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