1. ## Admissible Solutions and Rankine-Huoniot Shock Condition

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 $u_t + u^2u_x=0$ for $t> 0$ subject to the following initial conditions
$u=\begin{cases} 1 & x<0\\ 0 & 01\end{cases}$

My attempt: We see that $f(u) = \frac{1}{3}u^3$ and $\frac{dt}{dy} = 1\quad \frac{dx}{dy} = u^2$ so
$\frac{dt}{dx} = \begin{cases}1 & x<0\\ \infty & 01\end{cases}$.
Thus, initially we have two shock waves that should then combine to create a single shock wave. Using the Rankine-Hugoniot condition we have
$s = \frac{[f(u)]}{[u]} = \begin{cases} \frac{\frac{1}{3}\cdot 1^3 - \frac{1}{3}\cdot 0^3}{1 - 0} \\
\frac{\frac{1}{3}\cdot (-1)^3 - \frac{1}{3}\cdot 0^3}{-1 - 0} \end{cases} = \frac{1}{3}$
.
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: $f'(u^{\ell}) < f'(u^r)$ so there is actually a rarefaction wave at $x=1$. I am just having a hard time picturing this in my head now.
EDIT2: LOL, now I feel stupid!
$\frac{dt}{dx} = \begin{cases}1 & x<0\\ \infty & 01\end{cases}$.