= Semicircular pulse under cubic diffusion
{title2=$L\propto t^{1/4},\quad H\propto t^{-1/4}$}
= Cubic diffusion pulse
{synonym}
The source-type <similarity solution> of $h_t=D(h^2h_x)_x$ with whole-line conserved area $\mathcal V$ is $h=H(t)[1-x^2/L(t)^2]_+^{1/2}$, where $L=2(\mathcal V/\pi)^{1/2}(Dt)^{1/4}$ and $H=(\mathcal V/\pi)^{1/2}(Dt)^{-1/4}$. Its <volume flux> vanishes at the finite front, while the velocity tends to $\dot L=L/(4t)$. A reflecting half-line uses twice its prescribed area in these formulas.
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