Validity of the hyperbolic Whitham modulation equations in Sobolev spaces

TL;DR

This paper proves that the hyperbolic Whitham modulation equations accurately approximate the modulation of periodic wave trains in the defocusing nonlinear Schrödinger equation within Sobolev spaces, including higher-order corrections.

Contribution

It establishes the validity of hyperbolic Whitham equations in Sobolev spaces with rigorous error estimates and incorporates higher-order corrections into the modulation theory.

Findings

Validation of Whitham modulation equations in Sobolev spaces

Error estimates based on energy and existence arguments

Inclusion of higher-order corrections enhances approximation accuracy

Abstract

It is proved that modulation in time and space of periodic wave trains, of the defocussing nonlinear Schr\"odinger equation, can be approximated by solutions of the Whitham modulation equations, in the hyperbolic case, on a natural time scale. The error estimates are based on existence, uniqueness, and energy arguments, in Sobolev spaces on the real line. An essential part of the proof is the inclusion of higher-order corrections to Whitham theory, and concomitant higher-order energy estimates.