A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

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Pattern formation outside of equilibrium

The paper is concerned with the asymptotic behavior as t → ∞ of solutions u(x, t) of the equation ut—uxx—∞;(u)=O, x∈(—∞, ∞) , in the case ∞(0)=∞(1)=0, ∞′(0)<0, ∞′(1)<0. Commonly, a travelling front solution u=U(x-ct), U(-∞)=0, U(∞)=1, exists. The following types of global stability results for fronts and various combinations of them will be given.

The approach of solutions of nonlinear diffusion equations to travelling front solutions

Part I. Stationary waves: Scalar equation Leray-Schauder degree Existence of waves Structure of the spectrum Stability and approach to a wave Part II. Bifurcation of waves: Bifurcation of nonstationary modes of wave propagation Mathematical proofs Part III. Waves in chemical kinetics and combustion: Waves in chemical kinetics Combustion waves with complex kinetics Estimates and asymptotics of the speed of combustion waves Asymptotic and approximate analytical methods in combustion problems (supplement) Bibliography.

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Traveling wave solutions of parabolic systems

Singular perturbation theory of traveling waves in excitable media (a review)

Dissipative solitons

Periodic expanding target patterns of chemical activity are observed in thin layers of solution containing bromate, malonic acid and ferroin in dilute sulfuric acid. Commonly these patterns appear as thin blue (oxidized) rings propagating out from a central point into red (reduced) bulk medium. Recently, the opposite pattern has been observed: red waves of reduction propagating through an oxidized bulk medium. We discuss both of these patterns under the assumption that there is a heterogeneity at the center of the pattern—most likely a dust particle or a scratch on the glass—which changes the kinetics locally from a stable excitable steady state to a stable periodic oscillatory state. The temporal oscillation at the origin triggers waves of chemical activity which propagate radially into the excitable medium. Our approach is to combine recent advances in the mathematical description of traveling wave front solutions of reaction–diffusion equations with a realistic model of the kinetics of the reaction medium. The model we use, the Oregonator, is based on known features of the reaction mechanism, gives an acceptable qualitative and semiquantitative account of the reaction dynamics, and yet is simple enough to yield to analytic techniques developed primarily for scalar reaction–diffusion equations. With this approach we can account in some detail for most of the important features of target patterns in the Belousov–Zhabotinskii reaction. In particular, the distinction between trigger waves and phase waves is clarified by our analysis, and the novel properties of reducing waves in an oxidized medium appear as natural consequences of our model.

Target patterns in a realistic model of the Belousov–Zhabotinskii reaction

A mechanism is described, whereby stable sharply differentiated (dissipative) structures can evolve naturally within a mixture of reacting and diffusing substances. Our model has two reacting components, with one diffusion coefficient much smaller than the other. Unlike patterned states obtained by small amplitude analysis near uniform states, our structures have large amplitude and serve to divide the reactor into subregions, each corresponding to a distinct phase for the system. The evolution of the structured stationary state from an arbitrary initial distribution occurs in two stages. The first involves differentiation into subregions, and the second involves the migration of the boundaries of the subregions into a stable final configuration. A singular perturbation analysis and the theory of motion of wavefronts is used to deduce these qualitative properties.

Paul C. Fife

Papers

Target patterns in a realistic model of the Belousov–Zhabotinskii reaction

Pattern formation in reacting and diffusing systems