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<c_{ 0}(p) $ is orbitally unstable, while for $c>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 $c=c_{ 0}(p)$.
Categories: math.AP