Traveling front of the cubic-flux Burgers equation ($q=\beta [1+e^{-2(s-s_0)/(ϵc^2)}{]}^{-1/2},c=3/{\beta }^2$)

For a front written in $s=Z+c\theta$ connecting $0$ at $s=-\infty$ to nonzero $\beta$ at $s=+\infty$, one integration gives $ϵc^2{q}^{\prime }=q-c{q}^3/3$. 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 $-q^3/3$.