Depending on the nonlinear equation of motion and on the initial conditions, different regions of a front may dominate the propagation mechanism. The most familiar case is the so-called pushed front, whose speed is determined by the nonlinearities in the front region itself. Pushed dynamics is always found for fronts invading a linearly stable state. A pushed front relaxes exponentially in time towards its asymptotic shape and velocity, as can be derived by linear stability analysis. To calculate its response to perturbations, solvability analysis can be used. We discuss, why these methods and results in general do not apply to fronts, whose dynamics is dominated by the leading edge of the front. This can happen, if the invaded state is unstable. Leading edge dominated dynamics can occur in two cases: The first possibility is that the initial conditions are 'flat', i.e., decaying slower in space than $e^{-lambda^* x$ for $xto infty$ with $lambda^* $ defined below. The second and more important case is the one in which the initial conditions are 'steep', i.e., decay faster then $e^{-lambda^* x$. In this case, which is known as ``pulling'' or ``linear marginal stability'', it is as if the spreading leading edge is pulling the front along. In the central part of this paper, we analyze the convergence towards uniformly translating pulled fronts. We show, that when such fronts evolve from steep initial conditions, they have a universal relaxation behavior as time $ttoinfty$, which can be viewed as a general center manifold result for pulled front propagation. In particular, the velocity of a pulled front always relaxes algebraically like $v(t)=v^*-3/(2lambda^*t); left(1-sqrt{pi/big((lambda^*)^2Dtbig)right)+O(1/t^2)$, where the parameters $v^*$, $lambda^*$, 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 amplitude which one tracks to measure the front velocity. The interior of the front is essentially slaved to the leading edge, and develops universally as $phi(x,t)=Phi_{v(t)left(x-int^t dtau ;v(tau)right)+O(1/t^2)$, where $Phi_{v(x-vt)$ is a uniformly translating front solution with velocity $v$. We first derive our results in detail for the well known nonlinear diffusion equation of type $partial_t phi =partial_x^2phi+phi-phi^3$, where the invaded unstable state is $phi=0$, and then generalize our results to more general (sets of) partial differential equations with higher spatial or temporal derivatives, to {em p.d.e.'s with memory kernels, and also to difference equations occuring, e.g., in numerical finite difference codes. Our {it universal result for pulled fronts thus also implies independence of the precise nonlinearities, independence of the precise form of the dynamical equation, and independence 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 $Phi_v$ and the three saddle point parameters $v^*$, $lambda^*$, and $D$ of the linearized equation. 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 $ttoinfty$. 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.

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Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Ebert, U., & van Saarloos, W. (1999). Front propagation into unstable states : universal algebraic convergence towards uniformly translating pulled fronts. Modelling, Analysis and Simulation [MAS]. CWI.