I am following section 3.4 of Evan's book "Partial Differential Equations". In this section he considers the Burgers equation, $$u_t + \bigl(\frac{u^2}{2}\bigr)_x = 0$$ with initial data $$g(x) = \begin{cases} 0, & x < 0\\ 1, &0 \leq x \leq 1\\0, & x > 1\end{cases}$$ I have been able to follow how he obtains the solution for $0 \leq t \leq 2$, $$u(x,t) = \begin{cases} 0, & x < 0\\ \frac{x}{t}, & 0 <x < t\\ 1, & t < x < 1+\frac{t}{2}\\0, & x > 1 + \frac{t}{2}\end{cases}$$ However he notes for $t \geq 2$ this solution breaks down as the rarefraction wave and the curves stemming from the initial condition $g(x) = 0$ intersect. He uses the Rankine-Hugoniot jump condition to find that $$\dot{s}(t) = \sigma = \frac{x}{2t}$$
This is where he begins to lose me. He claims that this implies $$\dot{s}(t) = \frac{s(t)}{2}$$ with $s(2) = 2$. He claims by solving this ODE we find $$s(t) = \sqrt{2t}$$ and so the final solution for $t \geq 2$ is $$u(x,t) = \begin{cases} 0, & x < 0\\ \frac{x}{t}, & 0 < x < \sqrt{2t}\\0, & x > \sqrt{2t}\end{cases}$$
How does he obtain that ODE? Moreover, how does he conclude that the shock occurs at $t = 2$ (aside from graphically)?
