Orbital instability of solitary waves for the generalized BBM equation at the negative critical endpoint

Abstract

We study the orbital stability of solitary waves to the generalized Benjamin--Bona--Mahony equation $$ u_t+u_x+\kappa\frac{p+1}{2}(u^p)_x-u_{txx}=0, \qquad (t,x)\in\mathbb R^+\times\mathbb R, $$ where $\kappa\in\mathbb R\setminus\{0\}$ and $p\geq2$ is an integer. This equation admits solitary waves of the form $$ u(t,x)=\phi_c(x-ct). $$ At the negative critical endpoint speed $$ c=c_p^-= \frac{p-1}{2(p+1)} \left(1-\sqrt{\frac{p+3}{2}}\right)<0, $$ we prove that the corresponding solitary wave is orbitally unstable whenever either $\kappa<0$, or $\kappa>0$ and $p$ is even. The proof combines a codimension-two coercivity estimate, two-parameter modulation, and a corrected localized virial functional adapted to the degeneracy of the momentum slope at the endpoint.