Generalized KdV solitary wave

For $p>1$ and $c>0$, let $Q$ be a positive decaying solution of $Q^{\prime \prime }-Q+Q^p=0$. Thensatisfies $Q_c^{\prime \prime }-c{Q}_c+Q_c^p=0$, and $u(t,x)=Q_c(x-ct)$ is a traveling-wave solution of the Generalized Korteweg–De Vries equation.

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.