We performed a comprehensive study of the spike autosolitons: self-sustained solitary inhomogeneous states, in the classical reaction-diffusion system -the Gray-Scott model. We developed singular perturbation techniques based on the strong separation of the length scales to construct asymptotically the solutions in the form of a one-dimensional static autosolitons, higher-dimensional radially-symmetric static autosolitons, and two types of traveling autosolitons. We studied the stability of the static autosolitons in one and three dimensions and analyzed the properties of the static and the traveling autosolitons. A Analysis of Eq. (5.9) 87 B Analysis of Eq. (5.11) 91 References 94At the same time, there are many physical, chemical and biological systems for which the activator nullcline is Λ-or V-shaped [ Fig. 1(b)]. In this case the equation q(θ, η, A) = 0 for given A and η has only two roots: θ 1 corresponding to the stable state, and θ 2 corresponding to the unstable state in the system with η = const [9][10][11]16]. Among Λ-systems are many semiconductor and gas discharge structures, electron-hole and gas plasmas, radiation heated gas mixtures (see, for example, [9][10][11][12]16,25]). It is not difficult to see that the Brusselator and the Gray-Scott models are Λ-systems, and the Gierer-Meinhardt model is a V-system.Kerner and Osipov qualitatively showed that in Λ-systems the so-called spike ASs and more complex spike patterns can be excited [9,11,16,42,43]. They were the first to analyze the static spike ASs and strata in the Brusselator, the Gierer-Meinhardt model, and the electron-hole plasma [16,42]. They found that when ǫ ≪ 1 and α 1, the one-dimensional static spike AS can have small size of order l and huge amplitude which goes to infinity as ǫ → 0. Dubitskii, Kerner, and Osipov formulated the asymptotic procedure for finding the stationary solutions in Λ-systems for sufficiently small ǫ [11,42]. Recently, we showed that in another limiting case α ≪ 1 and ǫ ≫ 1 one can excite the one-dimensional traveling spike AS which also has small size and whose amplitude goes to infinity as α → 0 [44]. We also showed that, in contrast to the traveling patterns in N-systems, the velocity of this one-dimensional traveling spike AS can have huge values (c ≫ l/τ θ ) and that the inhibitor distribution varies stepwise in the front of the spike. Thus, one can see that the properties of the spike patterns forming in Λ-systems differ fundamentally from those of the domain patterns forming in N-systems. In particular, since the interface connecting the two stable states at η = const does not exist in Λ-systems, the size of the spike should be of the order of the smallest system length scale. For this reason, the concept of the interfacial dynamics developed for the domain patterns in N-systems is generally inapplicable to the description of spike patterns. Some properties of the one-dimensional static spike ASs and the main types of their instabilities in the simplified version of the Gray-Scott model have recently been ...
