= Traveling front of the cubic-flux Burgers equation
{title2=$q=\beta[1+e^{-2(s-s_0)/(\epsilon c^2)}]^{-1/2},\quad c=3/\beta^2$}
For a front written in $s=Z+c\theta$ connecting $0$ at $s=-\infty$ to nonzero $\beta$ at $s=+\infty$, one integration gives $\epsilon c^2q'=q-cq^3/3$. The endpoint condition fixes $c=3/\beta^2$. Setting $w=q^2$ gives a logistic equation and the displayed profile, with the sign of $q$ selected by $\beta$. The propagation speed in the $\theta$ coordinate is $-\beta^2/3$, agreeing with the <Rankine-Hugoniot condition> for flux $-q^3/3$.
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