## Differential and Integral Equations

### Blow-up and instability of a regularized long-wave-KP equation

#### Abstract

A regularized long-wave--Kadomtsev-Petviashvili equation of the form $$(u_{t}-u_{xxt}+u_{x}+u^{p}u_{x})_{x}-u_{yy}=0, \tag*{(*)}$$ is considered. It is shown that if $p\geq4$, certain initial data can lead to a solution that blows up in finite time. More precisely, under the above condition the solution cannot remain in the Sobolev class $H^{2}(\mathbb R)$ for all time. Also demonstrated here is the solitary-wave solutions $u(x, y, t) = \phi_c (x-ct, y)$, which exist if and only if $1\leq p <4$ and $c > 1$, when considered as solutions of the initial-value problem for (*), are nonlinearly unstable to perturbations of the initial data, if ${4\over3} <p <4$ and $1 < c < {4p\over 4+p}$.

#### Article information

Source
Differential Integral Equations, Volume 16, Number 9 (2003), 1131-1152.

Dates
First available in Project Euclid: 21 December 2012