The Cauchy problem for the following Boussinesq system, cases uₜ + vₓ + uuₓ = 0\\ vₜ - uₓₓₓ + uₓ + (uv)ₓ = 0 cases is considered. It is showed that this problem is locally well-posed in Hˢ(R) × Hˢ⁻¹(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.
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Jaime Angulo Pava (1999) studied this question.