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arXiv · 2309.00791

Instability of the solitary waves for the Generalized Benjamin-Bona-Mahony Equation

Rui Jia·Yifei Wu
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Abstract

In this work, we consider the generalized Benjamin-Bona-Mahony equation $$\partial_t u+\partial_x u+\partial_x( |u|^pu)-\partial_t \partial_x^{2}u=0, \quad(t,x) \in \mathbb{R} \times \mathbb{R}, $$ with $p>4$. This equation has the traveling wave solutions $\phi_{c}(x-ct), $ for any frequency $c>1.$ It has been proved by Souganidis and Strauss \cite{Strauss-1990} that, there exists a number $c_{0}(p)>1$, such that solitary waves $\phi_{c}(x-ct)$ with $1 c_{0}(p), $ $\phi_{c}(x-ct)$ is orbitally stable. The linear exponential instability in the former case was further proved by Pego and Weinstein \cite{Pego-1991-eigenvalue}. In this paper, we prove the orbital instability in the critical case $c=c_{0}(p)$.

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Rui Jia, Yifei Wu. 2023-09-02. Instability of the solitary waves for the Generalized Benjamin-Bona-Mahony Equation. https://arxiv.org/abs/2309.00791

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