Exact generalized Fisher front ($U(z)=(1+a{e}^{qz/\sqrt{2(q+2)}}{)}^{-2/q}$)

For $u_t=u_{xx}+u(1-u^q)$, $q>0$, the displayed decreasing travelling wave has speed $c=(q+4)/\sqrt{2(q+2)}$, with $a>0$ setting its translation. Writing $Y=U^{q/2}$ gives $U^{\prime }=-kU(1-Y)$, $k=\sqrt{2/(q+2)}$, which verifies the wave equation by equating constant and $Y$ coefficients. The explicit speed increases with $q$, the leading-edge decay rate decreases, and the filled-state decay rate $q/\sqrt{2(q+2)}$ increases. Aligning fronts at half height distinguishes these changes from a translation.