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

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

2007-10-17

arxiv.org/abs/0710.3379v2

PHYSICA D 237 (2008) 2423 - 2435

Nonlinear Sciences

Pattern Formation and Solitons

25 pages, 7 figures

Scientific paper

10.1016/j.physd.2008.03.031

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.

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