Stability of solitary-wave solutions to the Hirota-Satsumaequation

Jerry L. Bona; Didier Pilod

doi:10.3934/dcds.2010.27.1391

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2010, Volume 27, Issue 4: 1391-1413. Doi: 10.3934/dcds.2010.27.1391

This issue Previous Article Boundary layers in smooth curvilinear domains: Parabolic problems Next Article Singular perturbation systems with stochastic forcing and the renormalization group method

Stability of solitary-wave solutions to the Hirota-Satsuma equation

Jerry L. Bona^1, and
Didier Pilod^2,

1.
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607
2.
UFRJ, Institute of Mathematics, Federal University of Rio de Janeiro, P.O. Box 68530, CEP: 21945-970, Rio de Janeiro, RJ

Received: November 2009

Revised: February 2010

Published: December 2010

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Abstract

The evolution equation

$ u_t- $ u_xxt$ +u_x-$uu_t$ +u_x\int_x^{+\infty}u_tdx'=0, $ (1)

was developed by Hirota and Satsuma as an approximate model for unidirectional propagation of long-crested water waves. It possesses solitary-wave solutions just as do the related Korteweg-de Vries and Benjamin-Bona-Mahony equations. Using the recently developed theory for the initial-value problem for (1) and an analysis of an associated Liapunov functional, nonlinear stability of these solitary waves is established.

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Mathematics Subject Classification: Primary: 35Q53, 35B35, 76B25; Secondary: 35A15, 76B15.

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