International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 52020, 36 pages
doi:10.1155/2007/52020
Research Article

Existence and Orbital Stability of Cnoidal Waves for a 1D Boussinesq Equation

Jaime Angulo1 and Jose R. Quintero2

1Departamento de Matemática, Instituto de Matemática, Estatística e Computaçäo Científica, (IMECC), UNICAMP, CP 6065, Campinas CEP 13083-970, Säo Paulo, Brazil
2Departamento de Matemáticas, Universidad del Valle, Cali A. A. 25360, Colombia

Received 11 May 2006; Revised 1 October 2006; Accepted 7 February 2007

Academic Editor: Vladimir Mityushev

Copyright © 2007 Jaime Angulo and Jose R. Quintero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We will study the existence and stability of periodic travelling-wave solutions of the nonlinear one-dimensional Boussinesq-type equation Φtt−Φxx+aΦxxxx−bΦxxtt+ΦtΦxx+2ΦxΦxt=0. Periodic travelling-wave solutions with an arbitrary fundamental period T0 will be built by using Jacobian elliptic functions. Stability (orbital) of these solutions by periodic disturbances with period T0 will be a consequence of the general stability criteria given by M. Grillakis, J. Shatah, and W. Strauss. A complete study of the periodic eigenvalue problem associated to the Lame equation is set up.