= Frozen-recovery cubic front
{title2=$c=\sqrt{D/2}(r_1+r_3-2r_2)$}
For the scalar <reaction-diffusion equation> $v_t=Dv_{xx}-(v-r_1)(v-r_2)(v-r_3)$ with $D>0$ and $r_1<r_2<r_3$, an exact decreasing <travelling wave> is
$$
V(z)=r_1+\frac{r_3-r_1}{1+\exp[(r_3-r_1)(z-z_0)/\sqrt{2D}]},\qquad z=x-ct.
$$
Indeed $V'=-(V-r_1)(r_3-V)/\sqrt{2D}$, and substitution into $DV''+cV'-(V-r_1)(V-r_2)(V-r_3)=0$ leaves the factor $(r_1+r_3)/2-r_2-c/\sqrt{2D}$, proving the speed formula. A zero speed gives the <Maxwell balance for a scalar reaction-diffusion front>. In a <FitzHugh-Nagumo model> this calculation requires treating the <recovery variable> as frozen externally: retaining $w_t=\beta v-\gamma w$ with exactly constant $w$ forces $v$ to be constant.
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