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Modulational stability and instability of Korteweg-de Vries, Boussinesq, and Stokes nonlinear wavetrains

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Abstract

The modulational stability of both the Korteweg-de Vries (KdV) and the Boussinesq wavetrains is investigated using Whitham’s variational method. It is shown that both KdV and Boussinesq wavetrains are modulationally stable. This result seems to confirm why it is possible to transform the KdV equation into a nonlinear Schrödinger equation with a repulsive potential. Included is modulational instability of Stokes water waves based on the nonlinear Schrödinger equation in deep water. A brief discussion of Whitham’s averaged variational method is included to make the paper self-contained to some extent with application to instability of deep water waves.

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Acknowledgments

Author would like to thank Dr. Henrik Kalisch for suggesting to include Bridges and Mielke’s work in this paper.

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  1. Department of Mathematics, University of Texas-Pan American, Edinburg, TX, 78539, USA

    Lokenath Debnath

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  1. Lokenath Debnath
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Correspondence to Lokenath Debnath.

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This article is dedicated to the memory of Dr. Vladimir Varlamov.

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Debnath, L. Modulational stability and instability of Korteweg-de Vries, Boussinesq, and Stokes nonlinear wavetrains. Anal.Math.Phys. 2, 389–406 (2012). https://doi.org/10.1007/s13324-012-0042-5

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  • DOI: https://doi.org/10.1007/s13324-012-0042-5

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