(a) Given a smooth vector field

$$V=V_{1}(x, u) \frac{\partial}{\partial x}+\phi(x, u) \frac{\partial}{\partial u}$$

on $\mathbb{R}^{2}$ define the prolongation of $V$ of arbitrary order $N$.

Calculate the prolongation of order two for the group $S O(2)$ of transformations of $\mathbb{R}^{2}$ given for $s \in \mathbb{R}$ by

$$g^{s}\left(\begin{array}{l} u \\ x \end{array}\right)=\left(\begin{array}{l} u \cos s-x \sin s \\ u \sin s+x \cos s \end{array}\right)$$

and hence, or otherwise, calculate the prolongation of order two of the vector field $V=-x \partial_{u}+u \partial_{x}$. Show that both of the equations $u_{x x}=0$ and $u_{x x}=\left(1+u_{x}^{2}\right)^{\frac{3}{2}}$ are invariant under this action of $S O(2)$, and interpret this geometrically.

(b) Show that the sine-Gordon equation

$$\frac{\partial^{2} u}{\partial X \partial T}=\sin u$$

admits the group $\left\{g^{s}\right\}_{s \in \mathbb{R}}$, where

$$g^{s}:\left(\begin{array}{c} X \\ T \\ u \end{array}\right) \mapsto\left(\begin{array}{c} e^{s} X \\ e^{-s} T \\ u \end{array}\right)$$

as a group of Lie point symmetries. Show that there is a group invariant solution of the form $u(X, T)=F(z)$ where $z$ is an invariant formed from the independent variables, and hence obtain a second order equation for $w=w(z)$ where $\exp [i F]=w$.