 Generalized Korteweg–De Vries equation

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.

The Generalized Korteweg–de Vries (gKdV) equation is a significant extension of the classical Korteweg–de Vries (KdV) equation, which describes the evolution of shallow water waves in a canal.