BOLETÍN ELECTRÓNICO CIENTÍFICO
DEL NODO BRASILERO
DE INVESTIGADORES COLOMBIANOS
Número 2(Artículo 20), 2000

TÍTULO
ORBITAL STABILITY OF SOLITARY WAVE SOLUTIONS FOR AN INTERACTION EQUATION OF SHORT AND LONG DISPERSIVE WAVES

TIPO:

AUTOR: Jaime Angulo Pava1angulo@ime.unicamp.br & José Fabio B. Montenegro2

IDIOMA: Inglés

DIRECCIÓN PARA CONTACTO

1IMECC, Universidade Estadual de Campinas,
Campinas, CEP 13083-970, São Paulo, Brazil
2Departamento de Matemática, Campus do Pici, Universidade Federal do Ceará,
Fortaleza, CEP 60455-000, Ceará, Brazil

ENTIDADES QUE FINANCIARON LA INVESTIGACIÓN: The second author is supported by CNPq, a Brazilian government agency that supports the development of science and technology.

PALABRAS CLAVE:

ABSTRACT Download gziped postscript of the complete work (110KB) or See html version
We study the existence and orbital stability of solitary wave solutions for an interaction equation between a long internal wave and a short surface wave in a two layer fluid. If the short wave term is denoted by $u=u(x,t):\Bbb R\times \Bbb R\to \Bbb C$ and the long wave term by $v=v(x,t):\Bbb R\times \Bbb R\to \Bbb R$, the phenomena of interaction is described by the following equation ( Funakoshi and Oikawa (J. Phys. Soc. Japan, $\bold {52}$ (1983), 1982-1995),

\begin{displaymath} \cases iu_t+u_{xx}=\alpha vu,\\ v_t+\gamma \Cal Hv_{xx}=\beta (\vert u\vert^2)_x, \endcases \end{displaymath}

where $\alpha, \beta >0$, $\gamma \in \Bbb R$ and $\Cal H$ is the Hilbert transform. Via the Implicit Function Theorem we show the existence de smooth real solutions $\phi$ and $\psi$ of the system

\begin{displaymath} \cases \phi''-\sigma\phi=\alpha \,\psi\phi\\ \gamma\Cal H\psi'-c\psi=\beta \phi^2, \endcases \end{displaymath}

where $\sigma, c>0$ and $\gamma$ in some neighbourhood of zero. Moreover, using perturbation theory of closed operators on Hilbert spaces we show that the functions $\Phi(\xi)=e^{ic\xi/2}\phi(\xi)$ and $\Psi(\xi)=\psi(\xi)$ are solutions orbitally stable in $H^1(\Bbb R)\times H^{\frac12}(\Bbb R)$, at least when $\gamma$ is negative near zero .
BECNBIC,2(20)2000