Generalized Korteweg–De Vries equation ($u_t+(u_{xx}+u^p{)}_x=0$)

The generalized Korteweg–De Vries equation is the nonlinear partial differential equationThe cases below the exponent $p=5$ are energy-subcritical in $H^1(\mathbb{R})$.

Sufficiently regular decaying solutions conserveBoth identities follow by integration by parts from the divergence form of the equation.

For $p<5$, the one-dimensional Gagliardo-Nirenberg interpolation inequality givesThe exponent of $\Vert u_x{\Vert }_2$ is below two, so the conserved mass and energy control the $H^1$ norm. The blowup alternative then extends every local $H^1$ solution globally.

If $Q^{\prime \prime }-Q+Q^p=0$, 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 $p<5$, a 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.