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For the Nonlinear Schrodinger equation in dimension 2, the existence of aglobal minimizer of the energy at fixed momentum has been established byBethuel-Gravejat-Saut. This minimizer is a travelling wave for the NonlinearSchrodinger equation. For large momentums, the propagation speed is small andthe minimizer behaves like two well separated vortices. In that limit, we showthe uniqueness of this minimizer, up to the invariances of the problem, henceproving the orbital stability of this travelling wave. This work is a follow upto two previous papers, where we constructed and studied a particulartravelling wave of the equation. We show a uniqueness result on this travellingwave in a class of functions that contains in particular all possibleminimizers of the energy.

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