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

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.