Pattern forming pulled fronts: bounds and universal convergence

We analyze the dynamics of pattern forming fronts which propagate into an unstable state, and whose dynamics is of the pulled type, so that their asymptotic speed is equal to the linear spreading speed v^*. We discuss a method that allows to derive bounds on the front velocity, and which hence can be used to prove for, among others, the Swift-Hohenberg equation, the Extended Fisher-Kolmogorov equation and the cubic Complex Ginzburg-Landau equation, that the dynamically relevant fronts are of the pulled type. In addition, we generalize the derivation of the universal power law convergence of the dynamics of uniformly translating pulled fronts to both coherent and incoherent pattern forming fronts. The analysis is based on a matching analysis of the dynamics in the leading edge of the front, to the behavior imposed by the nonlinear region behind it. Numerical simulations of fronts in the Swift-Hohenberg equation are in full accord with our analytical predictions.Comment: 27 pages, 9 figure

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 unstable state come in two classes: pulled and pushed. ``Pulled'' fronts are ``pulled along'' by the spreading of linear perturbations about the unstable state, so their asymptotic speed $v^*$ equals the spreading speed of linear perturbations of the unstable state. The central result of this paper is that the velocity of pulled fronts converges universally for time $t\to\infty$ like $v(t)=v^*-3/(2\lambda^*t) + (3\sqrt{\pi}/2) D\lambda^*/(D{\lambda^*}^2t)^{3/2}+O(1/t^2)$. The parameters $v^*$, $\lambda^*$, and $D$ are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state. The interior of the front is essentially slaved to the leading edge, and we derive a simple, explicit and universal expression for its relaxation towards $\phi(x,t)=\Phi^*(x-v^*t)$. 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 F-KPP diffusion equation, and extended to much more general (sets of) equations (p.d.e.'s, difference equations, integro-differential equations etc.). 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, and (iv) of the precise initial conditions. Our simulations confirm all our analytical predictions in every detail. A consequence of the slow algebraic relaxation is the breakdown of various perturbative schemes due to the absence of adiabatic decoupling.Comment: 76 pages Latex, 15 figures, submitted to Physica D on March 31, 1999 -- revised version from February 25, 200

Large and small Density Approximations to the thermodynamic Bethe Ansatz

We provide analytical solutions to the thermodynamic Bethe ansatz equations in the large and small density approximations. We extend results previously obtained for leading order behaviour of the scaling function of affine Toda field theories related to simply laced Lie algebras to the non-simply laced case. The comparison with semi-classical methods shows perfect agreement for the simply laced case. We derive the Y-systems for affine Toda field theories with real coupling constant and employ them to improve the large density approximations. We test the quality of our analysis explicitly for the Sinh-Gordon model and the $(G_2^{(1)},D_4^{(3)})$-affine Toda field theory.Comment: 19 pages Latex, 2 figure

Surprising Aspects of Fluctuating "Pulled" Fronts

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 $\ln^{-2}N$. However, this convergence is seen only for very high values of $N$, while there can be significant deviations from it when $N$ is not too large. Pulled fronts are fronts that propagate into an unstable state, and the asymptotic front speed is equal to the linear spreading speed $v^*$ of infinitesimal perturbations around the unstable state. In this paper, we consider front propagation in a simple stochastic lattice model. 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 ``halt-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 occupied lattice site''. These probability distributions are obtained by matching the solution of the far tip with the uniformly translating solution behind in a mean-field type approximation, and the results for the probability distributions compare well to the results of stochastic numerical simulations. This approach allows one to deal with much smaller values of $N$ than it is required to have the $\ln^{-2}N$ asymptotics to be valid.Comment: 12 pages, 6 figures, intended proceedings for 3rd International Conference Unsolved Problems of Noise (UPoN) and fluctuations in physics, biology and high technology 2002; references update

Hydrodynamics of operator spreading and quasiparticle diffusion in interacting integrable systems

We address the hydrodynamics of operator spreading in interacting integrable lattice models. In these models, operators spread through the ballistic propagation of quasiparticles, with an operator front whose velocity is locally set by the fastest quasiparticle velocity. In interacting integrable systems, this velocity depends on the density of the other quasiparticles, so equilibrium density fluctuations cause the front to follow a biased random walk, and therefore to broaden diffusively. Ballistic front propagation and diffusive front broadening are also generically present in non-integrable systems in one dimension; thus, although the mechanisms for operator spreading are distinct in the two cases, these coarse grained measures of the operator front do not distinguish between the two cases. We present an expression for the front-broadening rate; we explicitly derive this for a particular integrable model (the "Floquet-Fredrickson-Andersen" model), and argue on kinetic grounds that it should apply generally. Our results elucidate the microscopic mechanism for diffusive corrections to ballistic transport in interacting integrable models.Comment: Published versio

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.