I thought these problems were pretty routine, and then I came across one that I am not able to do.

Find the admissible solution to the equation $\displaystyle u_t + u^2u_x=0$ for $\displaystyle t> 0$ subject to the following initial conditions

$\displaystyle u=\begin{cases} 1 & x<0\\ 0 & 0<x<1\\ -1 & x>1\end{cases}$

My attempt: We see that $\displaystyle f(u) = \frac{1}{3}u^3$ and $\displaystyle \frac{dt}{dy} = 1\quad \frac{dx}{dy} = u^2$ so

$\displaystyle \frac{dt}{dx} = \begin{cases}1 & x<0\\ \infty & 0<x<1\\ -1 & x>1\end{cases}$.

Thus, initially we have two shock waves that should then combine to create a single shock wave. Using the Rankine-Hugoniot condition we have

$\displaystyle s = \frac{[f(u)]}{[u]} = \begin{cases} \frac{\frac{1}{3}\cdot 1^3 - \frac{1}{3}\cdot 0^3}{1 - 0} \\

\frac{\frac{1}{3}\cdot (-1)^3 - \frac{1}{3}\cdot 0^3}{-1 - 0} \end{cases} = \frac{1}{3}$.

This doesn't make sense to me, because I expected the second shock to be in the opposite direction. I must be doing something wrong.

EDIT: $\displaystyle f'(u^{\ell}) < f'(u^r)$ so there is actually a rarefaction wave at $\displaystyle x=1$. I am just having a hard time picturing this in my head now.

EDIT2: LOL, now I feel stupid!

$\displaystyle \frac{dt}{dx} = \begin{cases}1 & x<0\\ \infty & 0<x<1\\ 1 & x>1\end{cases}$.