Justification of and long-wave correction to Davey-Stewartson systems from quadratic hyperbolic systems

T Colin,D Lannes

doi:10.3934/dcds.2004.11.83

Abstract

We prove that the Davey-Stewartson approximation (which degenerates into a cubic Schrödinger equation in $1D$) furnishes a good approximation for the exact solution of a wide class of quadratic hyperbolic systems. This approximation remains valid for large times of logarithmic order. We also consider the general case where the polarized component of the mean field needs not to be well-prepared. This is possible by adding to the Davey-Stewarston approximation a long-wave correction, which consists of a wave freely propagated by the long-wave operator associated to the original system.

Highlights

Some of the examples of the above references are nonlinear hyperbolic systems with quadratic nonlinearity; for the other examples, such quadratic hyperbolic systems appear as a crucial step in the derivation of the Davey-Stewartson system
Schrodinger equation) is studied for a wide class of nonlinear hyperbolic systems with quadratic nonlinearity. The systems of this class are of the form
A long-wave correction must be added to the Davey-Stewartson approximation

Summary

Introduction

Davey-Stewartson system are universal models in physics and mechanics, see for example [4], [5], [3] for water waves theory, [7] for internal gravity waves, [13, 14]. For ferromagnetism, [17] for nonlinear optics and [16] for plasma theory. Some of the examples of the above references are nonlinear hyperbolic systems with quadratic nonlinearity (in ferromagnetism for instance); for the other examples, such quadratic hyperbolic systems appear as a crucial step in the derivation of the Davey-. Schrodinger equation) is studied for a wide class of nonlinear hyperbolic systems with quadratic nonlinearity. The systems of this class are of the form

Eu ε

The equation for

Then one computes t