(i) Let $u(x, t)$ satisfy the Burgers equation

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

where $\nu$ is a positive constant. Consider solutions of the form $u=u(X)$, where $X=x-U t$ and $U$ is a constant, such that

$$u \rightarrow u_{2}, \quad \frac{\partial u}{\partial X} \rightarrow 0 \quad \text { as } \quad X \rightarrow-\infty ; \quad u \rightarrow u_{1}, \quad \frac{\partial u}{\partial X} \rightarrow 0 \quad \text { as } \quad X \rightarrow \infty$$

with $u_{2}>u_{1}$.

Show that $U$ satisfies the so-called shock condition

$$U=\frac{1}{2}\left(u_{2}+u_{1}\right)$$

By using the factorisation

$$\frac{1}{2} u^{2}-U u+A=\frac{1}{2}\left(u-u_{1}\right)\left(u-u_{2}\right)$$

where $A$ is the constant of integration, express $u$ in terms of $X, u_{1}, u_{2}$ and $\nu$.

(ii) According to shallow-water theory, river waves are characterised by the PDEs

$$\begin{gathered} \frac{\partial v}{\partial t}+v \frac{\partial v}{\partial x}+g \cos \alpha \frac{\partial h}{\partial x}=g \sin \alpha-C_{f} \frac{v^{2}}{h} \\ \frac{\partial h}{\partial t}+v \frac{\partial h}{\partial x}+h \frac{\partial v}{\partial x}=0 \end{gathered}$$

where $h(x, t)$ denotes the depth of the river, $v(x, t)$ denotes the mean velocity, $\alpha$ is the constant angle of inclination, and $C_{f}$ is the constant friction coefficient.

Find the characteristic velocities and the characteristic form of the equations. Find the Riemann variables and show that if $C_{f}=0$ then the Riemann variables vary linearly with $t$ on the characteristics.