(i) Show that the equation

$$\frac{\partial \phi}{\partial t}-\frac{\partial^{2} \phi}{\partial x^{2}}+1-\phi^{2}=0$$

has two solutions which are independent of both $x$ and $t$. Show that one of these is linearly stable. Show that the other solution is linearly unstable, and find the range of wavenumbers that exhibit the instability.

Sketch the nonlinear evolution of the unstable solution after it receives a small, smooth, localized perturbation in the direction towards the stable solution.

(ii) Show that the equations

$$\begin{array}{r} \frac{\partial u}{\partial x}+\frac{\partial v}{\partial x}=e^{-u+v} \\ -\frac{\partial u}{\partial y}+\frac{\partial v}{\partial y}=e^{-u-v} \end{array}$$

are a Bäcklund pair for the equations

$$\frac{\partial^{2} u}{\partial x \partial y}=e^{-2 u}, \quad \frac{\partial^{2} v}{\partial x \partial y}=0$$

By choosing $v$ to be a suitable constant, and using the Bäcklund pair, find a solution of the equation

$$\frac{\partial^{2} u}{\partial x \partial y}=e^{-2 u}$$

which is non-singular in the region $y<4 x$ of the $(x, y)$ plane and has the value $u=0$ at $x=\frac{1}{2}, y=0$.