Solution
= Solution
With $ξ=x-ct$, the equation is $Du\prime\prime+cu\prime+f(u)=0$. Since $f_r(u)-f_{1/2}(u)=u(1-u)(1/2-r)$ and $n_0\prime=-n_0(1-n_0)/\sqrt{2D}$, the same profile works with
$$
\boxed{c=\sqrt{D/2}(1-2r).}
$$
= Solution
With $ξ=x-ct$, the equation is $Du\prime\prime+cu\prime+f(u)=0$. Since $f_r(u)-f_{1/2}(u)=u(1-u)(1/2-r)$ and $n_0\prime=-n_0(1-n_0)/\sqrt{2D}$, the same profile works with
$$
\boxed{c=\sqrt{D/2}(1-2r).}
$$