Decay of solitary waves

TL;DR

This paper investigates the decay rates of solitary-wave solutions in certain non-linear dispersive equations, establishing exponential decay, symmetry, and crest properties, with decay rates depending on wave speed.

Contribution

It provides the first precise exponential decay rates for supercritical solitary waves in non-local dispersive equations with analytic Fourier symbols.

Findings

Supercritical solitary waves decay exponentially.

Solitary waves have only one crest.

Solitary waves are symmetric for some classes.

Abstract

In this paper we consider the decay rate of solitary-wave solutions to some classes of non-linear and non-local dispersive equations, including for example the Whitham equation and a Whitham--Boussinesq system. The dispersive term is represented by a Fourier multiplier operator that has a real analytic symbol, and we show that all supercritical solitary-wave solutions decay exponentially, and moreover provide the exact decay rate, which in general will depend on the speed of the wave. We also prove that solitary waves have only one crest and are symmetric for some class of equations.