So what I'm confused about is how you go about finding shock waves. So suppose we are given the Cauchy problem for Burgers' equation $u\_t + uu\_x = 0$ with $u(x, 0) = 1$ for $x \le 0$ and $u(x, 0) = 0$ for $x> 0$. Then using method of characteristics we get $x = x\_0 + t$ for $x \le 0$ and $x = x\_0$ for $x > 0$. How do we then proceed, given that the characteristic lines will intersect so there is no unique $x\_0$ such that $(x, t)$ are on only one characteristic line?