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

Abstract

Fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes: pulled fronts and pushed fronts. The term “pulled front” expresses that these fronts are “pulled along” by the spreading of linear perturbations about the unstable state. Accordingly, their asymptotic speed v * equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state. The central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towards v *. We show that when such fronts evolve from “sufficiently steep” initial conditions, which initially decay faster than e − λ * x for x→∞, they have a universal relaxation behavior as time t→∞: the velocity of a pulled front always relaxes algebraically like v(t)=v *−3/(2λ *t)+ 3 2 π Dλ */(Dλ * 2t) 3/2+ O(1/t 2) . The parameters v *, λ *, and D are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state. This front velocity is independent of the precise value of the front amplitude, which one tracks to measure the front position. The interior of the front is essentially slaved to the leading edge, and develops universally as φ(x,t)= Φ v(t)(x−∫ t dt′ v(t′))+ O(1/t 2) , where Φ 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 propagation is derived in detail for the well-known nonlinear diffusion equation of type ∂ t φ= ∂ x 2 φ+ φ− φ 3, where the invaded unstable state is φ=0. Even for this simple case, the subdominant t −3/2 term extends an earlier result of Bramson. Our analysis is then generalized to more general (sets of) partial differential equations with higher spatial or temporal derivatives, to PDEs with memory kernels, and also to difference equations such as those that occur in numerical finite difference codes. Our universal result for pulled fronts thus implies independence (i) of the level curve which is used to track the front position, (ii) of the precise nonlinearities, (iii) of the precise form of the linear operators in the dynamical equation, and (iv) of the precise initial conditions, as long as they are sufficiently steep. The only remainders of the explicit form of the dynamical equation are the nonlinear solutions Φ v and the three saddle point parameters v *, λ *, 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 the propagation mechanism and relaxation behavior of pulled fronts, if they are uniformly translating for t→∞. An immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation 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 propagation and convergence of fronts emerging from initial conditions which are not steep, as well as of pushed fronts. The latter relax exponentially fast to their asymptotic speed.