19. Asymptotic Factorisation of the Symplectic Form for N Well‑Separated Double Sine-Gordon Wobbling Kinks

We extend the asymptotic factorisation theorem for the canonical field‑theoretic symplectic form to $N$ widely separated double sine–Gordon wobbling kinks. Each wobbling kink carries, besides its translational degree of freedom, an \emph{approximate} internal shape mode derived from the two‑subkink model. Using a superposition ansatz that correctly accounts for both kink and antikink configurations, we prove that the pullback of the canonical symplectic form $\Omega$ factorises, up to exponentially small off‑diagonal corrections between different solitons, into a sum of single‑soliton–shape‑mode symplectic forms. Each such form is exactly the symplectic form of a moving wobbling kink (or antikink) as obtained in~\cite{Coupled}; it contains the free translational block $dP_i\wedge da_i$, the internal sector, and intra‑kink coupling terms that are perturbatively small in the non‑relativistic small‑amplitude regime. The inter‑kink corrections are bounded by $\mathcal{O}(e^{-\mu D_{\min}/2})$ where $\mu$ is the kink decay rate and $D_{\min}$ the minimal separation. The derivation is exact on the chosen ansatz; the internal mode is approximate, and its quantisation yields an effective quantum harmonic oscillator. The result provides the rigorous classical phase space for the quantum mechanics of multi‑kink states with excited internal modes in the asymptotic scattering regime.