A uniqueness result for the two vortex travelling wave in the Nonlinear Schrodinger equation

Abstract

For the Nonlinear Schrödinger equation in dimension 2, the existence of a global minimizer of the energy at fixed momentum has been established by Bethuel-Gravejat-Saut [7] (see also [14]). This minimizer is a travelling wave for the Nonlinear Schrödinger equation. For large momentums, the propagation speed is small and the minimizer behaves like two well separated vortices. In that limit, we show the uniqueness of this minimizer, up to the invariances of the problem, hence proving the orbital stability of this travelling wave. This work is a follow up to two previous papers [16], [15], where we constructed and studied a particular travelling wave of the equation. We show a uniqueness result on this travelling wave in a class of functions that contains in particular all possible minimizers of the energy.Besides the energy, the momentum is another quantity formally conserved by the (NLS) flow, and is associated with the invariance by translation of (NLS). Formally, the momentum of u is 1 2 Ê 2 Re(i∇uū) ∈ Ê 2 , but its precise definition requires some care in the energy space due to the condition at infinity (see [37] in dimension larger than two and [14] ) for instance, then the expression of the momentum reduces toRe i∇u(ū − 1) .In addition to the translation invariance, the (NLS) equation is also phase shift invariant, that is invariant by multiplication by a complex of modulus one, and rotation invariant. Travelling waves for (NLS)Following the works in the physical literature of Jones and Roberts (see [32], [31]), there has been a large amount of mathematical works on the question of existence and properties of travelling waves solutions in the (NLS) equation, that is solution offor some c > 0, corresponding to particular solutions of (NLS) of the form Ψ(t, x) = u(x 1 , x 2 + ct) (due to the rotational invariance, we may always assume that the traveling wave moves along the direction − − → e 2 ). We refer to [6] for an overview on these problems in several dimensions. A natural approach is to look at the minimizing problem for p > 0It was shown by Bethuel-Gravejat-Saut in [7] that there exists a minimizer to this problem.Theorem 1.2 ([7]) For any p > 0, there exists a non constant function u p ∈ E and c(u p ) > 0 such that P 2 (u p ) = p, u p is a solution to (TW c(up) )(u p ) = 0 and E(u p ) = E min (p).Furthermore, any minimizer for E min (p) is, up to a translation in x 1 , even in x 1 .The strategy is to look at the corresponding minimization problem on tori (this avoids the problems with the definition of the momentum) larger and larger, and then pass to the limit. For the minimizing problem E min (p), the compactness of minimizing sequences has been shown later on in [14] for the natural semi-distance onTheorem 1.3 ([14]) For any p > 0, and any minimizing sequence (u n ) n∈AE for E min (p), there exists a subsequence (u nj ) j∈AE , a sequence of translations (y j ) j∈AE and a non constant function u p ∈ E such that D 0 (u nj , u p ) → 0, P 2 (u nj ) → P 2 (u p ) = p and E(u nj ) → E(u p ) = E min (p) as ...