Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory

TL;DR

This paper derives an asymptotic formula for the amplitude distribution of solitary wave trains in fully nonlinear shallow-water theory, based on Whitham modulation properties, and confirms its accuracy with numerical simulations.

Contribution

It introduces a non-integrable analogue of the semi-classical distribution for solitary waves, applicable to non-integrable, fully nonlinear shallow-water systems.

Findings

Analytical formula matches numerical simulations

Applicable to non-integrable, fully nonlinear systems

Extends understanding of solitary wave train distributions

Abstract

We derive an asymptotic formula for the amplitude distribution in a fully nonlinear shallow-water solitary wave train which is formed as the long-time outcome of the initial-value problem for the Su-Gardner (or one-dimensional Green-Naghdi) system. Our analysis is based on the properties of the characteristics of the associated Whitham modulation system which describes an intermediate "undular bore" stage of the evolution. The resulting formula represents a "non-integrable" analogue of the well-known semi-classical distribution for the Korteweg-de Vries equation, which is usually obtained through the inverse scattering transform. Our analytical results are shown to agree with the results of direct numerical simulations of the Su-Gardner system. Our analysis can be generalised to other weakly dispersive, fully nonlinear systems which are not necessarily completely integrable.