## Advances in Differential Equations

- Adv. Differential Equations
- Volume 3, Number 2 (1998), 227-248.

### 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)$.

#### Article information

**Source**

Adv. Differential Equations, Volume 3, Number 2 (1998), 227-248.

**Dates**

First available in Project Euclid: 19 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1366399897

**Mathematical Reviews number (MathSciNet)**

MR1750417

**Zentralblatt MATH identifier**

0947.35037

**Subjects**

Primary: 35G10: Initial value problems for linear higher-order equations

Secondary: 35Q35: PDEs in connection with fluid mechanics 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30] 76B25: Solitary waves [See also 35C11]

#### Citation

Angulo Pava, Jaime. A dispersive system of long waves in weighted Sobolev spaces. Adv. Differential Equations 3 (1998), no. 2, 227--248. https://projecteuclid.org/euclid.ade/1366399897