Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg-de Vries Equation

TL;DR

This paper rigorously justifies the Whitham modulation equations for the generalized Korteweg-de Vries equation, linking spectral stability of periodic waves to the well-posedness of the modulation system.

Contribution

It extends Whitham's modulation theory to a broader class of equations with nonzero mean periodic waves, providing a rigorous spectral stability analysis.

Findings

Homogenized system accurately describes linearized dispersion near zero frequency.

Spectral stability near the origin is equivalent to local well-posedness of the Whitham system.

Generalization of modulation expansion for equations with nonzero mean.

Abstract

In this paper, we consider the spectral stability of spatially periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long-wavelength perturbations. Specifically, we extend the work of Bronski and Johnson by demonstrating that the homogenized system describing the mean behavior of a slow modulation (WKB) approximation of the solution correctly describes the linearized dispersion relation near zero frequency of the linearized equations about the background periodic wave. The latter has been shown by rigorous Evans function techniques to control the spectral stability near the origin, i.e. stability to slow modulations of the underlying solution. In particular, through our derivation of the WKB approximation we generalize the modulation expansion of Whitham for the KdV to a more general class of equations which admit periodic waves with nonzero mean. As a consequence, we will show that, assuming a particular non-degeneracy condition, spectral stability near the origin is equivalent with the local well-posedness of the Whitham system.