Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 32C (Integrable Systems)

Write $\xi =V_1$, $\eta =\varphi$, and $u_j=d^{j}u/d{x}^j$. On the jet space of a scalar ordinary differential equation, definewhere the total derivative operator isThe order-$N$ prolongation of a vector field is

In the coordinates $(x,u)$, the given action of the special orthogonal group isDifferentiating at $s=0$ gives the infinitesimal generator of a Lie point symmetryso $\xi =u$ and $\eta =-x$. The recursive formula givesandTherefore the second prolongation of the rotation generator for plane graphs is

For ${\Delta }_0=u_{xx}$,which vanishes when ${\Delta }_0=0$. Forwe obtainwhich likewise vanishes on the equation. This is the infinitesimal invariance criterion for both equations.

Geometrically, $u_{xx}=0$ describes straight lines. The signed curvature of a plane graph isso the second equation says $\kappa =1$ and describes consistently oriented arcs of unit circles. Rotation preserves straight lines, circles, and signed curvature, explaining both invariances.

The transformationsform a one-parameter group with infinitesimal generatorUnder this scaling, ${\partial }_X$ has weight $-1$ and ${\partial }_T$ has weight $1$, so $u_{XT}$ has weight zero, as does $\sin u$. Thus $u_{XT}=\sin u$ is invariant and $g^s$ is a Scaling symmetry of the sine-Gordon equation.

The productis invariant under the group. A group-invariant solution therefore has the form $u(X,T)=F(z)$. Sincethe Sine-Gordon equation reduces to

Now put $w=e^{iF}$. ThenSubstitution and multiplication by $iw$ giveHenceThis is the Painlevé III equation with parameters $(\alpha ,\beta ,\gamma ,\delta )=(1/2,-1/2,0,0)$.