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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.

Nonlinear analysis studies equations and variational problems for which the superposition principle fails. If $u$ and $v$ solve an equation, their sum or a scalar multiple need not solve it. For the ordinary differential equation $u^{\prime }=u^2$ with $u(0)=1$, the solution $u(t)=1/(1-t)$ exists for $t<1$ and becomes unbounded at $t=1$. Multiplying this solution by two changes the two sides by different factors, so it does not produce another solution of the same equation.

The questions extend beyond finding a formula: does a solution exist, is it unique, how long does it remain regular, and does it stay near a stable shape? Nonlinear partial differential equations such as the Generalized Korteweg–De Vries equation and the mass-critical focusing nonlinear Schrödinger equation combine local evolution with conservation laws, compactness and functional inequalities.

A variational method can construct a ground state by minimizing an energy functional. A virial identity can instead show that a regular solution cannot persist indefinitely. Scaling and sharp constants identify the thresholds at which the same conserved quantity changes from controlling an evolution to permitting concentration.

A Pohozaev identity is an integral relation obtained by testing a nonlinear partial differential equation against a scaling direction such as $x\cdot \nabla Q$. For a sufficiently regular, rapidly decaying real solution ofwrite $A=\int |\nabla Q{|}^2$, $B=\int |Q{|}^2$ and $P=\int |Q{|}^{p+1}$. The identity isOne derivation uses the stationary energy functional $S(Q)=A/2+B/2-P/(p+1)$. Evaluate it on $Q(x/\lambda )$: the three terms scale as ${\lambda }^{d-2}$, ${\lambda }^d$ and ${\lambda }^d$. Differentiating at $\lambda =1$ gives the displayed formula, with decay justifying the scaling variation. Direct integration by parts with $x\cdot \nabla Q$ gives the same identity.

Testing against $Q$ also gives $A+B=P$. Combining the two relations at $p=1+4/d$ yields the Pohozaev identity for the mass-critical NLS ground state. The usefulness is that a differential equation imposes exact constraints on global kinetic, mass and nonlinear integrals.

The porous medium equation $u_t=\Delta (u^m)$ with $m>1$ is a nonlinear partial differential equation whose diffusivity vanishes at $u=0$. Unlike the heat equation, it has compactly supported solutions with finite propagation speed.