We present a three-component reaction-diffusion system capable to support an arbitrary number of interacting traveling pulses in two spatial dimensions. Whereas a global coupling added to a twocomponent system is able to stabilize a single pulse, a fast and strongly diffusive third component can be used to stabilize multipulse solutions. We study two-pulse scattering including extinction and present a pulse generation process leading to a coherently propagating array. [S0031-9007(97)03097-4] PACS numbers: 82.20.Mj, 47.35. + i, 82.20.Wt Nonlinear reaction-diffusion (RD) systems are well suited to model a wide range of physical [1-3], chemical [4,5], and biological [6-8] pattern formation processes. In particular, stationary localized structures (single and multispot patterns) have been obtained in one-and two-dimensional systems in accordance with numerous experimental results. Recent observations in an ac gas discharge between two glass plates [1] established the long-time existence of an almost arbitrary number of moving spots limited only by the size of the system. Repulsion, annihilation, and generation of these spots has been observed. Since moving localized solutions which remain stable could not be obtained in the framework of two-component activator-inhibitor models, we propose a set of three RD equations capable to describe these phenomena at least in a qualitative manner. Our model system is simple enough to motivate the design of further experimental setups for the study of traveling spots in two dimensions. In order to motivate the construction of a three-component model for the description of traveling spots, we start with a short review of the localized structures that have been found using one-and two-dimensional RD models with one and two components, respectively.Simple front propagation is well described by onecomponent RD systems [9][10][11][12]. The interaction between two such fronts is attractive [13], destabilizing any multifront solution, as well as closed front lines in two dimensions. This restriction can be overcome by introducing a global inhibitory feedback which has proved to be able to change the character of front-front interactions to a repulsive type and thus to stabilize a single localized pattern. Nevertheless, multispot solutions are still unstable in such a model since an antisymmetrical evolution of two localized spots is not affected by a global term, and, therefore, cannot be suppressed. Instead, one of the spots grows, while the other one shrinks and vanishes [14]. It is well known that for one-dimensional and stationary two-dimensional problems the remedy lies in a distributed second component with inhibiting dynamics. In the limit of strong inhibitor diffusion front interaction is repulsive [13], permitting stable stationary multispot solutions on finite domains. There is also an intermediate range of inhibitor diffusion where the type of interaction oscillates between attractive and repulsive according to the distance between neighboring fronts [13] permitting more co...

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We present an analytical investigation of the bifurcation from stationary to traveling localized solutions in a three-component reaction-diffusion system of arbitrary dimension with one activator and two inhibitors. We show that increasing one of the inhibitors' time constants leads to such a bifurcation. For a limit case, which comprises the full range of stationary two-component patterns, the bifurcation is supercritical and no other bifurcation precedes it. Bifurcation points and velocities close to the branching point are predicted from the shape of the stationary solution. Existence and stability of the traveling solution are checked by means of multiple scales perturbation theory. Numerical simulations agree with the analytical results.

In two-dimensional reaction-diffusion systems localized, solitary structures, that we call self-organized quasiparticles or spots, can be found as stable and stationary solutions. Combinations of two or more spots can lead to rather complex patterns, that can be understood by treating them as particles. These particles can interact with the boundaries of the system as well as with each other in different ways, that depend essentially on the parameters of the system. The interaction can be described by an approximation based on the exponential decay of the spots apart from their centers. The calculations reduce the dynamics of the system to some equations for the velocities of the spots. In particular, there is a parameter range where the interaction of two spots oscillates with their distance, which gives rise to infinitely many bounded states, resembling molecules. Investigating more than two spots molecules of numerous shapes have been obtained.

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