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

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 [13]). 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 [15], [14], 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.