Rigorous derivation of the nonlinear Schrödinger equation and Davey–Stewartson systems from quadratic hyperbolic systems

Article type: Research Article

Authors: Colin, Thierry

Affiliations: Mathématiques Appliquées de Bordeaux, Université Bordeaux 1 and CNRS UMR 5466, 351 cours de la libération, 33405 Talence cedex, France E‐mail: colin@math.u‐bordeaux.fr

Abstract: In this paper, we derive rigorously the nonlinear Schrödinger equation and Davey–Stewartson systems from quadratic hyperbolic systems using nonlinear diffractive geometric optics. We construct approximate solutions and prove the convergence of the asymptotic expansion. The keys of this work is to consider only systems of a particular form introduced in [20], which include Maxwell–Bloch system, and which satisfy a “transparency” hypothesis.

Keywords: geometric optics, nonlinear Schrödinger equation, Davey–Stewartson systems

Journal: Asymptotic Analysis, vol. 31, no. 1, pp. 69-91, 2002