IJPAM: Volume 21, No. 4 (2005)

THE STABILITY OF SOLITARY WAVES FOR
A GENERALIZED BOUSSINESQ EQUATION

Shaoyong Lai$^1$, Yong Hong Wu$^2$, Jibiao He$^3$, Lin Qun$^4$
$^{1,2}$Department of Mathematics and Statistics
Curtin University of Technology
GPO BOX U1987, Perth, WA6845, AUSTRALIA
$^2$e-mail: yhwu@maths.curtin.edu.au
$^3$Department of Mathematics
Mian Yang Teacher's College
Sichuan, P.R. CHINA
$^4$Department of Mathematics
Sichuan Normal University, P.R. CHINA


Abstract.In this paper, we study an initial value problem for the following generalized Boussinesq equation \begin{equation*} u_{tt}-2bu_{txx}=-\alpha u_{xxxx}+u_{xx}-\gamma ^{2}u+\beta (f(u))_{xx}, \end{equation*} where $x\in R^{1}$, $t>0$, $b>0$, $\gamma \geq 0$, $\beta \in R^{1}$ and the function $f$ is a polynomial with $f(0)=0$. The conditions for the existence and uniqueness of a global solution to the problem in question are established in a Sobolev space. The results for sufficiently small $\beta $ confirm Bony and Saches' suggestion$^{[1]}$ that initial data lying relatively close to a stabe solitary wave could evolve into a global solution for some equations.

Received: May 16, 2005

AMS Subject Classification: 35K55, 35B40

Key Words and Phrases: Boussinesq equation, solitary wave, nonlinear equation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 21
Issue: 4