Rigorous justification of the Whitham modulation equations for equations of Whitham-type

TL;DR

This paper rigorously confirms that the Whitham modulation equations accurately predict the modulational stability of periodic waves in the generalized Whitham equation by linking hyperbolicity to spectral stability.

Contribution

It provides a rigorous justification connecting the hyperbolicity of Whitham equations with the spectral stability of periodic solutions in a Hamiltonian framework.

Findings

Hyperbolicity of Whitham equations is necessary for modulational stability.

Strict hyperbolicity ensures spectral stability in the Hamiltonian case.

The spectral stability criterion aligns with the modulational instability prediction.

Abstract

We prove that the modulational instability criterion of the formal Whitham modulation theory agrees with the spectral stability of long wavelength perturbations of periodic travelling wave solutions to the generalized Whitham equation. We use the standard WKB procedure to derive a quasi-linear system of three Whitham modulation equations, written in terms of the mass, momentum, and wave number of a periodic travelling wave solution. We use the same quantities as parameters in a rigorous spectral perturbation of the linearized operator, which allows us to track the bifurcation of the zero eigenvalue as the Floquet parameter varies. We show that the hyperbolicity of the Whitham system is a necessary condition for the existence of purely imaginary eigenvalues in the linearized system, and hence also a prerequisite for modulational stability of the underlying wave. Since the generalized Whitham equation has a Hamiltonian structure, we conclude that strict hyperbolicity is a sufficient condition for modulational stability.