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

TL;DR

This paper proves the uniqueness and orbital stability of a specific two-vortex travelling wave solution in the 2D Nonlinear Schrödinger equation, especially for large momentum where the solution resembles two separated vortices.

Contribution

It establishes the uniqueness of the energy minimizer traveling wave in the 2D NLS, extending previous constructions and analyzing the large momentum limit.

Findings

Uniqueness of the minimizer up to invariances

Orbital stability of the travelling wave

Behavior of the solution as two well-separated vortices at large momentum

Abstract

For the Nonlinear Schrodinger equation in dimension 2, the existence of a global minimizer of the energy at fixed momentum has been established by Bethuel-Gravejat-Saut. This minimizer is a travelling wave for the Nonlinear Schrodinger 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, 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.