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Solitary waves of a coupled Korteweg-de Vries system

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  • Grimshaw, Roger
  • Iooss, Gerard

Abstract

In the long-wave, weakly nonlinear limit a generic model for the interaction of two waves with nearly coincident linear phase speeds is a pair of coupled Korteweg-de Vries equations. Here we consider the simplest case when the coupling occurs only through linear non-dispersive terms, and for this case delineate the various families of solitary waves that can be expected. Generically, we demonstrate that there will be three families: (a) pure solitary waves which decay to zero at infinity exponentially fast; (b) generalized solitary waves which may tend to small-amplitude oscillations at infinity; and (c) envelope solitary waves which at infinity consist of decaying oscillations. We use a combination of asymptotic methods and the rigorous results obtained from a normal form approach to determine these solitary wave families.

Suggested Citation

  • Grimshaw, Roger & Iooss, Gerard, 2003. "Solitary waves of a coupled Korteweg-de Vries system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 62(1), pages 31-40.
    • Handle: RePEc:eee:matcom:v:62:y:2003:i:1:p:31-40
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    Cited by:

    1. Zayed, E.M.E. & Abourabia, A.M. & Gepreel, Khaled A. & El Horbaty, M.M., 2007. "Travelling solitary wave solutions for the nonlinear coupled Korteweg–de Vries system," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 292-306.
    2. Bona, J.L. & Dougalis, V.A. & Mitsotakis, D.E., 2007. "Numerical solution of KdV–KdV systems of Boussinesq equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 74(2), pages 214-228.

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