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

El, Gennady, Grimshaw, R.H.J. and Smyth, N.F. (2008) Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory. Physica D: Nonlinear Phenomena, 237 (19). pp. 2423-2435. ISSN 0167-2789

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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.

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