Semiclassical limit for nonlinear Schrödinger equation with potential. II

Abstract

This paper is dedicated to the semiclassical limit of the nonlinear focusing Schrödinger equation (NLS) with a potential with initial data in the form $Q(\frac{x-x_{0}}{\varepsilon })\,\mathrm{e}^{\mathrm{i}\frac{x\cdot\xi_{0}}{\varepsilon }}$ , where Q is the ground state of the associated unscaled elliptic problem. Using a refined version of the method introduced by Bronski and Jerrard [Math. Res. Lett. 7(2–3) (2000), 329–342], we prove that, up to a time-dependent phase shift, the initial shape is conserved with parameters that are transported by the classical flow.