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

G. A. El, R. H. J. Grimshaw, N. F. Smyth

Research output: Contribution to journalArticlepeer-review


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. (C) 2008 Elsevier B.V. All rights reserved.

Original languageEnglish
Pages (from-to)2423-2435
Number of pages13
JournalPhysica D: Nonlinear Phenomena
Issue number19
Publication statusPublished - 1 Oct 2008


  • shallow-water waves
  • soliton train
  • Whitham theory
  • undular bore


Dive into the research topics of 'Asymptotic description of solitary wave trains in fully nonlinear shallow-water theory'. Together they form a unique fingerprint.

Cite this