Bistable cubic reaction-diffusion equation ($u_t=D{u}_{xx}+u(1-u)(u-r)$)

The scalar equation $u_t=D{u}_{xx}+u(1-u)(u-r)$ with $D>0$ and $0<r<1$ has two stable homogeneous phases $u=0,1$ and an unstable intervening equilibrium $u=r$. Indeed the reaction derivatives at these three equilibria are $-r$, $-(1-r)$ and $r(1-r)$, respectively. A travelling wave can connect the two stable phases. At the equal-depth value $r=1/2$, its decreasing heteroclinic orbit is stationary. Away from this value the phases have unequal potential depths and the front propagates.

For the decreasing front connecting $1$ to $0$, the balanced stationary travelling-wave reduction of a reaction-diffusion system has first integral $D{U}_x^2/2-U^2(1-U{)}^2/4=0$. 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 $U_z=-U(1-U)/\sqrt{2}$, 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.