Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Fronts that start from a local perturbation and propagate into a linearly u nstable state come in two classes: pulled fronts and pushed fronts. The ter m "pulled front" expresses that these fronts are "pulled along" by the spre ading of linear perturbations about the unstable state. Accordingly, their asymptotic speed v* equals the spreading speed of perturbations whose dynam ics is governed by the equations linearized about the unstable state. The c entral result of this paper is the analysis of the convergence of asymptoti cally uniformly traveling pulled fronts towards v*. We show that when such fronts evolve from "sufficiently steep" initial conditions, which initially decay faster than e(-lambda *x) for x --> infinity, they have a universal relaxation behavior as time t --> infinity: the velocity of a pulled front always relaxes algebraically like v(t) = v* - 3/(2 lambda *t) + 3/2 root pi D lambda*/(D lambda*(2)t)(3/2) + O(1/t(2)). The parameters v*, lambda*, and D are determined through a saddle point analysis from the equation of moti on linearized about the unstable invaded state. This front velocity is inde pendent of the precise value of the front amplitude, which one tracks to me asure the front position. The interior of the front is essentially slaved t o the leading edge, and develops universally as phi>(*) over bar * (x, t) = Phi (v(t))(x - integral (t) dt' v(t')) + O(1/t(2)), where Phi (v)(x - vt) is a uniformly translating front solution with velocity v < v*. Our result, which can be viewed as a general center manifold result for pulled front p ropagation is derived in derail for the well-known nonlinear diffusion equa tion of type <partial derivative>(t)phi = partial derivative (2)(x)phi + ph i - phi (3), where the invaded unstable state is phi = 0. Even for this sim ple case, the subdominant t(-3/2) term extends an earlier result of Bramson . Our analysis is then generalized to more general (sets of) partial differ ential equations with higher spatial or temporal derivatives, to PDEs with memory kernels, and also to difference equations such as those that occur i n numerical finite difference codes. Our universal result for pulled fronts thus implies independence (i) of the level curve which is used to track th e front position, (ii) of the precise nonlinearities, (iii) of the precise form of the linear operators in the dynamical equation, and (iv) of the pre cise initial conditions, as long as they are sufficiently steep. The only r emainders of the explicit form of the dynamical equation are the nonlinear solutions Phi (v) and the three saddle point parameters v*, lambda*, and D. As our simulations confirm all our analytical predictions in every detail, it can be concluded that we have a complete analytical understanding of th e propagation mechanism and relaxation behavior of pulled fronts, if they a re uniformly translating for t --> infinity. An immediate consequence of th e slow algebraic relaxation is that the standard moving boundary approximat ion breaks down for weakly curved pulled fronts in two or three dimensions. In addition to our main result for pulled fronts, we also discuss the prop agation and convergence of fronts emerging from initial conditions which ar e not steep, as well as of pushed fronts. The latter relax exponentially fa st to their asymptotic speed. (C) 2000 Elsevier Science B.V. All rights res erved.