(i) The so-called breather solution of the sine-Gordon equation is

$$\phi(x, t)=4 \tan ^{-1}\left(\frac{\left(1-\lambda^{2}\right)^{\frac{1}{2}}}{\lambda} \frac{\sin \lambda t}{\cosh \left(1-\lambda^{2}\right)^{\frac{1}{2}} x}\right), \quad 0<\lambda<1$$

Describe qualitatively the behaviour of $\phi(x, t)$, for $\lambda \ll 1$, when $|x| \gg \ln (2 / \lambda)$, when $|x| \ll 1$, and when $\cosh x \approx \frac{1}{\lambda}|\sin \lambda t|$. Explain how this solution can be interpreted in terms of motion of a kink and an antikink. Estimate the greatest separation of the kink and antikink.

(ii) The field $\psi(x, t)$ obeys the nonlinear wave equation

$$\frac{\partial^{2} \psi}{\partial t^{2}}-\frac{\partial^{2} \psi}{\partial x^{2}}+\frac{d U}{d \psi}=0$$

where the potential $U$ has the form

$$U(\psi)=\frac{1}{2}\left(\psi-\psi^{3}\right)^{2} .$$

Show that $\psi=0$ and $\psi=1$ are stable constant solutions.

Find a steady wave solution $\psi=f(x-v t)$ satisfying the boundary conditions $\psi \rightarrow 0$ as $x \rightarrow-\infty, \psi \rightarrow 1$ as $x \rightarrow \infty$. What constraint is there on the velocity $v ?$