On the Cauchy problem for a Boussinesq-type system

Abstract

The Cauchy problem for the following Boussinesq system, $$ \begin{cases} u_t + v_x + uu_x = 0\\ v_t - u_{xxx} + u_{x} + (uv)_x = 0 \end{cases} $$ is considered. It is showed that this problem is locally well-posed in $H^s(\mathbb{R}) \times H^{s-1}(\mathbb{R})$ for any $s>3/2$. The proof involves parabolic regularization and techniques of Bona-Smith. It is also determined that the special solitary-wave solutions of this system are orbitally stable for the entire range of the wave speed. Combining these facts we can extend globally the local solution for data sufficiently close to the solitary wave.