Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 154 / 1

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

Differentiate the proposed energy and integrate the kinetic term by parts:Putting $F=u_{xx}+u^p$, the equation says $u_t=-F_x$, soHence KdV energy conservation gives

The one-dimensional Gagliardo-Nirenberg interpolation inequality yieldsMass conservation fixes $\Vert u{\Vert }_2$. If $X=\Vert u_x{\Vert }_2$, energy conservation therefore givesFor $p<5$, the second exponent is strictly smaller than two. The right side tends to infinity with $X$, so this inequality bounds $X$ uniformly throughout the lifespan. The conserved $L^2$ norm then bounds $\Vert u(t){\Vert }_{H^1}$. The stated blowup criterion rules out a finite endpoint, proving Global existence for the energy-subcritical generalized KdV equation.

Set $z=x-ct$ and $u(t,x)=Q_c(z)$. Substitution into the equation givesDecay at infinity makes the integration constant zero. Ifthen every term in $Q_c^{\prime \prime }-c{Q}_c+Q_c^p$ equals $c^{1+1/(p-1)}$ times the corresponding term in $Q^{\prime \prime }-Q+Q^p$. Thus the required Generalized KdV solitary wave is

For every $p<5$, the wave $Q_c$ has Orbital stability of a generalized KdV solitary wave in $H^1(\mathbb{R})$ modulo translation: for every $\epsilon >0$ there is $\delta >0$ such thatThe relevant stability slope has the correct sign because scaling giveswhich is strictly increasing in $c$ precisely for $p<5$. Together with the constrained variational characterization of $Q_c$, the conserved mass and energy provide a coercive Lyapunov function transverse to the translation direction.