One-diffuser reaction front ($U=(1+e^{-kz}{)}^{-1},V=v_0(1-U{)}^2$)

A propagating chemical front can persist when only one species diffuses. For $u_t=u(1-u-\beta v)+u_{xx}$, $v_t=-\delta uv$, set $z=x-ct$, $k^2=1-\delta /2$, $c=-\delta /(2k)$ and $\beta v_0=2-\delta$, with $0<\delta <2$. The displayed profiles satisfy both equations because $U^{\prime }=kU(1-U)$ and $V^{\prime }/V=-2kU$. They connect $(u,v)=(0,v_0)$ to $(1,0)$: the second concentration changes by local reaction, even without its own diffusion.