Traveling spike autosolitons in the Gray–Scott model

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Abstract

We developed singular perturbation techniques based on the strong separation of time and length scales to construct the solutions in the form of the traveling spike autosolitons (self-sustained solitary waves) in the Gray–Scott model of an autocatalytic reaction. We found that when the inhibitor diffusion is sufficiently slow, the ultrafast traveling spike autosolitons are realized in a wide range of parameters. When the diffusion of the inhibitor is sufficiently fast, the slower traveling spike autosolitons with the diffusion precursor are realized. We asymptotically calculated the main parameters such as speed and amplitude of these autosolitons as well as the regions of their existence in the Gray–Scott model. We also showed that in certain parameter regions the traveling spike autosolitons coexist with static.

Introduction

Self-organization and pattern formation in nonequilibrium systems are among the most fascinating phenomena in nonlinear physics [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Pattern formation is observed in various physical systems including fluids; gas and electron–hole plasmas; various semiconductor, superconductor and gas-discharge structures; some ferroelectric, magnetic and optical media; combustion systems (see, for example [5], [9], [10], [11], [12], [13], [14], [53], [54]), as well as in many chemical and biological systems (see, for example [1], [2], [3], [4], [5], [6], [7], [15], [55], [56]).

Self-organization is often associated with the destabilization of the homogeneous state of the system [1], [2], [5], [10], [11]. At the same time, when the homogeneous state of the system is stable, one can excite large-amplitude patterns, including autosolitons (ASs)  self-sustained solitary inhomogeneous states, by applying a sufficiently strong stimulus [8], [9], [10], [11], [16], [17], [18], [19], [57], [58], [59], [60]. Autosolitons are elementary objects in open dissipative systems away from equilibrium. They share the properties of both solitons and traveling waves (or autowaves, as they are also referred to [2], [6]). They are similar to solitons since they are localized objects whose existence is due to the nonlinearities of the system. On the other hand, from the physical point of view they are essentially different from solitons in that they are dissipative structures, that is, they are self-sustained objects which form in strongly dissipative systems as a result of the balance between the dissipation and pumping of energy or matter. This is the reason why, in contrast to solitons, their properties are independent of the initial conditions and are determined primarily by the nonlinearities of the system [8], [9], [10], [11].

A prototype model used to study pattern formation in nonequilibrium systems is a pair of reaction–diffusion equations of the activator–inhibitor typeτθ∂θ∂t=l2Δθ−q(θ,η,A),τη∂η∂t=L2Δη−Q(θ,η,A),where θ is the activator, η the inhibitor, τθ, l and τη, L are the time and the length scales of the activator and the inhibitor, respectively; A is the control (bifurcation) parameter; q and Q are certain nonlinear functions representing the activation and the inhibition processes. Examples of these equations for various physical systems are given in [9], [10], [11], [12], [13], [20] where the physical meaning of the variables θ and η and the nature of the activation and the inhibition processes are discussed. The well-known Brusselator [1] and the Gray–Scott [21] models of autocatalytic chemical reactions, the classical Gierer–Meinhardt model of morphogenesis [22], the FitzHugh–Nagumo [23], [61] and the piecewise-linear Rinzel–Keller model [24], [62] for the propagation of pulses in the nerve fibers are all special cases of , .

The fact that θ is the activator means that for certain parameters the uniform fluctuations of θ will grow when the value of η is fixed. From the mathematical point of view, this is given by the condition [9], [10], [11], [18], [60]qθ<0,where qθ=∂q/∂θ, for certain values of θ and η. On the other hand, the fact that η is the inhibitor means that its own fluctuations decay and that it damps the fluctuations of the activator. Mathematically, these conditions are expressed by [9], [10], [11], [18], [60]Qη>0,qηQθ<0for all values of θ and η, provided that the derivatives in Eq. (4) do not change sign.

Kerner and Osipov [8], [9], [10], [11], [16], [17], [18], [20], [57], [58], [59], [60] showed that the properties of patterns and self-organization scenarios in systems described by , are chiefly determined by the parameters ϵl/L and ατθ/τη and the shape of the nullcline of the equation for the activator, that is, the dependence η(θ) given by the equation q(θ,η,A)=0 for A=const. They demonstrated that depending on the shape of the activator nullcline the majority of systems can be divided into two fundamentally different classes: N-systems, for which the nullcline is N- or inverted N-shaped and, Λ- or V-systems, for which the nullcline is Λ- or V-shaped, respectively (see Fig. 1).

Most works devoted to the description of pattern formation on the basis of , deal with N-systems. In N-systems the equation q(θ,η,A)=0 has three roots: θ1,θ2, and θ3, for given values of A and η. The roots θ1 and θ3 correspond to the stable states and θ2 corresponds to the unstable state in the system with η=const. It is easy to see that the FitzHugh–Nagumo and the piecewise-linear models belong to N-systems. For these models it was shown [24], [25], [62], [63] that , with L=0 and α=τθ/τη≪1 have solutions in the form of the traveling waves (also called autowaves [2], [6], or traveling ASs [8], [9], [10], [11]). For α≪1 these ASs consist of fronts and backs in which the activator varies sharply while the inhibitor remains almost constant, separated by regions in which both the inhibitor and the activator vary smoothly. It was found that in N systems the speed of these ASs cannot exceed values of order l/τθ [9], [11], [18], [24], [25], [60], [62], [63]. In [16], [17], [18], [19], [57], [58], [59], [60] it was shown that in another limit Ll (or, more precisely, when ϵ=l/L≪1 and α≳1) N-systems admit solutions in the form of the stable static patterns including ASs (see also [9], [10], [11]). Furthermore, it was shown that in N-systems with ϵ≪1 and α≪1 one can excite static, pulsating, and traveling patterns [8], [9], [10], [11], [18], [20], [26], [27], [28], [29], [30], [31], [60], [64], [65], [66], [67].

On the other hand, 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: θ1 corresponding to the stable state, and θ2 corresponding to the unstable state in the system with η=const [9], [10], [11], [16], [57], [58]. 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], [20], [57], [58]). 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 [9], [11], [16], [32], [33], [57], [58] qualitatively showed that in Λ-systems the so-called spike ASs and more complex spike patterns can be excited. They analyzed the static spike ASs and strata in the Brusselator, the Gierer–Meinhardt model, and the electron–hole plasma [16], [32], [57], [58]. They found that when ϵ≪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 ϵ→0. Dubitskii et al. [11], [32] formulated the asymptotic procedure for finding the stationary solutions in Λ-systems for sufficiently small ϵ. In [34] we showed that in another limiting case α≪1 and ϵ≫1 one can excite the one-dimensional traveling spike AS which also has small size and whose amplitude goes to infinity as α→0. 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 (cl/τθ) 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 [35], [68], [69], [70]. At the same time, spike patterns including the spike ASs are observed experimentally in the nerve tissue [36], chemical reactions [5], [37], electron–hole plasma [38], [71], gas-discharge structures [39], [72], as well as numerically in the simulations of the Brusselator, the Gierer–Meinhardt, and the Gray–Scott models [1], [15], [22], [40], [41], [55], [56], [73].

Let us emphasize that ϵ or α are the natural small parameters in the systems under consideration. Their relative smallness is in fact a necessary condition for the feasibility of any patterns [9], [10], [11]. Indeed, if the inverse were true, that is, if both the characteristic time and length scales of the variation of the inhibitor were much smaller than those of the activator, the inhibitor would easily damp all the deviations of the activator from the homogeneous steady state, making the formation of any kinds of persistent patterns impossible. On the other hand, the fact that we must have either ϵ≲1 or α≲1 for the patterns to be feasible implies that it is advantageous to consider the asymptotic limits ϵ≪1 and/or α≪1, which should result in a significant simplification of the original highly nonlinear problem. Note that this kind of approach has been successfully applied to a variety of pattern formation problems (see, for example [42], [74], [75], [76]).

This paper is one in a series of papers devoted to an asymptotic study of the spike ASs in the Gray–Scott model of an autocatalytic chemical reaction. We chose the Gray–Scott model because it has an advantage of relatively simple nonlinearities, which in many cases allow to obtain explicit analytic results. Also, because of this one can expect a certain degree of universality of pattern formation exhibited by it. In this paper we concentrate on the traveling spike ASs, so we will study the one-dimensional Gray–Scott model with α≪1 and different values of ϵ. We will develop formal asymptotic methods for the description of these patterns and study their major properties.

The outline of our paper is as follows. In Section 2 we introduce the model we will study, in Section 3 we asymptotically construct the solutions in the form of two types of traveling spike ASs, in Section 4 we compare our results with the numerical simulations, and in Section 5 we draw conclusions.

Section snippets

The model

The Gray–Scott model describes the kinetics of a simple autocatalytic reaction in an unstirred flow reactor. The reactor is a narrow space between two porous walls. Substance Y whose concentration is kept fixed outside of the reactor is supplied through the walls into the reactor with the rate k0 and the products of the reaction are removed from the reactor with the same rate. Inside the reactor Y undergoes the reaction involving an intermediate species X:2X+Yk13X,Xk2inert.The first reaction

Traveling spike autosolitons

According to the general qualitative theory of ASs, for α=τθ/τη≪1 and ϵ sufficiently large, , should have solutions that propagate with a constant speed without decay  traveling ASs [9], [10], [11]. As we will show below, in the Gray–Scott model traveling ASs are realized for sufficiently small α and have the shape of narrow spikes of high amplitude which strongly depends on α.

The equations describing an AS traveling with constant speed c along the x-axis take the form,d2θdz2+cdθdz+Aθ2η−θ=0,ϵ−2d

Numerical simulations

The numerical simulations of , confirm the conclusions of Section 3 about the existence of the traveling spike ASs. A sufficiently strong localized stimulus applied to one boundary of the system of large but finite size in the homogeneous state results in the formation of a traveling spike AS. The properties as well as the behavior of this AS mainly depend on the relationship between α and ϵ.

Fig. 8(a) shows the distributions of θ and η in the form of an ultrafast traveling spike AS for ϵ=∞, α

Conclusion

In conclusion, we have asymptotically constructed solutions in the form of the traveling spike ASs in the Gray–Scott reaction–diffusion model of an autocatalytic chemical reaction. We found that the traveling spike ASs exist in the Gray–Scott model in a wide range of the system’s parameters and can be excited even in the system only weakly away from thermal equilibrium.

The properties of the traveling spike ASs in the Gray–Scott model are mainly determined by the ratio of the diffusion

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