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Amplitude equations for weakly nonlinear surface waves in variational problems

Sylvie Benzoni-Gavage 1, 2 Jean-François Coulombel 2
1 MMCS - Modélisation mathématique, calcul scientifique
ICJ - Institut Camille Jordan [Villeurbanne]
2 Equations aux dérivées partielles
LMJL - Laboratoire de Mathématiques Jean Leray
Abstract : Among hyperbolic Initial Boundary Value Problems (IBVP), those coming from a variational principle 'generically' admit linear surface waves, as was shown by Serre [J. Funct. Anal. 2006]. At the weakly nonlinear level, the behavior of surface waves is expected to be governed by an amplitude equation that can be derived by means of a formal asymptotic expansion. Amplitude equations for weakly nonlinear surface waves were introduced by Lardner [Int. J. Engng Sci. 1983], Parker and co-workers [J. Elasticity 1985] in the framework of elasticity, and by Hunter [Contemp. Math. 1989] for abstract hyperbolic problems. They consist of nonlocal evolution equations involving a complicated, bilinear Fourier multiplier in the direction of propagation along the boundary. It was shown by the authors in an earlier work [Arch. Ration. Mech. Anal. 2012] that this multiplier, or kernel, inherits some algebraic properties from the original IBVP. These properties are crucial for the (local) well-posedness of the amplitude equation, as shown together with Tzvetkov [Adv. Math., 2011]. Properties of amplitude equations are revisited here in a somehow simpler way, for surface waves in a variational setting. Applications include various physical models, from elasticity of course to the director-field system for liquid crystals introduced by Saxton [Contemp. Math. 1989] and studied by Austria and Hunter [Commun. Inf. Syst. 2013]. Similar properties are eventually shown for the amplitude equation associated with surface waves at reversible phase boundaries in compressible fluids, thus completing a work initiated by Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009].
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  • HAL Id : hal-01208479, version 1
  • ARXIV : 1510.01119

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Sylvie Benzoni-Gavage, Jean-François Coulombel. Amplitude equations for weakly nonlinear surface waves in variational problems. Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics., Sep 2015, Rome, Italy. ⟨hal-01208479⟩

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