Modulational instability in a full-dispersion shallow water model

TL;DR

This paper introduces a new bidirectional shallow water model combining water wave dispersion with Boussinesq equations, analyzing its modulational instability and effects of surface tension, supported by spectral analysis and numerical validation.

Contribution

It extends the Whitham equation to bidirectional propagation and provides a spectral stability analysis of small periodic waves in this new model.

Findings

Small periodic waves are spectrally unstable to long wavelength perturbations beyond a critical wave number.

The linear operator has infinitely many imaginary eigenvalue collisions, but they do not cause instability at leading order.

Surface tension influences the modulational instability behavior.

Abstract

We propose a shallow water model which combines the dispersion relation of water waves and the Boussinesq equations, and which extends the Whitham equation to permit bidirectional propagation. We establish that its sufficiently small, periodic wave train is spectrally unstable to long wavelength perturbations, provided that the wave number is greater than a critical value, like the Benjamin-Feir instability of a Stokes wave. We verify that the asso- ciated linear operator possesses infinitely many collisions of purely imaginary eigenvalues, but they do not contribute to instability away from the origin in the spectral plane to the leading order in the amplitude parameter. We discuss the effects of surface tension on the modulational instability. The results agree with those from formal asymptotic expansions and numerical computations for the physical problem.