Waves and patterns

The most common kind of chemical wave is the single propagating front, where, in an unstirred medium, there is a relatively sharp boundary between 

When we studied the emergence of temporal oscillations in Chapter 2, we found that it was useful to examine whether a small perturbation to a steady state would grow or decay. We now attempt a similar linear stability analysis of a system in which diffusion, as well as reaction, can occur. First, consider the general reaction-diffusion equation  

In order to solve a partial differential equation like eq. (6.1), we need to know the boundary conditions. In most situations of interest—for example, in a Petri dish with an impermeable wall—the appropriate condition to use is the zero-flux boundary condition  

Just as in the case of purely temporal perturbations, it is useful to examine the behavior of a solution to eq. (6,1) composed of a steady-state solution plus an infinitesimal perturbation. Here, the steady-state solution is uniform in space, that is, c is independent of x, y, and z, and is constant in time, that is, J ( c ) = 0. We chose our temporal perturbation to be u , an eigenfunction of the time derivative operator 3/9/. Here, we choose the spatiotemporal perturbation to be a product of eigenfunctions of the time derivative and the Laplacian operators  

To make things more concrete (and somewhat simpler), consider a system that has only one spatial dimension, that is, a line of length L, If we put the boundaries of the system at z = L/2, then condition (6.2) is simply the requirement that M, /z vanish at the boundary points. It is not difficult to see that the function 

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Kapral R and Showalter K (eds) 1995 Chemical Waves and Patterns (Dordrecht Kluwer)... 

Multi-author volume surveying chemical wave and pattern formation, an up-to-date introduction for those entering the field. 

Kapral, R. and Showalter, K. (1994) Chemical Waves and Patterns, Kluwer, London. 

Raymond Kapral, Simulating Chemical Waves and Patterns. 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

I have returned to my original field, and now I am working with chemical waves and patterns. During the past four years, we have obtained a lot of interesting results. However, I would like to resume work on biological kinetics, and I believe that I will be able to do this in the rather near future. 

V. Krinsky and H. Swinney (eds) Wave and patterns in biological and chemical excitable media (North-Holland, Amsterdam, 1991). 

Borckmans P, Dewel G, De Wit A and Walgraef D Turing bifurcations and pattern selection Chemical Waves and Patterns eds R Kapral and K Showalter (Dordrecht Kluwer) ch 10, pp 323-63... 

Winfree A T 1994 Chemical Waves and Patterns ed R Kapral and K Showalter (Dordrecht Kluwer) p 3 Plesser T, Muller S C and Fless B 1990 J. Chem. Phys. 94 7501... 

R. Kapral and K. Showalter (eds.). Chemical Waves and Pattern Formation, Kluwer, 1995. 

Lengyel, I., Epstein, I.R. The chemistry behind the first experimental chemical examples of Turing patterns. In Kapral, R., Showalter, K. (eds.) Chemical Waves and Patterns, pp. 297-322. Kluwer Academic Publishers, Dordrecht (1995)... 

Vanag, V.K. Waves and patterns in reaction-diffusion systems. Belousov-Zhabotinsky reaction in water-in-oil microemulsions. Phys. Usp. 47(9), 923-941 (2004). http //dx.doi. org/10.1070/PU2004v047n09ABEH001742... 

Another set of pattern formation phenomena involve stationary, or Turing patterns (77), which arise in systems where an inhibitor species diffuses much more rapidly than an activator species. These patterns, which are often invoked as a mechanism for biological pattern formation, were first found experimentally in the chlorite-iodide-malonic acid reaction (72). Examples of typical spot and stripe patterns appear in Figure 3. Recently, experiments in reverse microemulsions have given rise not only to the waves and patterns described above, but to a variety of novel behaviors, including standing waves and inwardly moving spirals, as well (75). 

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