5,519 research outputs found
Instability of the solitary waves for the generalized Boussinesq equations
In this work, we consider the following generalized Boussinesq equation
\begin{align*}
\partial_{t}^2u-\partial_{x}^2u+\partial_{x}^2(\partial_{x}^2u+|u|^{p}u)=0,\qquad
(t,x)\in\mathbb R\times \mathbb R, \end{align*} with . This
equation has the traveling wave solutions , with the
frequency and satisfying \begin{align*}
-\partial_{xx}{\phi}_{\omega}+(1-{\omega^2}){\phi}_{\omega}-{\phi}_{\omega}^{p+1}=0.
\end{align*} Bona and Sachs (1988) proved that the traveling wave
is orbitally stable when . Liu (1993) proved the orbital instability under the conditions
or . In this paper, we prove
the orbital instability in the degenerate case .Comment: 29 page
- β¦