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

Abstract

We will study the existence and stability of periodic travelling-wave solutions of the nonlinear one-dimensional Boussinesq-type equationΦtt−Φxx+aΦxxxx−bΦxxtt+ΦtΦxx+2ΦxΦxt=0. Periodic travelling-wave solutions with an arbitrary fundamental periodT0will be built by using Jacobian elliptic functions. Stability (orbital) of these solutions by periodic disturbances with periodT0will 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.