Generalized Solitary Waves in the Gravity-Capillary Whitham Equation

TL;DR

This paper proves the existence of small amplitude solitary waves in a gravity-capillary Whitham equation, showing they decay to small periodic waves and are modulationally stable for certain Bond numbers.

Contribution

It establishes the existence and stability of solitary waves in a unidirectional shallow water model with full dispersion, extending previous analytical and numerical results.

Findings

Existence of small amplitude solitary waves for Bond numbers 0 to 1/3.

Solitary waves decay to small periodic waves at infinity.

Numerical evidence supports modulational stability of these waves.

Abstract

We study the existence of traveling wave solutions to a unidirectional shallow water model which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between $0$ and ${1}/{3}$) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave. We also present numerical evidence, based on the recent analytical work of Hur \& Johnson, that the asymptotic end states are modulationally stable for all Bond numbers between $0$ and $1/3$.