1,T1, k]]))]+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2])//Simplify;

(*Let us simplify this further by hand*)

((ϕr (-2 π T0+((Fpprime[y1m,T1,k])/(Fprime[y1m,T1,k]))+((2 Fprime[y1m,T1,k])/(x1p-F[y1m,T1,k]))+2 k Log[(1/2)((2σ2)/ϵ)((2(u-y1m+σ1) (x1p-F[-S1+u,T1,k]) (Sqrt[((Fprime[y1m,T1,k])/(Fprime[u-σ1,T1,k]))]))/(ϵ (x1p-F[y1m,T1,k])))(((((m (u-σ2) (σ1-σ2) F[u-σ1,T1,k])/(ϵ (Sqrt[Fprime[u-σ1,T1,k]])))))^(-1))]))/(4 GN))

(1/(4 GN))ϕr (-2 π T0+((Fpprime[y1m,T1,k])/(Fprime[y1m,T1,k]))+((2 Fprime[y1m,T1,k])/(x1p-F[y1m,T1,k]))+2 k Log[2σ2(((u-y1m+σ1) (x1p-F[-S1+u,T1,k]) (Sqrt[((Fprime[y1m,T1,k])/(Fprime[u-σ1,T1,k]))]))/( (x1p-F[y1m,T1,k])))(((((12 π (Es/(c )) (u-σ2) (σ1-σ2) F[u-σ1,T1,k])/( (Sqrt[Fprime[u-σ1,T1,k]])))))^(-1))])

(1/(4 GN))ϕr (-2 π T0+((Fpprime[y1m,T1,k])/(Fprime[y1m,T1,k]))+((2 Fprime[y1m,T1,k])/(x1p-F[y1m,T1,k]))+2 k Log[2σ2(((u-y1m+σ1) (x1p-F[-S1+u,T1,k]) (Sqrt[Fprime[y1m,T1,k]]))/( (x1p-F[y1m,T1,k])))(((12 π (Es/(c )) (u-σ2) (σ1-σ2) F[u-σ1,T1,k]))^(-1))])

(1/(4 GN))ϕr (-2 π T0+((Fpprime[y1m,T1,k])/(Fprime[y1m,T1,k]))+((2 Fprime[y1m,T1,k])/(x1p-F[y1m,T1,k]))+2 k Log[2σ2(((u-y1m+σ1) (x1p-F[-S1+u,T1,k]) (Sqrt[Fprime[y1m,T1,k]]))/( (x1p-F[y1m,T1,k])))((( (π^2) ((((T1^2)-(T0^2)))/(k )) (u-σ2) (σ1-σ2) F[u-σ1,T1,k]))^(-1))])

(*Now let us reinstate the correct parameter combination for the introduced parameter m)

(1/2)2((σ2)/ϵ)((2(u-y1m+σ1) (x1p-G[-S1+u,T1,k]) (Sqrt[((Gprime[y1m,T1,k])/(Gprime[u-σ1,T1,k]))]))/(ϵ (x1p-G[y1m,T1,k])((m (u-σ2) (σ1-σ2) G[u-σ1,T1,k])/(ϵ (Sqrt[Gprime[u-σ1,T1,k]])))))/.m->12 π (Es/(c ϵ))/.Es->((c π)/(12k))((T1^2)-(T0^2))//Simplify

(*Now we are going to use that*)

subsfλ={f[u,k]-> t∞(1-2μ ν λ),Df[u,k]-> 2π T1 (E^(-((k u)/2)))2t∞ μ ν λ(1+Dfu2  ((k^2)/2)),DDf[u,k]->- 4 (π^2) (T1^2)(E^(-k u))2t∞ μ ν λ(1+(((E^((k u)/2)) k)/(4  π T1))+DDfu2  ((k^2)/2)),DDDf[u,k]->(E^(-((3k u)/2))) 8 (π^3) (T1^3) 2t∞ μ ν λ(1+((3 (E^((k u)/2)) k)/(4 π T1))k+DDDfu2  ((k^2)/2))};
subsfτ={f[y1m,k]-> t∞(1-2μ  τ),Df[y1m,k]-> 2π T1 (E^(-(Y1m/2)))2t∞ μ τ(1+Dfy2  ((k^2)/2)),DDf[y1m,k]->- 4 (π^2) (T1^2)(E^(-Y1m))2t∞ μ τ(1+(((E^(Y1m/2)) k)/(4  π T1))+DDfy2  ((k^2)/2)),DDDf[y1m,k]->(E^(-((3Y1m)/2))) 8 (π^3) (T1^3) 2t∞ μ τ(1+((3 (E^(Y1m/2)) k)/(4 π T1))+DDDfy2  ((k^2)/2))};

(*And we also know that (Subscript[x, QL]) is located at*)
/.x1p->+(1/3) (-t∞(1-2μ ν λ)+4t∞)
/.y1m->(u-σ1)-(Log[(8  π T1)/(3 ν k)]/(2 (π T1)))

(*We also know that*)
/.Dfu2-> -(1/(16 π T1 π T1 (E^(-((k u)/2)))))/.DDfu2-> ((-(E^((k u)/2)))/(8π T1 π T1))

(*Let us do this by hand*)

(1/(4 GN))ϕr (-2 π T0+(((- 4 (π^2) (T1^2)(E^(-ν k y1m))2t∞  μ τ(1+(((E^(Y1m/2)) k)/(4  π T1))ν+DDfy2 (ν^2) ((k^2)/2)))/(2π T1 (E^(-((ν k y1m)/2)))2t∞  μ τ(1+Dfy2 (ν^2) ((k^2)/2))))+((4π T1 (E^(-((ν k y1m)/2)))2t∞  μ τ(1+Dfy2 (ν^2) ((k^2)/2)))/(x1p- t∞(1-2μ τ))))+2 ν k Log[2σ2(((u-y1m+σ1) (x1p-t∞(1-2μ ν λ)) (Sqrt[2π T1 (E^(-((ν k y1m)/2)))2t∞  μ τ(1+Dfy2 (ν^2) ((k^2)/2))]))/( (x1p- t∞(1-2ν τ))))((( (π^2) ((((T1^2)-(T0^2)))/(ν k )) (u-σ2) (σ1-σ2) t∞(1-2μ ν λ)))^(-1))])

(*Let us now simplify this further and use the location of x1m*)

(x1p- t∞(1-2μ τ))/.x1p->+(1/3) (-t∞(1-2μ ν λ)+4t∞)//Simplify

(x1p-t∞(1-2μ ν λ)) /.x1p->+(1/3) (-t∞(1-2μ ν λ)+4t∞)//Simplify

(u-y1m+σ1)/.y1m->(u-σ1)-(Log[(8  π T1)/(3 ν k)]/(2 (π T1)))//Simplify

(1/(4 GN))ϕr (-2 π T0+(((- 4 (π^2) (T1^2)(E^(-ν k y1m))2t∞  μ τ(1+(((E^(Y1m/2)) k)/(4  π T1))ν+DDfy2 (ν^2) ((k^2)/2)))/(2π T1 (E^(-((ν k y1m)/2)))2t∞  μ τ(1+Dfy2 (ν^2) ((k^2)/2))))+((4π T1 (E^(-((ν k y1m)/2)))2t∞  μ τ(1+Dfy2 (ν^2) ((k^2)/2)))/((2/3) t∞ μ(λ ν+3 τ))))+2 μ k Log[2σ2(((2 σ1+((Log[((8 π T1)/(3 k ν))])/(2 π T1))) (8/3) t∞ λ μ ν (Sqrt[2π T1 (E^(-((ν k y1m)/2)))2t∞  μ τ(1+Dfy2 (ν^2) ((k^2)/2))]))/((2/3) t∞ μ (λ ν+3 τ)(π^2) ((((T1^2)-(T0^2)))/(ν k )) (u-σ2) (σ1-σ2) t∞(1-2μ ν λ)))])

(*This can be simplified even further*)

Assuming[{T1>0,t∞>0,k>0,τ>0,ν>0},2σ2(((2 σ1+((Log[((8 π T1)/(3 k ν))])/(2 π T1))) (8/3) t∞ λ μ ν (Sqrt[2π T1 (E^(-((ν k y1m)/2)))2t∞  μ τ(1+Dfy2 (ν^2) ((k^2)/2))]))/((2/3) t∞ μ (λ ν+3 τ)(π^2) ((((T1^2)-(T0^2)))/(ν k )) (u-σ2) (σ1-σ2) t∞(1-2μ ν λ)))]//Simplify

Assuming[{T1>0,t∞>0,k>0,τ>0,ν>0},((- 4 (π^2) (T1^2)(E^(-ν k y1m))2t∞  μ τ(1+(((E^(Y1m/2)) k)/(4  π T1))ν+DDfy2 (ν^2) ((k^2)/2)))/(2π T1 (E^(-((ν k y1m)/2)))2t∞  μ τ(1+Dfy2 (ν^2) ((k^2)/2))))+((4π T1 (E^(-((ν k y1m)/2)))2t∞  μ τ(1+Dfy2 (ν^2) ((k^2)/2)))/((2/3) t∞ μ(λ ν+3 τ)))]//Simplify

(1/(4 GN))ϕr (-2 π T0+(((4 (E^(-(1/2) k y1m ν)) π T1 (-1-(((E^(Y1m/2)) k ν)/(4 π T1))-(1/2) DDfy2 (k^2) (ν^2)+((3 (((2+Dfy2 (k^2) (ν^2)))^2) τ)/(2 λ ν+6 τ))))/(2+Dfy2 (k^2) (ν^2))))+2μ k Log[((4 (Sqrt[2]) k λ (ν^2) σ2 (Sqrt[(E^(-(1/2) k y1m ν)) T1 t∞ μ (2+Dfy2 (k^2) (ν^2)) τ]) (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]))/((π^(5/2)) T1 (-(T0^2)+(T1^2)) t∞ (-1+2 λ μ ν) (u-σ2) (-σ1+σ2) (λ ν+3 τ)))])

(*Now let us expand this expression in μ*)

Series[((4 (Sqrt[2]) k λ (ν^2) σ2 (Sqrt[(E^(-(1/2) k y1m ν)) T1 t∞ μ (2+Dfy2 (k^2) (ν^2)) τ]) (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]))/((π^(5/2)) T1 (-(T0^2)+(T1^2)) t∞ (-1+2 λ μ ν) (u-σ2) (-σ1+σ2) (λ ν+3 τ))),{μ,0,1}]

(1/(4 GN))ϕr (-2 π T0+(((4 (E^(-(1/2) k y1m ν)) π T1 (-1-(((E^(Y1m/2)) k ν)/(4 π T1))-(1/2) DDfy2 (k^2) (ν^2)+((3 (((2+Dfy2 (k^2) (ν^2)))^2) τ)/(2 λ ν+6 τ))))/(2+Dfy2 (k^2) (ν^2))))+2 ν k Log[-(((((4 ((Sqrt[2]) k λ (ν^2) σ2 (Sqrt[(E^(-(1/2) k y1m ν)) T1 t∞ (2+Dfy2 (k^2) (ν^2)) τ]) (4 π T1 σ1+Log[((8 π T1)/(3 k ν))])) (Sqrt[μ]))/((π^(5/2)) T1 (-(T0^2)+(T1^2)) t∞ (u-σ2) (-σ1+σ2) (λ ν+3 τ)))))^( ))(1+O[(μ^(1/2))]) ])

(*We find*)

(1/(4 GN))ϕr (-2 π T0+(((4 (E^(-(1/2) k y1m ν)) π T1 (-1-(((E^(Y1m/2)) k ν)/(4 π T1))-(1/2) DDfy2 (k^2) (ν^2)+((3 (((2+Dfy2 (k^2) (ν^2)))^2) τ)/(2 λ ν+6 τ))))/(2+Dfy2 (k^2) (ν^2))))+2ν k Log[-((4 ((Sqrt[2]) k λ (ν^2) σ2 (Sqrt[(E^(-(1/2) k y1m ν)) T1 t∞ (2+Dfy2 (k^2) (ν^2)) τ]) (4 π T1 σ1+Log[((8 π T1)/(3 k ν))])) (Sqrt[μ]))/((π^(5/2)) T1 (-(T0^2)+(T1^2)) t∞ (u-σ2) (-σ1+σ2) (λ ν+3 τ)))]+0)

(*Next, we perform a series in small ν*)
Series[-((4 ((Sqrt[2]) k λ (ν^2) σ2 (Sqrt[(E^(-(1/2) k y1m ν)) T1 t∞ (2+Dfy2 (k^2) (ν^2)) τ]) (4 π T1 σ1+Log[((8 π T1)/(3 k ν))])) (Sqrt[μ]))/((π^(5/2)) T1 (-(T0^2)+(T1^2)) t∞ (u-σ2) (-σ1+σ2) (λ ν+3 τ))),{ν,0,3}]//Simplify

Series[((4 (E^(-(1/2) k y1m ν)) π T1 (-1-(((E^(Y1m/2)) k ν)/(4 π T1))-(1/2) DDfy2 (k^2) (ν^2)+((3 (((2+Dfy2 (k^2) (ν^2)))^2) τ)/(2 λ ν+6 τ))))/(2+Dfy2 (k^2) (ν^2)))/.Dfy2-> -(1/(16 π T1 π T1 (E^(-(Y1m/2)))))/.DDfy2-> ((-(E^(Y1m/2)))/(8π T1 π T1)),{ν,0,2}]//Simplify

(1/(4 GN))ϕr (-2 π T0+2 π T1-(((8 π T1 λ+3 (E^(Y1m/2)) k τ+6 k π T1 y1m τ) ν)/(6 τ))+(((-9 (E^(Y1m/2)) (k^2) (τ^2)+36 (E^(Y1m/2)) (k^2) π T1 y1m (τ^2)+4 (π^2) (T1^2) (((4 λ+3 k y1m τ))^2)) (ν^2))/(144 π T1 (τ^2)))+2ν k Log[-((8 (k λ (Sqrt[μ]) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ν))])) (ν^2))/(3 ((π^(5/2)) (-(T0^2)+(T1^2)) (u-σ2) (-σ1+σ2) (Sqrt[T1 t∞ τ]))))+((2 k T1 t∞ λ (Sqrt[μ]) σ2 (4 λ+3 k y1m τ) (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]) (ν^3))/(9 (π^(5/2)) (-(T0^2)+(T1^2)) (u-σ2) (-σ1+σ2) (((T1 t∞ τ))^(3/2))))]+0)

Series[Log[-((8 (k λ (Sqrt[μ]) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ))])) (ν^2))/(3 ((π^(5/2)) (-(T0^2)+(T1^2)) (u-σ2) (-σ1+σ2) (Sqrt[T1 t∞ τ]))))+((2 k T1 t∞ λ (Sqrt[μ]) σ2 (4 λ+3 k y1m τ) (4 π T1 σ1+Log[((8 π T1)/(3 k ))]) (ν^3))/(9 (π^(5/2)) (-(T0^2)+(T1^2)) (u-σ2) (-σ1+σ2) (((T1 t∞ τ))^(3/2))))],{ν,0,1}]//Simplify

(1/(4 GN))ϕr (-2 π T0+2 π T1-(((8 π T1 λ+3 (E^(Y1m/2)) k τ+6 k π T1 y1m τ) ν)/(6 τ))+(((-9 (E^(Y1m/2)) (k^2) (τ^2)+36 (E^(Y1m/2)) (k^2) π T1 y1m (τ^2)+4 (π^2) (T1^2) (((4 λ+3 k y1m τ))^2)) (ν^2))/(144 π T1 (τ^2)))+2ν k (Log[-((8 k λ (Sqrt[μ]) (ν^2) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k))]))/(3 (π^(5/2)) (-(T0^2)+(T1^2)) (u-σ2) (-σ1+σ2) (Sqrt[T1 t∞ τ])))]+(((-4 λ-3 k y1m τ) ν)/(12 τ)))+0)

(*The correction is of the order (ν^3)*)

(*Next we use the relation between τ and λ*)

(1/(4 GN))ϕr (-2 π T0+2 π T1-(((8 π T1 λ+3 (E^(Y1m/2)) k τ+6 k π T1 y1m τ) ν)/(6 τ))+(((-9 (E^(Y1m/2)) (k^2) (τ^2)+36 (E^(Y1m/2)) (k^2) π T1 y1m (τ^2)+4 (π^2) (T1^2) (((4 λ+3 k y1m τ))^2)) (ν^2))/(144 π T1 (τ^2)))+2ν k (Log[-((8 k λ (Sqrt[μ]) (ν^2) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]))/(3 (π^(5/2)) (-(T0^2)+(T1^2)) (u-σ2) (-σ1+σ2) (Sqrt[T1 t∞ τ])))]+(((-4 λ-3 k y1m τ) ν)/(12 τ))))/.τ->λ((((((3(E^(ν k y1m/2))) k )/(8 π T1))))^(-1))/.Y1m->ν k y1m//Simplify

(*Let us insert the remaining terms*)

(*We expand further*)

Series[(E^(ν k (y1m/2))),{ν,0,2}]

Series[ (E^((k y1m ν)/2)) (-5+(E^((k y1m ν)/2))) (k^2) (ν^2)+8 k π T1 ν (-k y1m ν+(E^((k y1m ν)/2)) (-2+k y1m ν))+4 (π^2) T1 (-8 T0+T1 (8-4 k y1m ν+(k^2) (y1m^2) (ν^2))),{ν,0,2}]//Simplify

32 (π^2) T1 (-T0+T1)-16 (k π T1 (1+π T1 y1m)) ν+4 (k^2) (-1-2 π T1 y1m+(π^2) (T1^2) (y1m^2)) (ν^2)+32 k π T1 ν Log[-((2 (Sqrt[(2/3)]) k λ (Sqrt[μ]) (ν^2) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]))/((π^3) (-(T0^2)+(T1^2)) (Sqrt[(((E^(-(1/2) k y1m ν)) (T1^2) t∞ λ)/k)]) (u-σ2) (-σ1+σ2)))]

(*We can simplify this furter and take the exponential outside the log and substitute the rest of the expressions*)
32 (π^2) T1 (-T0+T1)-16 (k π T1 (1+π T1 y1m)) ν+4 (k^2) (-1-2 π T1 y1m+(π^2) (T1^2) (y1m^2)) (ν^2)+32 k π T1 ν ((1/4) k y1m ν+ Log[-((2 (Sqrt[(2/3)]) k(Sqrt[k]) λ (Sqrt[μ]) (ν^2) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]))/((π^3) (-(T0^2)+(T1^2)) ((((T1^2) t∞ λ ))^(1/2))(u-σ2) (-σ1+σ2)))])/.y1m->(u-σ1)-Log[(8  π T1)/(3 ν k)]/(2 (π T1))/.λ->(1/(μ ν)) (E^(-4π (T1/(ν k))(1-(E^(-ν k(((u-σ1))/2))))))//Simplify

4 (8 (π^2) T1 (-T0+T
1)-2 k π T1 ν (2+2 π T1 (u-σ1)-Log[((8 π T1)/(3 k ν))])+(k^2) (ν^2) (-1-2 π T1 u+2 π T1 σ1+Log[((8 π T1)/(3 k ν))]+(1/4) (((2 π T1 (-u+σ1)+Log[((8 π T1)/(3 k ν))]))^2))+k ν (-k ν Log[((8 π T1)/(3 k ν))]+2 π T1 (k ν (u-σ1)-4((2(1-(E^(-(1/2) k ν (u-σ1)))) π T1)/(k ν)) +4 Log[-((2 (Sqrt[(2/3)]) (k^(3/2)) (Sqrt[μ]) (Sqrt[(( (T1^2) t∞)/( ν))]) (ν^2) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]))/((π^3) (T1^2) (-(T0^2)+(T1^2)) t∞ (u-σ2) (-σ1+σ2)))])))

Series[8 (π^2) T1 (-T0+T1)-2 k π T1 ν (2+2 π T1 (u-σ1)-Log[((8 π T1)/(3 k ν))])+(k^2) (ν^2) (-1-2 π T1 u+2 π T1 σ1+Log[((8 π T1)/(3 k ν))]+(1/4) (((2 π T1 (-u+σ1)+Log[((8 π T1)/(3 k ν))]))^2))+k ν (-k ν Log[((8 π T1)/(3 k ν))]+2 π T1 (k ν (u-σ1)-4((2(1-(E^(-(1/2) k ν (u-σ1)))) π T1)/(k ν)) +4 Log[-((2 (Sqrt[(2/3)]) (k^(3/2)) (Sqrt[μ]) (Sqrt[(( (T1^2) t∞)/( ν))]) (ν^2) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]))/((π^3) (T1^2) (-(T0^2)+(T1^2)) t∞ (u-σ2) (-σ1+σ2)))])),{ν,0,2}]

8 (π^2) (T1-T0) T1-2 (2 k π T1+6 k (π^2) (T1^2) u-6 k (π^2) (T1^2) σ1-k π T1 Log[((8 π T1)/(3 k ν))]-4 k π T1 Log[-((2 (Sqrt[(2/3)]) (k^(3/2)) (((((8 π T1)/3)))^(-3/2))(((((8 π T1)/3)))^(3/2))(Sqrt[μ]) (Sqrt[(((T1^2) t∞)/ν)]) (ν^2) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]))/((π^3) (T1^2) (-(T0^2)+(T1^2)) t∞ (u-σ2) (-σ1+σ2)))]) ν+(1/4) (-4 (k^2)+12 (k^2) (π^2) (T1^2) (u^2)-24 (k^2) (π^2) (T1^2) u σ1+12 (k^2) (π^2) (T1^2) ((σ1)^2)-4 (k^2) π T1 u Log[((8 π T1)/(3 k ν))]+4 (k^2) π T1 σ1 Log[((8 π T1)/(3 k ν))]+(k^2) ((Log[((8 π T1)/(3 k ν))])^2)) (ν^2)

8 (π^2) (T1-T0) T1-2 (2 k π T1+6 k (π^2) (T1^2) u-6 k (π^2) (T1^2) σ1-k π T1 Log[((8 π T1)/(3 k ν))]+6k π T1 Log[(((8 π T1)/(3k)))]-4 k π T1 Log[-((2 (Sqrt[(2/3)])  (((((8 π T1)/3)))^(3/2))(Sqrt[μ]) (Sqrt[(((T1^2) t∞)/ν)]) (ν^2) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]))/((π^3) (T1^2) (-(T0^2)+(T1^2)) t∞ (u-σ2) (-σ1+σ2)))]) ν+(1/4) (-4 (k^2)+12 (k^2) (π^2) (T1^2) (u^2)-24 (k^2) (π^2) (T1^2) u σ1+12 (k^2) (π^2) (T1^2) ((σ1)^2)-4 (k^2) π T1 u Log[((8 π T1)/(3 k ν))]+4 (k^2) π T1 σ1 Log[((8 π T1)/(3 k ν))]+(k^2) ((Log[((8 π T1)/(3 k ν))])^2)) (ν^2)

Collect[{8 (π^2) (T1-T0) T1, (2 k π T1+6 k (π^2) (T1^2) u-6 k (π^2) (T1^2) σ1-k π T1 Log[((8 π T1)/(3 k ν))]+6k π T1 Log[(((8 π T1)/(3k)))]-4 k π T1 Log[-((2 (Sqrt[(2/3)])  (((((8 π T1)/3)))^(3/2))(Sqrt[μ]) (Sqrt[(((T1^2) t∞)/ν)]) (ν^2) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]))/((π^3) (T1^2) (-(T0^2)+(T1^2)) t∞ (u-σ2) (-σ1+σ2)))]),-4 (k^2)+12 (k^2) (π^2) (T1^2) (u^2)-24 (k^2) (π^2) (T1^2) u σ1+12 (k^2) (π^2) (T1^2) ((σ1)^2)-4 (k^2) π T1 u Log[((8 π T1)/(3 k ν))]+4 (k^2) π T1 σ1 Log[((8 π T1)/(3 k ν))]+(k^2) ((Log[((8 π T1)/(3 k ν))])^2)},Log[((8 π T1)/(3 k))]]//Simplify

(*So we find*)

8 (π^2) (T1-T0) T1-2  ν k π T1 (2+6 π T1 u-6 π T1 σ1+6 Log[((8 π T1)/(3 k))]-Log[((8 π T1)/(3 k ν))]-4 Log[-((64 (T1^(3/2)) (Sqrt[μ]) ν σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ν))]))/(9 (π^(3/2)) (-(T0^2)+(T1^2)) (Sqrt[(((T1^2) t∞)/ν)]) (u-σ2) (-σ1+σ2)))])+(1/4)  (ν^2)(k^2) (4 (-1+3 (π^2) (T1^2) (((u-σ1))^2))+4 π T1 (-u+σ1) Log[((8 π T1)/(3 k ν))]+((Log[((8 π T1)/(3 k ν))])^2))

(*let us try to solve this equation recursively*)

Solve[8 (π^2) (T1-T0) T1-2  ν k π T1 (2+6 π T1 u-6 π T1 σ1+6 Log[((8 π T1)/(3 k))]-Log[((8 π T1)/(3 k ))]-4 Log[(((*-*)64 (T1^(3/2)) (Sqrt[μ]) ν σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ))])π T1)/(9 (π^(3/2)) (-(T0^2)+(T1^2)) (Sqrt[(((T1^2) t∞)/ν)]) (*π T1 (u-σ2) *)(-σ1+σ2)))])==0,u]//Simplify

Assuming[{ν>0,T1>0,T0>0,k>0},Series[((-4 π T0+4 π T1-2 k ν+6 k π T1 ν σ1-5 k ν Log[((8 π T1)/(3 k))]+4 k ν Log[((64 (T1^(5/2)) (Sqrt[μ]) ν σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k))]))/(9 (Sqrt[π]) ((T0^2)-(T1^2)) (Sqrt[(((T1^2) t∞)/ν)]) (σ1-σ2)))])/(6 k π T1 ν)),{ν,0,1}]]//Simplify

uScrosszero=((2 (T1-T0))/(3 k T1 ν))+((-2+6 π T1 σ1-5 Log[((8 π T1)/(3 k))]+4 Log[((64 (Sqrt[μ]) (((T1 ν))^(3/2)) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k))]))/(9 (Sqrt[π]) ((T0^2)-(T1^2)) (Sqrt[t∞]) (σ1-σ2)))])/(6 π T1));

Solve[8 (π^2) (T1-T0) T1-2  ν k π T1 (2+6 π T1 u-6 π T1 σ1+6 Log[((8 π T1)/(3 k))]-Log[((8 π T1)/(3 k ))]-4 Log[((-64 (T1^(3/2)) (Sqrt[μ]) ν σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k ))]))/(9 (π^(3/2)) (-(T0^2)+(T1^2)) (Sqrt[(((T1^2) t∞)/ν)]) (uScrosszero-σ2) (-σ1+σ2)))]) ==0,u]//Simplify

Assuming[{ν>0,T1>0,k>0,T0>0},Series[((-4 π T0+4 π T1-2 k ν+6 k π T1 ν σ1-5 k ν Log[((8 π T1)/(3 k))]+4 k ν Log[-((64 (T1^(3/2)) (Sqrt[μ]) ν σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k))]))/(9 (π^(3/2)) (-(T0^2)+(T1^2)) (Sqrt[(((T1^2) t∞)/ν)]) (uScrosszero-σ2) (-σ1+σ2)))])/(6 k π T1 ν)),{ν,0,1}]]//Simplify

(*Next step is to find out where the derivatives cross*)

ScrossConn[σ1_, σ2_, u_,ϕr_,ϵ_,GN_, m_, k_]:=2 k ((ϕr)/(4GN))Log[(m/ϵ) (u-σ2)(((σ1-σ2)F[u-σ1,T1, k])/(Sqrt[Fprime[u-σ1,T1, k]]))]+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2]/.ϕ0->vvϕ0;

Slatetrivial[xqm_, xqp_, σ1_,σ2_, u_, m_, k_]:=(((ϕ0+ϕ[x1p,y1m,k]))/(4 GN))+2 k ((ϕr)/(4GN))Log[Abs[(2/ϵ)(((((u+σ1-y1m) (x1p-ff[u-S1,k]) ))/(((x1p-ff[y1m,k]) )))) Sqrt[(((ffd[y1m,k] ))/((ffd[u-σ1,k] )))]]]+2 k ((ϕr)/(4GN))Log[Abs[2((σ2)/ϵ)]]

(*This means that*)
Slatetrivialminuscrossconnected=(((ϕ0+ ϕr (Fpprime[y1m,T1,k]/Fprime[y1m,T1,k]+2 Fprime[y1m,T1,k]/(x1p-F[y1m,T1,k]))))/(4 GN))+2 k ((ϕr)/(4GN))Log[(2/ϵ)(((((u+σ1-y1m) (x1p-F[u-S1,T1,k]) ))/(((x1p-F[y1m,T1,k]) )))) Sqrt[(((Fprime[y1m,T1,k] ))/((Fprime[u-σ1,T1,k] )))]]+2 k ((ϕr)/(4GN))Log[2((σ2)/ϵ)]-(2 k ((ϕr)/(4GN))Log[(m/ϵ) (u-σ2)(((σ1-σ2)F[u-σ1,T1, k])/(Sqrt[Fprime[u-σ1,T1, k]]))]+(((ϕ0+ϕr 2 π T0))/(4 GN))+2 k ((ϕr)/(4GN))Log[2])//Simplify

(*Simplifying leads to*)

((ϕr (-2 π T0+((Fpprime[y1m,T1,k])/(Fprime[y1m,T1,k]))+((2 Fprime[y1m,T1,k])/(x1p-F[y1m,T1,k]))+2 k Log[2σ2(((u-y1m+σ1) (x1p-F[-S1+u,T1,k]) (Sqrt[Fprime[y1m,T1,k]]))/( (x1p-F[y1m,T1,k])))(((12 π (Es/(c )) (u-σ2) (σ1-σ2) F[u-σ1,T1,k]))^(-1))]))/(4 GN))

(*Let us now take a derivative*)
D[(1/(4 GN))ϕr (-2 π T0+((Fpprime[y1m,T1,k])/(Fprime[y1m,T1,k]))+((2 Fprime[y1m,T1,k])/(x1p-F[y1m,T1,k]))+2 k Log[2σ2(((u-y1m+σ1) (x1p-F[-σ1+u,T1,k]) (Sqrt[Fprime[y1m,T1,k]]))/( (x1p-F[y1m,T1,k])))(((12 π (Es/(c )) (u-σ2) (σ1-σ2) F[u-σ1,T1,k]))^(-1))]),u]//Simplify

(*as always, we use the following expansions*)

subsfλ={f[u,k]-> t∞(1-2μ ν λ),Df[u,k]-> 2π T1 (E^(-((k u)/2)))2t∞ μ ν λ(1+Dfu2  ((k^2)/2)),DDf[u,k]->- 4 (π^2) (T1^2)(E^(-k u))2t∞ μ ν λ(1+(((E^((k u)/2)) k)/(4  π T1))+DDfu2  ((k^2)/2)),DDDf[u,k]->(E^(-((3k u)/2))) 8 (π^3) (T1^3) 2t∞ μ ν λ(1+((3 (E^((k u)/2)) k)/(4 π T1))k+DDDfu2  ((k^2)/2))};
subsfτ={f[y1m,k]-> t∞(1-2μ  τ),Df[y1m,k]-> 2π T1 (E^(-(Y1m/2)))2t∞ μ τ(1+Dfy2  ((k^2)/2)),DDf[y1m,k]->- 4 (π^2) (T1^2)(E^(-Y1m))2t∞ μ τ(1+(((E^(Y1m/2)) k)/(4  π T1))+DDfy2  ((k^2)/2)),DDDf[y1m,k]->(E^(-((3Y1m)/2))) 8 (π^3) (T1^3) 2t∞ μ τ(1+((3 (E^(Y1m/2)) k)/(4 π T1))+DDDfy2  ((k^2)/2))};

(*Since we want to know when this becomes zero, it is sufficient to look at the numerator of this expression*)

Series[(*(k/(2 GN))ϕr*) (((x1p (y1m-σ1-σ2) t∞(1-2μ ν λ)+(-y1m+σ1+σ2) ((( t∞(1-2μ ν λ)))^2)-x1p (u-y1m+σ1) (u-σ2) 2π T1 (E^(-((ν k (u-σ1))/2)))2t∞ μ ν λ(1+Dfu2 (ν^2) ((k^2)/2))))/(1(* (u-y1m+σ1) (u-σ2) (x1p-F[u-σ1,T1,k]) F[u-σ1,T1,k]*)))/.y1m->-(Log[(8 (E^(-2 π T1 (u-σ1))) π T1)/(3 ν k)]/(2 (π T1)))/.x1p->+(1/3) (-t∞(1-2μ ν λ)+4t∞),{ν,0,1}]//Simplify

(*Let us perform an expansion*)

Series[(1/3) ((t∞)^2) λ μ (-8 (σ1+σ2)-((4 Log[((8 (E^(-2 π T1 (u-σ1))) π T1)/(3 k ν))])/(π T1))-3 (E^(-(1/2) k ν (u-σ1))) (2+Dfu2 (k^2) (ν^2)) (u-σ2) (2 π T1 (u+σ1)+Log[((8 (E^(-2 π T1 (u-σ1))) π T1)/(3 k ν))])),{ν,0,1}]//Simplify

(*Now we can solve this for u*)

Solve[(*6 (π^2) (T1^2) (u^2)+4 π T1 σ1+6 (π^2) (T1^2) u σ1+4 π T1 σ2-6 (π^2) (T1^2) u σ2-6 (π^2) (T1^2) σ1 σ2-4 π T1 (u-σ1)*)+2 Log[((8  π T1)/(3 k ν))](*-2 π T1 (u-σ1)3 π T1 u*)+3 π T1 u Log[((8  π T1)/(3 k ν))](*+2 π T1 (u-σ1)3 π T1 σ2*) -3 π T1 σ2 Log[((8  π T1)/(3 k ν))] ==0,u]//Simplify

Solve[6 (π^2) (T1^2) (u^2)+4 π T1 σ1+6 (π^2) (T1^2) u σ1+4 π T1 σ2-6 (π^2) (T1^2) u σ2-6 (π^2) (T1^2) σ1 σ2-4 π T1 (u-σ1)+2 Log[((8  π T1)/(3 k ))]-2 π T1 (u-σ1)3 π T1 u+3 π T1 u Log[((8  π T1)/(3 k ))]+2 π T1 (u-σ1)3 π T1 σ2 -3 π T1 σ2 Log[((8  π T1)/(3 k ))] ==0,u]//Simplify

Series[((4 π T1 (-σ2+σ1 (-2+3 π T1 σ2))+(-2+3 π T1 σ2)Λ Log[((8 π T1)/(3 k))])/(π T1 (-4+12 π T1 σ1+Λ 3 Log[((8 π T1)/(3 k))]))),{Λ,∞,2}]//Simplify

(*Now we have to equate these two times in order to find the upper bound*)

Solve[((2 (-T0+T1))/(3 k T1 ))+((-2+6 π T1 σ1-5 Log[((8 π T1)/(3 k))]+4 Log[-((64 (Sqrt[T1])   σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k))]))/(9 (π^(3/2)) (-(T0^2)+(T1^2)) (Sqrt[t∞]) (uScrosszero-σ2) (-σ1+σ2)))])/(6 π T1))==((4 π T1 (-σ2+σ1 (-2+3 π T1 σ2))+(-2+3 π T1 σ2) Log[((8 π T1)/(3 k))])/(π T1 (-4+12 π T1 σ1+3 Log[((8 π T1)/(3 k))]))),k]

(*This is hard to solve*)

Solve[((2 (-T0+T1))/(3 k T1 ))+((-2+6 π T1 σ1(*-5 Log[((8 π T1)/(3 k))]+4 Log[-((64 (Sqrt[T1])   σ2 (4 π T1 σ1+Log[((8 π T1)/(3 k))]))/(9 (π^(3/2)) (-(T0^2)+(T1^2)) (Sqrt[t∞]) (uScrosszero-σ2) (-σ1+σ2)))]*))/(6 π T1))==(-(2/(3 π T1))+σ2),k]//Simplify

ksecondzero=((2 π (T0-T1))/(1+3 π T1 (σ1-σ2)));

Solve[((2 (-T0+T1))/(3 k T1 ))+((-2+6 π T1 σ1-5 Log[((8 π T1)/(3 ksecondzero))]+4 Log[-((64 (Sqrt[T1])   σ2 (4 π T1 σ1+Log[((8 π T1)/(3 ksecondzero))]))/(9 (π^(3/2)) (-(T0^2)+(T1^2)) (Sqrt[t∞]) (uScrosszero-σ2) (-σ1+σ2)))])/(6 π T1))==((4 π T1 (-σ2+σ1 (-2+3 π T1 σ2))+(-2+3 π T1 σ2) Log[((8 π T1)/(3 ksecondzero))])/(π T1 (-4+12 π T1 σ1+3 Log[((8 π T1)/(3 ksecondzero))]))),k]//Simplify//Simplify

(*Upper bound on (Es/(c(T1+T0)))*)
(( π)/(12k))(T1-T0)/.k->(4 π (T0-T1) (-4+12 π T1 σ1+3 Log[((8 π T1)/(3 ksecondzero))]))/(-15 ((Log[((8 π T1)/(3 ksecondzero))])^2)-2 Log[((8 π T1)/(3 ksecondzero))] (-13+3 π T1 (7 σ1+3 σ2)-6 Log[-((64 (Sqrt[T1]) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 ksecondzero))]))/(9 (π^(3/2)) (-(T0^2)+(T1^2)) (Sqrt[t∞]) (uScrosszero-σ2) (-σ1+σ2)))])+8 (1+9 (π^2) (T1^2) σ1 (σ1-σ2)+3 π T1 σ2+(-2+6 π T1 σ1) Log[-((64 (Sqrt[T1]) σ2 (4 π T1 σ1+Log[((8 π T1)/(3 ksecondzero))]))/(9 (π^(3/2)) (-(T0^2)+(T1^2)) (Sqrt[t∞]) (uScrosszero-σ2) (-σ1+σ2)))]))/.t∞->(1/(π T1))//Simplify