Semiclassical limit for nonlinear Schrödinger equation with potential. II

Article type: Research Article

Authors: Keraani, Sahbi

Affiliations: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, France E-mail: [email protected]

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.

Keywords: Schrödinger equation, ground state, stability, semiclassical limit, Wigner measure, WKB method

Journal: Asymptotic Analysis, vol. 47, no. 3-4, pp. 171-186, 2006