  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.
