Advances in Differential Equations

On the instability of solitary waves solutions of the generalized Benjamin equation

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

Abstract

This work is concerned with instability properties of solutions $u(x,t)=\phi(x-ct)$ of the equation $u_t+(u^p)_x + l H u_{xx}+u_{xxx}=0$ in $\mathbb R$, where $p\in \mathbb N$, $p\geqq 2$, and $H$ is the Hilbert transform. Here, $\phi$ will be a solution of the pseudo-differential equation $\phi''+l H \phi'-c\phi=-\phi^{p}$ solving a certain variational problem. We prove that the set $$ \Omega_{\phi}=\{\phi(\cdot+y) : y\in \mathbb R\;\} $$ is unstable by the flow of the evolution equation above provided $l$ is small, $c>\frac14 l^2$ and $p>5$. Moreover, the trajectories used to exhibit instability are global and uniformly bounded. Finally, we extend these results for a natural generalization of the evolution equation above with general forms of competing dispersion, in particular, we obtain instability results for some Korteweg-de Vries type equations without requiring spectral conditions.

Article information

Source
Adv. Differential Equations Volume 8, Number 1 (2003), 55-82.

Dates
First available in Project Euclid: 19 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355926868

Mathematical Reviews number (MathSciNet)
MR1946558

Zentralblatt MATH identifier
1038.35088

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B35: Stability 35Q51: Soliton-like equations [See also 37K40] 35R25: Improperly posed problems 76B25: Solitary waves [See also 35C11] 76E30: Nonlinear effects

Citation

Angulo Pava, Jaime. On the instability of solitary waves solutions of the generalized Benjamin equation. Adv. Differential Equations 8 (2003), no. 1, 55--82. https://projecteuclid.org/euclid.ade/1355926868.


Export citation