A remark on the stability of solitary waves for a 1-D Benney-Luke equation.

Abstract

The aim of this work is to analyze the stability of solitary waves of the one-dimensional Benney-Luke equation. We prove that the null solution is asymptotically stable due to the presence of a Hamiltonian structure and to the existence of invariant quantities with regard to time. In the case of a non-null solitary wave, we find the Hamiltonian structure but the verification that some quantities are conserved with respect to time has turned out to be a difficult numerical calculation. We show that the criteriom of stability and orbital instability of M. Grillakis, J. Shatah and W. Strauss is not applicable and we present numerical evidence for unstable solitary waves.