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Instability of the solitary waves for the generalized Boussinesq equations

Li, Bing; Ohta, Masahito; Wu, Yifei; Xue, Jun
Peer reviewed, Journal article
Accepted version
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URI
https://hdl.handle.net/11250/2734264
Date
2020
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  • Institutt for matematiske fag [1792]
  • Publikasjoner fra CRIStin - NTNU [26728]
Original version
SIAM Journal on Mathematical Analysis. 2020, 52 (4), 3192-3221.   https://doi.org/10.1137/18M1199198
Abstract
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\R\times \R, \end{align*} with $0<p<\infty$. This equation has the traveling wave solutions $\phi_\omega(x-\omega t)$, with the frequency $\omega\in (-1,1)$ and $\phi_\omega$ 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 $\phi_\omega(x-\omega t)$ is orbitally stable when $0<p<4,$ $\frac p4<\omega^2<1$. Liu (1993) proved the orbital instability under the conditions $0<p<4,$ $\omega^2<\frac p4$ or $p\ge 4,$ $\omega^2<1$. In this paper, we prove the orbital instability in the degenerate case $0<p<4,\omega^2=\frac p4$.
Publisher
Society for Industrial and Applied Mathematics
Journal
SIAM Journal on Mathematical Analysis

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