Frozen-recovery cubic front ($c=\sqrt{D/2}(r_1+r_3-2r_2)$)

For the scalar reaction-diffusion equation $v_t=D{v}_{xx}-(v-r_1)(v-r_2)(v-r_3)$ with $D>0$ and $r_1<r_2<r_3$, an exact decreasing travelling wave isIndeed $V^{\prime }=-(V-r_1)(r_3-V)/\sqrt{2D}$, and substitution into $D{V}^{\prime \prime }+c{V}^{\prime }-(V-r_1)(V-r_2)(V-r_3)=0$ 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.