Characteristic transformation for conical ice drainage

Putting $y=x^{5/3}$ and $w=x^{1/3}h$ transforms $h_t+x^{-1}(xD{h}^3{)}_x=0$ into the scalar conservation lawAlong smooth characteristic curves, $w$ is constant and $y$ grows at speed $5D{w}^2$. Starting from $h_0(s)$ givesAfter characteristic crossing, the Rankine-Hugoniot condition and entropy solution select the physical front.