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Weakly nonlinear wavepackets in the Korteweg–de Vries equation: the KdV/NLS connection

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  • Boyd, John P.
  • Chen, Guan-Yu

Abstract

If the initial condition for the Korteweg–de Vries (KdV) equation is a weakly nonlinear wavepacket, then its evolution is described by the nonlinear Schrödinger (NLS) equation. This KdV/NLS connection has been known for many years, but its various aspects and implications have been discussed only in asides. In this note, we attempt a more focused and comprehensive discussion including such as issues as the KdV-induced long wave pole in the nonlinear coefficient of the NLS equation, the derivation of NLS from KdV through perturbation theory, resonant effects that give the NLS equation a wide range of applicability, and numerical illustrations. The multiple scales/nonlinear perturbation theory is explicitly extended to two orders beyond that which yields the NLS equation; the wave envelope evolves under a generalized-NLS equation which is third order in space and quintically-nonlinear.

Suggested Citation

  • Boyd, John P. & Chen, Guan-Yu, 2001. "Weakly nonlinear wavepackets in the Korteweg–de Vries equation: the KdV/NLS connection," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 55(4), pages 317-328.
  • Handle: RePEc:eee:matcom:v:55:y:2001:i:4:p:317-328
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    Cited by:

    1. Groesen, E.van & Westhuis, J.H., 2002. "Modelling and simulation of surface water waves," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 59(4), pages 341-360.

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