On the asymptotic stability of N-soliton solutions of the defocusing nonlinear Schrödinger equation

We consider the Cauchy problem for the defocusing nonlinear Schrödinger (NLS) equation for finite density type initial data. Using the dbar generalization of the nonlinear steepest descent method of Deift and Zhou, we derive the leading order approximation to the solution of NLS for large times in the solitonic region of space–time, |x| < 2t, and we provide bounds for the error which decay as t → ∞for a general class of initial data whose difference from the non vanishing background possesses a fixed number of finite moments and derivatives. Using properties of the scattering map of NLS we derive, as a corollary, an asymptotic stability result for initial data that are sufficiently close to the N-dark soliton solutions of NLS.