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Variational approximation for wave propagation in continuum and discrete media

 

L. A. Cisneros-Ake

 

Departmento de Matemáticas, Escuela Superior de Física y Matemáticas, Instituto Politécnico Nacional, Unidad Profesional Adolfo López Mateos Edificio 9, México 07738 D.F., México. e-mail: cisneros@esfm.ipn.mx.

 

Received 27 August 2012;
Aaccepted 16 April 2013.

 

Abstract

We develop a variational approximation for wave propagation in continuum and discrete media based on the modulation of wave profiles described by appropriate trial functions. We illustrate the method by considering an application to the theory of dislocation of materials. We first consider the continuum approximation of the model and reproduce the exact traveling known solution. We then consider the fully discrete non integrable model and obtain an approximate solution based on trial functions with functional form similar to the exact solution of the continuum. The description of this discrete approximate solution is in terms of a discrete nonlinear dispersion relation between the wave parameters. In this last situation we compare the numerical and variational solutions at the stationary case. We thus illustrate the usage of a variational asymptotic approximation to study nonlinear problems and we contrast the differences and difficulties between continuum and discrete problems.

Keywords: Modulation theory; average Lagrangian; trial function.

PACS: 62.20.mm; 63.20.Pw.

 

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Acknowledgments

The author thanks the financial support from COFAA-IPN, IPN-CGPI-20130803 and Conacyt project 177246. Thanks are also expressed to the anonymous referees for their useful comments, which substantially improved this work, and to professor Tim Minzoni for helpful discussions.

 

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