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

The Generalized Korteweg–De Vries equation is the nonlinear partial differential equationFor the displayed real power, one commonly takes an integer $p\ge 2$; noninteger models need a specified real nonlinearity such as $|u{|}^{p-1}u$. The third spatial derivative produces dispersion, while the nonlinear term changes propagation with amplitude. The familiar Korteweg-De Vries equation corresponds, after coefficient rescaling, to $p=2$.

Its scaling is $u(t,x)\mapsto {\lambda }^{2/(p-1)}u({\lambda }^{3}t,\lambda x)$. Therefore the squared L2 norm scales by ${\lambda }^{4/(p-1)-1}$. The quintic case $p=5$ is mass-critical; the integer cases $p=2,3,4$ are mass-subcritical. This is the threshold behind global existence for the mass-subcritical generalized KdV equation.

For sufficiently smooth real solutions of the Generalized Korteweg–De Vries equation with enough decay to remove boundary terms, the mass and energyare constant in time. To check mass conservation, multiply the equation by $2u$ and use integration by parts. The linear term gives $\int u{u}_{xxx}=0$, and the nonlinear term is a total derivative proportional to $(u^{p+1}{)}_x$.

For energy conservation, put $F=u_{xx}+u^p$. Differentiating the energy functional givesbecause $u_t=-F_x$. Extending these identities to rougher solution classes requires the corresponding approximation and local well-posedness theory; the displayed computation is the smooth-decay argument.

For the integer powers $p=2,3,4$, the Generalized Korteweg–De Vries equation is mass-subcritical. Its Gagliardo-Nirenberg interpolation inequality isThe exponent $\alpha =(p-1)/2$ is strictly below two. Since KdV mass and energy conservation fixes $\Vert u{\Vert }_2$ and $E(u)$, the gradient size $X=\Vert u_x{\Vert }_2$ obeys $X^2/2\le |E(u_0)|+C_{M(u_0)}{X}^{\alpha }$. Young's inequality for products absorbs the lower power into $X^2/4$ and leaves a uniform bound for $X$.

Under the standard local $H^1(\mathbb{R})$ solution theory with a continuation time controlled by the H1 norm, this bound extends the solution globally. At $p=5$, the nonlinear estimate has gradient exponent two, so this absorption argument requires additional information such as a small-mass threshold. The scaling identifies this as an L2 norm threshold.

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.