 Unit-mass source solution for a quartic flux

The nonnegative scalar conservation law ${\rho }_t+({\rho }^4/4{)}_x=0$ has a unit-mass source-type entropy solution with $f(\eta )={\eta }^{1/3}$ on $0<\eta <(4/3{)}^{3/4}$ and zero elsewhere. Mass conservation gives the scaling exponents $\alpha +\beta =0$, while flux balance gives $\alpha -1=4\alpha -\beta$, so $\alpha =-1/4$, $\beta =1/4$. The reduced ODE integrates to $f^4-\eta f=0$ on the positive branch adjoining vacuum. The integral ${\int }_0^{{\eta }_F}{\eta }^{1/3}d\eta =1$ fixes the front. Its speed ${\eta }_F{t}^{-3/4}/4$ equals the Rankine-Hugoniot speed ${\rho }_L^3/4$, between the two limiting characteristic speeds, so the front is compressive. The initial trace is a unit Dirac delta distribution, not a smooth finite-width density.