(i) Establish two conservation laws for the $\mathrm{MKdV}$ equation

$$\frac{\partial u}{\partial t}+u^{2} \frac{\partial u}{\partial x}+\frac{\partial^{3} u}{\partial x^{3}}=0$$

State sufficient boundary conditions that $u$ should satisfy for the conservation laws to be valid.

(ii) The equation

$$\frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x}(\rho V)=0$$

models traffic flow on a single-lane road, where $\rho(x, t)$ represents the density of cars, and $V$ is a given function of $\rho$. By considering the rate of change of the integral

$$\int_{a}^{b} \rho d x,$$

show that $V$ represents the velocity of the cars.

Suppose now that $V=1-\rho$ (in suitable units), and that $0 \leqslant \rho \leqslant 1$ everywhere. Assume that a queue is building up at a traffic light at $x=1$, so that, when the light turns green at $t=0$,

$$\rho(x, 0)=\left\{\begin{array}{l} 0 \text { for } x<0 \text { and } x>1 \\ x \text { for } 0 \leqslant x<1 . \end{array}\right.$$

For this problem, find and sketch the characteristics in the $(x, t)$ plane, for $t>0$, paying particular attention to those emerging from the point $(1,0)$. Show that a shock forms at $t=\frac{1}{2}$. Find the density of cars $\rho(x, t)$ for $0<t<\frac{1}{2}$, and all $x$.