Rarefaction wave

Remarks

The function is continuous, since $G(F'(u_l)) = u_l$ and $G(F'(u_r)) = u_r$. Inside the fan, $F'(u) = x/t$, so $u_t = -\tfrac{x}{t^2} G'(x/t)$ and $u_x = \tfrac1t G'(x/t)$. Hence $u_t + F'(u)\, u_x = 0$ there, and $u$ is a weak solution with the jump initial values $u_l$ for $x < 0$ and $u_r$ for $x > 0$. It depends only on $x/t$, so it is self-similar.

Example: for Burgers' equation, $F(u) = u^2/2$, so $F'(u) = u$ and $G$ is the identity. The rarefaction wave from $0$ to $1$ is $u = 0$ for $x < 0$, $u = x/t$ for $0 \le x \le t$, and $u = 1$ for $x > t$. Check: $u_t + u u_x = -x/t^2 + (x/t)(1/t) = 0$ inside the fan.

Non-example: if $u_l > u_r$, then $F'(u_l) t > F'(u_r) t$ and the middle interval is empty. The formula then gives no fan, and the admissible solution is a shock wave instead.