A dispersive system of long waves in weighted Sobolev spaces

Abstract

In this article we treat the Cauchy problem for the dispersive system of long waves, $$ \begin{cases} \partial_t u + \partial_{x}v + u\partial_{x}u = 0\\ \partial_t v - \partial_{x}^3u + \partial_{x}u + \partial_{x}(uv) = 0, \end{cases} $$ in weighted Sobolev spaces. It is shown that this problem is locally well-posed in $H^s(\Bbb R)\times H^{s-1}(\Bbb R) \cap L^2_{\gamma}(\Bbb R)\times H^{-1}_{\gamma}(\Bbb R)$, for $s>3/2$ and $0\leqq \gamma \leqq s$. The proof involves parabolic regularization and techniques of Bona-Smith. It is also determined, using the orbital stability of the special solitary-wave solutions of this system, that we can extend globally the local solution for data sufficiently close to the solitary wave in the norm $ H^1(\Bbb R)\times L^{2}(\Bbb R)$.