= Unit-mass source solution for a quartic flux
{title2=$\rho=t^{-1/4}f(x/t^{1/4})$}
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 $\eta_Ft^{-3/4}/4$ equals the <Rankine-Hugoniot condition>[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.
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