Non-existence for travelling waves with small energy for the Gross–Pitaevskii equation in dimension N (cid:2) 3

We prove that the Ginzburg–Landau energy of non-constant travelling waves of the Gross–Pitaevskii equation has a lower positive bound, depending only on the dimension, in any dimension larger or equal to three. In particular, we conclude that there are no non-constant travelling waves with small energy. To cite this article: A. de Laire, C. R. Acad. Sci. Paris, Ser. I 347 (2009). © 2009 Académie


Introduction
The Gross-Pitaevskii equation i∂ whose Hamiltonian is the Ginzburg-Landau energy given by appears as a relevant model in several areas of physics: superfluidity, superconductivity, non-linear optics and the Bose-Einstein condensation (see e.g.[4][5][6]8]).In this work, we investigate the energy of travelling waves to this equation, i.e. solutions of the form Here, the parameter c ∈ R corresponds to the speed of the travelling waves.Using complex conjugation, we may restrict to the case c 0. The equation for the profile v is given by

Main result
A result of Tarquini [9] states that there exists a minimal value E(N, c) for the Ginzburg-Landau energy of travelling waves, depending only on N and c.This lower bound for the energy functional implies that non-constant finite energy solutions of (2) of sufficiently small energy, with respect to their speed, are excluded in dimension N 2. Furthermore E(N, c) → 0 as c → √ 2. This result has been recently improved by Béthuel, Gravejat and Saut [1] in dimension three, proving that there exists some universal positive bound for the energy functional for non-constant travelling waves.
Our aim is to extend the result of Béthuel, Gravejat and Saut [1] in any dimension larger than three, and therefore also to improve the non-existence theorem of Tarquini [9].More precisely, our main result is Theorem 2.1.Let N 3.There exists some positive constant E(N ), depending only on N, such that any non-constant finite energy solution v of (2) satisfies E(v) E(N ).In particular, there are no non-constant solutions of (2) with small energy.

Proof of main result
In dimension N 3, it follows from [3] that the speed of non-constant finite energy solutions of (2) satisfy 0 < c √ 2. From Lemma 3 in [9], we deduce that 1 − |v| 2 ) , where K(N) is a positive constant, depending only on N .Therefore, choosing a possibly smaller constant E(N ), we may assume that v satisfies We recall that v is a smooth function (see e.g.[2]), and then in view of (3), v may be expressed as v = ρe iϕ , where ρ and ϕ are scalar functions, and ϕ is defined modulo a multiple of 2π .Defining also the quantity η = 1 − ρ 2 , we have Applying the Fourier transform to (4), we obtain where R 0 = |∇v| 2 + η 2 , R j = η∂ j ϕ, j ∈ {1, . . ., N}, and Now we recall two results of Béthuel, Gravejat and Saut.The first one corresponds to Lemma 2.9 in [1], and the second one is an immediate extension to R N of some part of the argument used in Lemma 2.15 (see inequality (2.65) in [1]).Lemma 3.1.Let v be a non-constant finite energy solution to (2) satisfying (3).Then, Lemma 3.2.For any 1 < q < ∞, there exists a positive constant K(N, q), depending only on N and q, such that We denote L c the operator given by L c (f ) = L c f , ∀f ∈ S(R N ).We recall that in the case that there exists a constant K such that L c (f ) L q (R N ) K f L p (R N ) , L c is called a Fourier multiplier from L p to L q .We notice that identity (5) implies that η is the value of the multiplier operator associated to L c , evaluated in the function F given by (6), that is In order to complete the proof of Theorem 2.1, we need the following lemma, whose proof we postpone to the next section: given by ( 7) is a Fourier multiplier from L p to L q , with 1 p = 1 q + α.More precisely, there exists a positive constant K(N, α, q), depending only on N , α and q, such that In view of (8), applying Lemma 3.3, with α = 2 2N −1 and q = 2, we deduce that there exists a positive constant K(N), depending only on N , such that Combining Lemmas 3.1, 3.2 and (10), we conclude that 4 , which finishes the proof of Theorem 2.1.

Proof of Lemma 3.3
Here we use the standard multi-index notation, i.e where P m,c is a two-variable polynomial of degree m + 1.More precisely, for x, y ∈ R, where {γ m,i,j } m i,j =1 and γ m are polynomial functions of the variable c.Furthermore, in the case , we have In particular, λ m is a well defined and bounded function on (0, √ Proof.The differentiability of L c is immediate.The case m = 1 is checked explicitly, since we have We fix now m, with 1 < m N .Let us suppose that (12) and ( 13) are valid for some 1 n < m.We take any r = (r 1 , . . ., r N ) ∈ {0, 1} N such that |r| = n + 1 and define j * = max{1 j N | r j = 1}.Then j * > 1, and we consider r = (r 1 , . . ., rN ) ∈ {0, 1} N given by ri = r j (1 − δ i,j * ), i, j ∈ {1, . . ., N}.Therefore, |r| = n and we have, where Using this inductive argument, we conclude the first part of the lemma, that is, identities (12) and (13).In order to deduce, in the case k 1 = 1, that the coefficients of lower terms are explicitly given by ( 14) and (15), we use the same inductive argument but we replace the polynomial expression (18) by the following one P n+1,c (x, y) = γn (c)x n+2 + n+1 i,j =0 2 i+j n γn,i,j (c)x i y j − 4nα n (c)x − 2c 2 α n (c) + 4(n + 1)β n (c) y, for some { γn,i,j } n i,j =1 , γn , polynomial functions of the variable c.The formulas ( 14) and (15) allow us to finish the induction.Finally we notice that identity (16) is an immediate consequence of ( 14) and (15). 2 An important property that follows from identities (14)-( 16) is that for small values of ξ , we may compute an explicit bound for P m,c , that is