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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.

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 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 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.

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.

Write the Generalized Korteweg–De Vries equation asFor a sufficiently regular solution, integration by parts givesThus KdV mass conservation givesis conserved.