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

We study the orbital stability of solitary waves to the generalized Benjamin–Bona–Mahony equation $$u_t+u_x+\kappa \frac{p+1}{2}(u^p{)}_x-u_{txx}=0,(t,x)\in {\mathbb{R}}^{+}\times \mathbb{R},$$ where $\kappa \in \mathbb{R}\setminus \{0\}$ and $p\ge 2$ is an integer. This equation admits solitary waves of the form $$u(t,x)={\phi }_c(x-ct)\mathrm{.}$$ At the negative critical endpoint speed $$c=c_p^{-}=\frac{p-1}{2(p+1)}(1-\sqrt{\frac{p+3}{2}})<0,$$ 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.