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On the Amplitude Equations for Weakly Nonlinear Surface Waves

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Abstract

Nonlocal generalizations of Burgers’ equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185–202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3–4):1463–1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3–4):303–320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220–2240, 2011). In this article, we show how the verification of Hunter’s stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.

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Authors and Affiliations

  1. Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43 boulevard du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Sylvie Benzoni-Gavage

  2. Laboratoire de mathématiques Jean Leray (UMR CNRS 6629), CNRS and Université de Nantes, 2 rue de la Houssinière, BP 92208, 44322, Nantes Cedex 3, France

    Jean-François Coulombel

Authors
  1. Sylvie Benzoni-Gavage
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  2. Jean-François Coulombel
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Correspondence to Sylvie Benzoni-Gavage.

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Communicated by C. Dafermos

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Benzoni-Gavage, S., Coulombel, JF. On the Amplitude Equations for Weakly Nonlinear Surface Waves. Arch Rational Mech Anal 205, 871–925 (2012). https://doi.org/10.1007/s00205-012-0522-7

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  • DOI: https://doi.org/10.1007/s00205-012-0522-7

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