Rarefaction Wave in nonlinear PDE $u_t + (u^{3/2})_x = 0$

Question

I am dealing with the following PDE: $$u_t + \big(u^{\frac{3}{2}}\big)_{x} = 0$$ subject to: $$u(x,0) = \begin{cases} 1 & x\leq 0 \\ 4 & 0\leq x\leq 10 \\ 1 & x > 10 \end{cases}$$

The solution at $u(x,1)$ is known to be: $$u(x,1) = \begin{cases} 1 & x < \frac{3}{2}\\ \left(\frac{2x}{3}\right)^2 & \frac{3}{2} < x <3\\ 4 & 3<x<10 + \frac{7}{3}\\ 1 & x > 10 + \frac{7}{3} \end{cases}$$ I have worked through everything, and understand the solution aside from the rarefaction part. I know that one must use $\frac{x}{t}$ as a characteristic equation, but I cannot understand how they jumped from that to $(\frac{2x}{3})^2$ in the final solution.

Thank you.

Answer 1

The general procedure is described here. In the present case, we have the conservation law $u_t + f(u)_x = 0$ with the flux $f: u\mapsto u^{3/2}$. The corresponding characteristic speed is $f'(u) = \tfrac32 u^{1/2}$. Considering a rarefaction wave starting at $x=0$, we have $$ u(x,t) = (f')^{-1}(x/t) = \big(\tfrac23 x/t\big)^2 $$ for $\tfrac32 t < x< 3 t$. Setting $t=1$ gives the result.