= Generalized Korteweg–De Vries equation
{c}
{title2=$u_t+(u_{xx}+u^p)_x=0$}
The <Generalized Korteweg–De Vries equation> is the <nonlinear partial differential equation>
$$
u_t+\partial_x(u_{xx}+u^p)=0,\qquad x\in\mathbb R.
$$
For the displayed real power, one commonly takes an integer $p\ge2$; 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 $u(t,x)\mapsto\lambda^{2/(p-1)}u(\lambda^3t,\lambda x)$. 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>.
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