Decay and symmetry of solitary waves

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 that either decays/grows, and we show that all supercritical/subcritical 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.