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Nonlinear Counterpropagating Waves, Multisymplectic Geometry, and the Instability of Standing Waves

Bridges, T. J. and Laine-Pearson, Fiona E. (2004) Nonlinear Counterpropagating Waves, Multisymplectic Geometry, and the Instability of Standing Waves the <i>SIAM Journal on Applied Mathematics</i>, 64. pp. 2096-2120.

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Abstract

Standing waves are a fundamental class of solutions of nonlinear wave equations with a spatial reflection symmetry, and they routinely arise in optical and oceanographic applications. At the linear level they are composed of two synchronized counterpropagating periodic traveling waves. At the nonlinear level, they can be defined abstractly by their symmetry properties. In this paper, general aspects of the modulational instability of standing waves are considered. This problem has difficulties that do not arise in the modulational instability of traveling waves. Here we propose a new geometric formulation for the linear stability problem, based on embedding the standing wave in a four-parameter family of nonlinear counterpropagating waves. Multisymplectic geometry is shown to encode the stability properties in an essential way. At the weakly nonlinear level we obtain the surprising result that standing waves are modulationally unstable only if the component traveling waves are modulation unstable. Systems of nonlinear wave equations will be used for illustration, but general aspects will be presented, applicable to a wide range of Hamiltonian PDEs, including water waves.

Item Type: Article
Additional Information: Published in the SIAM Journal on Applied Mathematics, Volume 64, Number 6, pp. 2096-2120. © 2004 Society for Industrial and Applied Mathematics.
Uncontrolled Keywords: modulation instability, variational principles, periodic waves, hyperbolic PDEs, water waves
Divisions: Faculty of Engineering and Physical Sciences > Mathematics
Depositing User: Mr Adam Field
Date Deposited: 27 May 2010 14:40
Last Modified: 07 Nov 2013 14:11
URI: http://epubs.surrey.ac.uk/id/eprint/1395

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