(i) Find a travelling wave solution of unchanging shape for the modified Burgers equation (with $\alpha>0$ )

$$\frac{\partial u}{\partial t}+u^{2} \frac{\partial u}{\partial x}=\alpha \frac{\partial^{2} u}{\partial x^{2}}$$

with $u=0$ far ahead of the wave and $u=1$ far behind. What is the velocity of the wave? Sketch the shape of the wave.

(ii) Explain why the method of characteristics, when applied to an equation of the type

$$\frac{\partial u}{\partial t}+c(u) \frac{\partial u}{\partial x}=0$$

with initial data $u(x, 0)=f(x)$, sometimes gives a multi-valued solution. State the shockfitting algorithm that gives a single-valued solution, and explain how it is justified.

Consider the equation above, with $c(u)=u^{2}$. Suppose that

$$u(x, 0)=\left\{\begin{array}{ll} 0 & x \geq 0 \\ 1 & x<0 \end{array} .\right.$$

Sketch the characteristics in the $(x, t)$ plane. Show that a shock forms immediately, and calculate the velocity at which it moves.