A uniqueness result for the two-vortex traveling 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 and Saut (2009) (see also work of Chiron and Mariş (2017)). This minimizer is a traveling wave for the nonlinear Schrödinger equation. For large momenta, 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 traveling wave. This work is a follow up to two previous papers, where we constructed and studied a particular traveling wave of the equation. We show a uniqueness result on this traveling wave in a class of functions that contains in particular all possible minimizers of the energy.