A remark on the stability of solitary waves for a 1-D Benney-Luke equation
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.
Key words and phrases: Solitary wave, Stability theory,
Primary: 74J35, Secondary 74H55, 74H15.