= One-diffuser reaction front
{title2=$U=(1+e^{-kz})^{-1},\quad V=v_0(1-U)^2$}
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 $U'=kU(1-U)$ and $V'/V=-2kU$. They connect $(u,v)=(0,v_0)$ to $(1,0)$: the second <concentration> changes by local reaction, even without its own <diffusion>.
Back to article page
