We study the dynamics of reaction-diffusion fronts under the influence of multiplicative noise. An approximate theoretical scheme is introduced to compute the velocity of the front and its diffusive wandering due to the presence of noise. The theoretical approach is based on a multiple scale analysis rather than on a small noise expansion and is confirmed with numerical simulations for a wide range of the noise intensity. We report on the possibility of noise sustained solutions with a continuum of possible velocities, in situations where only a single velocity is allowed without noise. ͓S1063-651X͑98͒03611-3͔

We present a phase-field model for the dynamics of the interface between two inmiscible fluids with arbitrary viscosity contrast in a rectangular Hele-Shaw cell. With asymptotic matching techniques we check the model to yield the right Hele-Shaw equations in the sharp-interface limit, and compute the corrections to these equations to first order in the interface thickness. We also compute the effect of such corrections on the linear dispersion relation of the planar interface. We discuss in detail the conditions on the interface thickness to control the accuracy and convergence of the phase-field model to the limiting Hele-Shaw dynamics. In particular, the convergence appears to be slower for high viscosity contrasts.

We study the effects of external noise in a one-dimensional model of front propagation. Noise is introduced through the fluctuations of a control parameter leading to a multiplicative stochastic partial differential equation. Analytical and numerical results for the front shape and velocity are presented. The linear-marginal-stability theory is found to increase its range of validity in the presence of external noise. As a consequence noise can stabilize fronts not allowed by the deterministic equation. [S0031-9007(96) PACS numbers: 03.40. Kf, 05.40.+j, 47.20.Ky, 47.54.+r The problem of front propagation has been receiving a great deal of attention in recent years due to its relevance to a large variety of systems in nonlinear physics, chemistry, and biology [1]. Here we will focus on the simplest case in which a globally stable state invades an unstable or metastable state. This problem has been extensively studied in the recent literature [2][3][4][5][6][7][8] particularly concerning the issue of velocity selection.On the other hand, in the last few years there has been a growing interest in the theoretical study of the role of fluctuations in front propagation [7,[9][10][11][12][13][14], and in particular there have been some experiments on the effects of stochastic turbulence in front propagation in the context of chemical fronts [15]. These studies have been basically concerned with the modification of the front velocity and the spreading of the front due to fluctuations.Internal [9][10][11][12] and external [13,14] fluctuations have been introduced in particular models using both Langevin [9,11,13,14] and master equation formalisms [10,12], but no systematic studies have been carried out concerning the modification of the well established selection criteria of the deterministic case. For internal fluctuations mostly numerical studies of different situations have obtained distinct effects on the front propagation. The case with the most direct comparison with the present work [9] found no change in the front velocity. On the other hand, previous analytical approaches for external fluctuations [13,14] have been based on small noise perturbative expansions which turn out to have a rather small range of validity for our purposes.Here we will introduce a new approach which relies on a physically intuitive picture of the problem but which is nonperturbative. As the accompanying numerical simulations will show, our theoretical approach gives an accurate quantitative description for a very broad range of noise intensities and allows for a general discussion of selection criteria in the presence of external fluctuations.We focus our study on the simplest prototypical equation for front propagation dynamics, and we introduce fluctuations via a Langevin equation. In our study, noise is assumed to be of external origin and is thus introduced as a stochastic spatiotemporal variation of a control parameter. For example, in an experimental situation such as a nematic liquid crystal in the presence of a magnetic field...

PACS. 47.55.Mh -Flows through porous media. PACS. 68.35.Ct -Interface structure and roughness. PACS. 05.40.-a -Fluctuation phenomena, random processes, noise, and Brownian motion..Abstract. -We study the forced fluid invasion of an air-filled model porous medium at constant flow rate, in 1+1 dimensions, both experimentally and theoretically. We focus on the non-local character of the interface dynamics, due to liquid conservation, and its effect on the scaling properties of the interface upon roughening. Specifically, we study the limit of large flow rates and weak capillary forces. Our theory predicts a roughening behaviour characterized at short times by a growth exponent β1 = 5/6, a roughness exponent α1 = 5/2, and a dynamic exponent z1 = 3, and by β2 = 1/2, α2 = 1/2, and z2 = 1 at long times, before saturation. This theoretical prediction is in good agreement with the experiments at long times.The ensemble of experiments, theory, and simulations provides evidence for a new universality class of interface roughening in 1 + 1 dimensions.

A diffusion-reaction model for the growth of bacterial colonies is presented. The often observed cooperative behavior developed by bacteria which increases their motility in adverse growth conditions is here introduced as a nonlinear diffusion term. The presence of this mechanism depends on a response which can present hysteresis. By changing only the concentrations of agar and initial nutrient, numerical integration of the proposed model reproduces the different patterns shown by Bacillus subtilis OG-01.

We implement a phase-field simulation of the dynamics of two fluids with arbitrary viscosity contrast in a rectangular Hele-Shaw cell. We demonstrate the use of this technique in different situations including the linear regime, the stationary Saffman-Taylor fingers and the multifinger competition dynamics, for different viscosity contrasts. The method is quantitatively tested against analytical predictions and other numerical results. A detailed analysis of convergence to the sharp interface limit is performed for the linear dispersion results. We show that the method may be a useful alternative to more traditional methods.

Spiral chemical waves subjected to a spatiotemporal random excitability are experimentally and numerically investigated in relation to the light-sensitive Belousov-Zhabotinsky reaction. Brownian motion is identified and characterized by an effective diffusion coefficient which shows a rather complex dependence on the time and length scales of the noise relative to those of the spiral. A kinematically based model is proposed whose results are in good qualitative agreement with experiments and numerics. PACS numbers: 82.40.Bj, 05.45.Jn, 47.54. + r Spirals are generic structures in extended nonequilibrium systems. They are characteristic of many reactiondiffusion systems [1], the most paradigmatic experimental example being the Belousov-Zhabotinsky (BZ) reaction [2], and they have been observed in systems as complex as the heart muscle associated to cardiac fibrillation [3,4]. Spiral patterns appear also as elementary solutions of the complex Ginzburg-Landau (CGL) equation [5].Beyond the standard description of spiral waves, their response to spatial and/or temporal forcing has been largely analyzed. Temporal resonance [6], drift of vortices due to parameter gradients [7,8] or external fields [9], and anchoring on localized defects [10] are among the most studied effects.On the other hand, the influence of random heterogeneities on extended excitable systems has recently attracted much attention. Noise as an initiator of new spatial structures [11][12][13], or sustaining wave propagation in subexcitable media [14][15][16], is a subject of much theoretical and experimental interest. Complementarily, the role of superimposed disorder on preexisting spatiotemporal patterns has been examined, in relation to propagating pulses [17], to the dynamics of CGL spirals [18] and 3D structures [19].In this paper, we study the effect of a spatiotemporal structured noise on the motion of a spiral wave for the photosensitive BZ reaction. In the absence of randomness, the spiral tip rotates quasirigidly around its core, with no net translational mobility. When the noise is switched on, Brownian diffusion of the spiral is observed, characterized by a nonmonotonous dependence on the parameters of the noise. These observations are confirmed numerically using a two-variable Oregonator model. The analysis is completed by proposing a simple theoretical model based on a kinematic approach [20], capturing the basic features observed in experiments.Experiments were carried out in a Petri dish of 9 cm diameter. The catalyst, ruthenium bipyridil, is immobilized in a thin, 1 mm thick, film of silica gel, prepared as in [21].A solution of catalyst-free BZ reaction (initial concentrations 0.18M KBr, 0.33M malonic acid, 0.39M NaBrO 3 , and 0.50M H 2 SO 4 ) was poured onto the gel. The temperature was kept constant at 25 6 1 ± C. Spatiotemporal noise is introduced by projecting on to the Petri dish the desired patterned illumination, controlling the excitability of the system, by means of a video projector (SONY CPJ-D500). Experiments were capt...

We develop an algorithm to simulate a Gaussian stochastic process that is non-δ-correlated in both space and time coordinates. The colored noise obeys a linear reaction-diffusion Langevin equation with Gaussian white noise. This equation is exactly simulated in a discrete Fourier space.Peer ReviewedPreprin