Existence and stability for the travelling waves of the Benjamin equation

Abstract

In the seminal work of Benjamin,\cite{Ben} in the late 70's, he has derived the ubiquitous Benjamin model, which is a reduced model in the theory of water waves. Notably, it contains two parameters in its dispersion part and under some special circumstances, it turns into the celebrated KdV or the Benjamin-Ono equation, During the90's, there was renewed interest in it. Benjamin, \cite{Ben1}, \cite{Ben2} studied the problem for existence of solitary waves, followed by works of Bona-Chen, \cite{BC}, Albert-Bona-Restrepo, \cite{ABR}, Pava, \cite{Pava1}, who have showed the existence of travelling waves, mostly by variational, but also bifurcation methods. Some results about the stability became available, but unfortunately, those were restricted to either small waves or Benjamin model, close to a distinguished (i.e. KdV or BO) limit. Quite recently, in \cite{ADM}, Abdallah, Darwich and Molinet, proved existence, orbital stability and uniqueness results for these waves, but only for large values of $\f{c}{\ga^2}>>1$. In this article, we present an alternative constrained maximization procedure for the construction of these waves, for the full range of the parameters, which allows us to ascertain their spectral stability. Moreover, we extend this construction to all $L^2$ subcritical cases (i.e. power nonlinearities $(|u|^{p-2}u)_x$, $2<p\leq 6$). Finally, we propose a different procedure, based on a specific form of the Sobolev embedding inequality, which works for all powers $2<p<\infty$, but produces some unstable waves, for large $p$. Some open questions and a conjecture regarding this last result are proposed for further investigation.