 Orbital stability of a generalized KdV solitary wave

For the integer powers $p=2,3,4$ and $c>0$, the positive decaying Generalized KdV solitary wave is orbitally stable in $H^1(\mathbb{R})$ modulo spatial translations. The variational slope has the stable sign becauseis strictly increasing in $c$. Conserved mass and energy then furnish a Lyapunov functional transverse to the translation orbit.