Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 3 / 32B (Integrable Systems)

For $k,n>0$, substitute $x^{\prime }=\alpha x$, $t^{\prime }=\beta t$, $u^{\prime }=\gamma u$ and compare the two distinct right-hand terms. Invariance requires $\beta ={\alpha }^{2k+1}$ and ${\gamma }^n={\alpha }^{-2k}$. Taking $\alpha =e^s$ gives the Lie point symmetryTranslations in $x$ and $t$ are also Lie point symmetries, generated by $X={\partial }_x$ and $T={\partial }_t$. Direct differentiation givesso their span is a three-dimensional Lie algebra.

The printed nonnegative-integer range also includes cases where the generic scaling would divide by zero. If $k>0,n=0$, the equation is linear, $u_t=u_x+{\partial }_x^{2k+1}u$. Amplitude scaling $(x,t,u)\mapsto (x,t,e^{s}u)$ is a symmetry, with $D=u{\partial }_u$. Along with $X,T$ all brackets are zero, giving an abelian three-dimensional Lie algebra. A nontrivial pure rescaling of space and time cannot preserve both derivative orders in this case.

If $k=0$, the equation becomes $u_t=(u^n+1)u_x$, including $u_t=2u_x$ when $n=0$. In either case choose $(x,t,u)\mapsto (e^{s}x,e^{s}t,u)$, with $D=x{\partial }_x+t{\partial }_t$. Then $[D,X]=-X$, $[D,T]=-T$, and $[X,T]=0$. This covers every allowed pair $(k,n)$; when both vanish, there is also an independent amplitude scaling.

Assume periodic boundary conditions or sufficient decay so that integrations by parts have no boundary terms. Use the Hamiltonian operator $\mathcal{J}={\partial }_x$ and the functionalFor a variation $u+\epsilon v$, integrate the second term by parts $k$ times. The two factors $(-1{)}^k$ cancel, givingThus $\boxed{u_t=\mathcal{J}\delta H/\delta u}$ is exactly the equation, including $k=0$ and $n=0$. The bracket $\{F,G\}=\int (\delta F/\delta u){\partial }_x(\delta G/\delta u)dx$ is antisymmetric by integration by parts. Since this differential operator has coefficients independent of $u$, its associated constant Poisson bracket satisfies the Jacobi identity. This supplies a genuine Hamiltonian field equation, not just a conserved scalar functional.