Whitham modulation equations and application to small dispersion asymptotics and long time asymptotics of nonlinear dispersive equations

TL;DR

This paper reviews Whitham modulation equations for KdV and applies them to analyze the long-time and small dispersion asymptotics of nonlinear dispersive equations, providing insights into wave behavior over extended periods.

Contribution

It introduces the application of Whitham modulation equations to describe asymptotic behaviors of KdV solutions in new regimes.

Findings

Effective description of long-time KdV behavior

Asymptotic characterization in small dispersion limit

Enhanced understanding of nonlinear wave evolution

Abstract

We review the theory of modulation equations or Whitham equations for the travelling wave solution of KdV. We then apply the Whitham modulation equations to describe the long-time asymptotics and small dispersion asymptotics of the KdV solution