## Advances in Differential Equations

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

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

#### 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.