Advances in Differential Equations

A dispersive system of long waves in weighted Sobolev spaces

Jaime Angulo Pava

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 19 April 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


Angulo Pava, Jaime. A dispersive system of long waves in weighted Sobolev spaces. Adv. Differential Equations 3 (1998), no. 2, 227--248.

Export citation