= Logistic front of a bistable cubic equation
{title2=$U=[1+e^{(x-ct-q_0)/\sqrt{2D}}]^{-1}$}
For the decreasing front connecting $1$ to $0$, the balanced stationary <travelling-wave reduction of a reaction-diffusion system> has first integral $DU_x^2/2-U^2(1-U)^2/4=0$. Choosing the negative derivative gives $U_x=-U(1-U)/\sqrt{2D}$ and hence $U=[1+e^{(x-q)/\sqrt{2D}}]^{-1}$. This logistic profile remains an exact <travelling wave> when the reaction parameter differs from $1/2$: using $z=(x-ct-q_0)/\sqrt D$, its derivative is $U_z=-U(1-U)/\sqrt2$, and the wave equation is satisfied for $c=\sqrt{D/2}(1-2r)$. Thus the speed changes with the potential bias while the shape, apart from <translation>, does not.
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