The Tale of Two Tails
The Tale of Two Tails
We discuss formation of patterns due to Fisher-KPP reaction appended with fast or slow diffusions听ut=[D(u)ux]x+u(1鈭抲)ut=[D(u)ux]x+u(1鈭抲)
where听D(u)=鈳р帺鈳ㄢ帾鈳猽,1,1u,slow diffusionStandard Fisher-KPPfast (logarithmic)D(u)={u,slow diffusion1,Standard Fisher-KPP1u,fast (logarithmic)
In the听Fast Diffusion听case the problem of travelling waves, TW, is mapped into a linear problem with the propagation speed听位位听being selected by the far away boundary condition(s). Imposing the natural convective b.c.;听ux+hu=0ux+hu=0, leads the system into a heating (cooling)TW for听h<1h<1听(1<h1<h) and if听h=1h=1听the system relaxes into an equilibrium. We derive explicit solutions听of both expanding and听collapsing formations that quench within a finite time. The latter being a unique feature of the fast diffusion.
In the Slow Diffusion case wherein听D(u)=uD(u)=u, unfolding a hidden symmetry we map the problem into听it a purely diffusive process听and thus demonstrate that both the semi-compact Travelling Kinks and certain expanding formations are strong attractors of their respective initial excitations.
Curiously enough, though slow and fast processes are completely different processes, one can map both problems into new variables which reveal new families of explicit solutions used in their analysis.