A proof of validity for multiphase Whitham modulation theory

Abstract

It is proved that approximations which are obtained as solutions of the multiphase Whitham modulation equations stay close to solutions of the original equation on a natural time scale. The class of nonlinear wave equations chosen for the starting point is coupled nonlinear Schrödinger equations. These equations are not in general integrable, but they have an explicit family of multiphase wavetrains that generate multiphase Whitham equations, which may be elliptic, hyperbolic, or of mixed type. Due to the change of type, the function space set-up is based on Gevrey spaces with initial data analytic in a strip in the complex plane. In these spaces a Cauchy–Kowalevskaya-like existence and uniqueness theorem is proved. Building on this theorem and higher-order approximations to Whitham theory, a rigorous comparison of solutions, of the coupled nonlinear Schrödinger equations and the multiphase Whitham modulation equations, is obtained.