On the Modulation of Wave Trains in the Ostrovsky Equation
Mathew A. Johnson, Jeffrey Oregero, Wesley R. Perkins
TL;DR
This work studies modulational dynamics of general amplitude periodic traveling waves for the Ostrovsky equation, which models rotating-fluid wave propagation. It develops a formal Whitham modulation framework to obtain a two-dimensional dispersionless system for slow wave-number and momentum evolution, and then provides a rigorous spectral perturbation analysis to connect hyperbolicity of the Whitham system with spectral modulational stability near the origin. The main contribution is proving that strict hyperbolicity of the Whitham modulation matrix $\mathcal{W}(\phi)$ guarantees spectral stability to long-wavelength perturbations, while ellipticity implies spectral instability, with a precise reduction to a $2\times2$ matrix $\mathcal{M}_0(\phi)$ governing the near-origin spectrum. The analysis links the formal modulation predictions to rigorous Bloch-spectral data for general amplitude wave trains and includes discussion of the reduced Ostrovsky case ($\beta=0$), highlighting the broader applicability and significance of modulational analysis in rotating-fluid models.
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.