On the Modulation of Wave Trains in the Ostrovsky Equation

TL;DR

This paper investigates the modulational stability of wave trains in the Ostrovsky equation by deriving Whitham modulation equations and linking their hyperbolicity or ellipticity to spectral stability or instability of the waves.

Contribution

It provides a rigorous connection between the hyperbolicity of Whitham modulation equations and spectral stability of wave trains in the Ostrovsky equation, extending understanding beyond small-amplitude waves.

Findings

Hyperbolicity of Whitham system implies spectral stability.

Ellipticity of Whitham system implies spectral instability.

Established a direct link between modulation equations and spectral stability analysis.

Abstract

We consider the nonlinear wave modulation of arbitrary amplitude periodic traveling wave solutions of the Ostrovsky equation, which arises as a model for the unidirectional propagation of small-amplitude, weakly nonlinear surface and internal gravity waves in a rotating fluid of finite depth. While the modulation of such waves with asymptotically small amplitudes of oscillation (the so-called Stokes waves) has been studied in several works, our goal is to understand the modulational dynamics of general amplitude wave trains. To this end, we first use Whitham's theory of modulations to derive a dispersionless system of quasilinear partial differential equations that is expected to model the slow evolution of the fundamental characteristics of a given wave train. In practice, the modulational stability or instability of a given wave train is considered to be determined by the hyperbolicity or ellipticity, respectively, of the resulting system of Whitham modulation equations. Using rigorous spectral perturbation theory we then study the spectral (linearized) stability problem for a given wave train solution of the Ostrovsky equation, directly connecting the hyperbolicity or ellipticity of the associated Whitham system to the rigorous spectral stability problem for the underlying wave. Specifically, we prove that strict hyperbolicity of the Whitham system implies spectral stability near the origin in the spectral plane, i.e. so-called spectral modulational stability, while ellipticity implies spectral instability of the underlying wave train.