Instability and long-time evolution of cnoidal wave solutions for a Benney–Luke equation

We investigate numerically the stability of periodic traveling wave solutions (cnoidal waves) for a generalized Benney–Luke equation. By using a high-accurate Fourier spectral method, we find different kinds of evolution depending on the period of the perturbation. A cnoidal wave solution with period T is orbitally stable with regard to perturbations having the same period T, within certain range of wave velocities. This is a fact proved recently by Angulo and Quintero Existence and orbital stability of cnoidal waves for a 1D boussinesq equation, International Journal of Mathematics and Mathematical Sciences (2007), in press, doi:10.1155/2007/52020] and our numerical experiments are consistent with their theory. In the present work we show numerically that cnoidal waves with period T become unstable when perturbed by small amplitude disturbances whose period is an integer multiple of T. Particularly, if the period of the perturbation is 2T, the evolution of the deviation of the solution from the orbit of the cnoidal wave is found to be approximately a time-periodic function. In other cases, the numerical experiments indicate a non-periodic behavior.