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

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 remaind

Subdiffusive fluctuations of 'pulled' fronts with multiplicative noise

We study the propagation of a ``pulled'' front with multiplicative noise that is created by a local perturbation of an unstable state. Unlike a front propagating into a metastable state, where a separation of time scales for sufficiently large $t$ creates a diffusive wandering of the front position about its mean, we predict that for so-called pulled fronts, the fluctuations are subdiffusive with root mean square wandering Delta(t) sim t^{1/4, {em not t^{1/2. The subdiffusive behavior is confirmed by numerical simulations: For $tle 600$, these yield an effective exponent slightly larger than $1/4$

The diffusion coefficient of propagating fronts with multiplicative noise

Recent studies have shown that in the presence of noise both fronts propagating into a metastable state and so-called pushed fronts propagating into an unstable state, exhibit diffusive wandering about the average position. In this paper we derive an expression for the effective diffusion coefficient of such fronts, which was motivated before on the basis of a multiple scale ansatz. Our systematic derivation is based on the decomposition of the fluctuating front into a suitably positioned average profile plus fluctuating eigenmodes of the stability operator. While the fluctuations of the front position in this particular decomposition are a Wiener process on all time scales, the fluctuations about the time averaged front profile relax exponentially.Comment: 4 page

Streamer Propagation as a Pattern Formation Problem

Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Universal Algebraic Relaxation of Velocity and Phase in Pulled Fronts generating Periodic or Chaotic States

We investigate the asymptotic relaxation of so-called pulled fronts propagating into an unstable state. The ``leading edge representation'' of the equation of motion reveals the universal nature of their propagation mechanism and allows us to generalize the universal algebraic velocity relaxation of uniformly translating fronts to fronts, that generate periodic or even chaotic states. Such fronts in addition exhibit a universal algebraic phase relaxation. We numerically verify our analytical predictions for the Swift-Hohenberg and the Complex Ginzburg Landau equation.Comment: 4 pages Revtex, 2 figures, submitted to Phys. Rev. Let

The Weakly Pushed Nature of "Pulled" Fronts with a Cutoff

The concept of pulled fronts with a cutoff $\epsilon$ has been introduced to model the effects of discrete nature of the constituent particles on the asymptotic front speed in models with continuum variables (Pulled fronts are the fronts which propagate into an unstable state, and have an asymptotic front speed equal to the linear spreading speed $v^*$ of small linear perturbations around the unstable state). In this paper, we demonstrate that the introduction of a cutoff actually makes such pulled fronts weakly pushed. For the nonlinear diffusion equation with a cutoff, we show that the longest relaxation times $\tau_m$ that govern the convergence to the asymptotic front speed and profile, are given by $\tau_m^{-1} \simeq [(m+1)^2-1] \pi^2 / \ln^2 \epsilon$, for $m=1,2,...$.Comment: 4 pages, 2 figures, submitted to Brief Reports, Phys. Rev.

Fronts with a Growth Cutoff but Speed Higher than v∗v^*

Fronts, propagating into an unstable state $\phi=0$, whose asymptotic speed $v_{\text{as}}$ is equal to the linear spreading speed $v^*$ of infinitesimal perturbations about that state (so-called pulled fronts) are very sensitive to changes in the growth rate $f(\phi)$ for $\phi \ll 1$. It was recently found that with a small cutoff, $f(\phi)=0$ for $\phi < \epsilon$, $v_{\text{as}}$ converges to $v^*$ very slowly from below, as $\ln^{-2} \epsilon$. Here we show that with such a cutoff {\em and} a small enhancement of the growth rate for small $\phi$ behind it, one can have $v_{\text{as}} > v^*$, {\em even} in the limit $\epsilon \to 0$. The effect is confirmed in a stochastic lattice model simulation where the growth rules for a few particles per site are accordingly modified.Comment: 4 pages, 4 figures, to appear in Rapid Comm., Phys. Rev.

Statistics at the tip of a branching random walk and the delay of traveling waves

We study the limiting distribution of particles at the frontier of a branching random walk. The positions of these particles can be viewed as the lowest energies of a directed polymer in a random medium in the mean-field case. We show that the average distances between these leading particles can be computed as the delay of a traveling wave evolving according to the Fisher-KPP front equation. These average distances exhibit universal behaviors, different from those of the probability cascades studied recently in the context of mean field spin-glasses.Comment: 4 pages, 2 figure

Fluctuating "Pulled" Fronts: the Origin and the Effects of a Finite Particle Cutoff

Recently it has been shown that when an equation that allows so-called pulled fronts in the mean-field limit is modelled with a stochastic model with a finite number $N$ of particles per correlation volume, the convergence to the speed $v^*$ for $N \to \infty$ is extremely slow -- going only as $\ln^{-2}N$. In this paper, we study the front propagation in a simple stochastic lattice model. A detailed analysis of the microscopic picture of the front dynamics shows that for the description of the far tip of the front, one has to abandon the idea of a uniformly translating front solution. The lattice and finite particle effects lead to a ``stop-and-go'' type dynamics at the far tip of the front, while the average front behind it ``crosses over'' to a uniformly translating solution. In this formulation, the effect of stochasticity on the asymptotic front speed is coded in the probability distribution of the times required for the advancement of the ``foremost bin''. We derive expressions of these probability distributions by matching the solution of the far tip with the uniformly translating solution behind. This matching includes various correlation effects in a mean-field type approximation. Our results for the probability distributions compare well to the results of stochastic numerical simulations. This approach also allows us to deal with much smaller values of $N$ than it is required to have the $\ln^{-2}N$ asymptotics to be valid.Comment: 26 pages, 11 figures, to appear in Phys. rev.

Propagation and Structure of Planar Streamer Fronts

Streamers often constitute the first stage of dielectric breakdown in strong electric fields: a nonlinear ionization wave transforms a non-ionized medium into a weakly ionized nonequilibrium plasma. New understanding of this old phenomenon can be gained through modern concepts of (interfacial) pattern formation. As a first step towards an effective interface description, we determine the front width, solve the selection problem for planar fronts and calculate their properties. Our results are in good agreement with many features of recent three-dimensional numerical simulations. In the present long paper, you find the physics of the model and the interfacial approach further explained. As a first ingredient of this approach, we here analyze planar fronts, their profile and velocity. We encounter a selection problem, recall some knowledge about such problems and apply it to planar streamer fronts. We make analytical predictions on the selected front profile and velocity and confirm them numerically. (abbreviated abstract)Comment: 23 pages, revtex, 14 ps file