ON THE TRANSITION FROM INITIAL DATA TO TRAVELING WAVES IN THE FISHER-KPP EQUATION

The Fisher-KPP equation u(t) = u(xx) + u(1 - u) has a travelling wave solution for all speeds greater than or equal to 2. Initial data that decrease monotonically from 1 to 0 on - infinity < x < infinity, with u(x, 0) = O-s(e(-Ex)) as x --> infinity, are Known to evolve to a trav elling wave, whose speed depends on zeta. Here, it is shown that the r elationship between wave speed and zeta can be recovered by linearizin g the Fisher-KPP equation about u = O and explicitly, solving the line ar equation. Moreover, the calculation predicts that in the case zeta > I, the solution for u,iu itself evolves to a transition wave, moving ahead of the (minimum speed) u wave at the greater speed of 2 zeta. B ehind this transition, u(x)/u = - x/(2t), while ahead of it, u,iu = - zeta The paper goes on to discuss the potential applications of the me thod to systems of coupled reaction-diffusion equations.