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.
