Spiral waves solutions in reaction-diffusion equations with symmetries. Analysis through specific models

Journal of Physics A: Mathematical and General Spiral waves solutions in reaction-diffusion equations with symmetries. Analysis through specific models To cite this article: J F R Archilla et al 1997 J. Phys. A: Math. Gen. 30 4259 View the article online for updates and enhancements. You may also like Overview of progress in European medium sized tokamaks towards an integrated plasma-edge/wall solution H. Meyer, T. Eich, M. Beurskens et al. - Event reconstruction for KM3NeT/ORCA using convolutional neural networks S. Aiello, A. Albert, S. Alves Garre et al. - 1H MRS of a boron neutron capture therapy 10B-carrier, L-pboronophenylalanine–fructose complex, BPA–F: phantom studies at 1.5 and 3.0 T S Heikkinen, A Kangasmäki, M Timonen et al. - This content was downloaded from IP address 150.214.182.36 on 10/02/2022 at 08:49 J. Phys. A: Math. Gen. 30 (1997) 4259–4271. Printed in the UK PII: S0305-4470(97)72465-X Spiral waves solutions in reaction-diffusion equations with symmetries. Analysis through specific models J F R Archilla†, J L Romero‡, F Romero Romero§and F Palmero† †Departamento de F´ ısica Aplicada, Universidad de Sevilla, PO Box 1065, Sevilla, Spain ‡Departamento de Matem´ aticas Universidad de C´ adiz, PO Box 40, Puerto Real (C´ adiz), Spain §Departamento de FAMN, Universidad de Sevilla, PO Box 1065, Sevilla, Spain Received 1 March 1996, in final form 7 August 1996 Abstract. Symmetries of nonlinear reaction-diffusion equations determine the existence of regular rotating spiral waves. They are only a consequence of kinetics processes and molecular diffusion. We prove the existence of these waves as invariant solutions of reaction-diffusion models with appropiate Lie point symmetries. 1. Introduction Nonlinear reaction-diffusion equations have been widely studied. These equations arise naturally as description models of many evolution problems in the real world, as in chemistry [1], biology [2], ecology [3], etc. Sometimes models not related directly to nature have been proposed with the main object of finding various kinds of cooperative behaviour, as is the case of the so-called tri-molecular model of Lefever and Prigogine [4]. Among the structures which can be found in systems modelled by reaction-diffusion equations we must mention regular rotating spiral patterns. They occur in a wide variety of biological, physiological and chemical contexts, and the Belousov–Zhabotinskii reaction [5–7] provides a classic example. The mathematical study of spirals in excitable media has followed several approachs. Some authors [8–12] have found asymptotic solutions which represent spiral waves far from a fixed origin. The far field of the spiral is viewed as a modification of periodic plane waves, but no analysis is given to show that these asymptotic spirals correspond to solutions that are smooth at the origin. In this approach, arguments have been advanced that additional kinematic rules must be present to produce and possibly maintain spiral waves in the core of the spiral. For a certain class of models some authors [13] have demonstrated that rotating logarithmic spiral waves can be maintained by reaction and diffusion alone. They proved the existence via the Schauder fixed point theorem applied to a certain class of the λ−ω systems introduced by Kopell and Howard [14]. The importance of these models lies in the fact that they arise naturally as the dominant part in the asymptotic analysis of many general reaction-diffusion systems [13]. One-armed Archimedian spiral waves were obtained by Greenberg [15], and Hagan [16] obtained both one-armed and multi-armed Archimedian spiral waves. They used formal asymptotic expansions methods for two different classes of λ−ωsystems. 0305-4470/97/124259+13$19.50 c 1997 IOP Publishing Ltd 4259 4260 JFRArchilla et al Another aproach [17,18] makes use of singular perturbation theory to examine the detailed motion of spiral interfaces between excited and recovering regions. The analysis results in a free-boundary problem for the shape and frequency of rotation of the spiral. Although this reduced problem is still unsolved in general, recently Keener [19] has solved this free-boundary problem in the special case where the spiral is rotationally symmetric. These spirals arise when the excitable medium is described by antisymmetric dynamics. This paper deals with the application of Lie group theory to nonlinear reaction-diffusion systems of λ−ωtype. Although group analysis of differential equations has been applied a great deal in many fields of mathematical physics [20–24], much less analysis has been used in connection with reaction-diffusion systems. In a previous paper [25] we have applied Lie theory of transformation groups to the study of λ−ωreaction-diffusion systems in two-dimensional media. Our study proves that they are invariant with respect to a fiveparameter symmetry group. Multiple types of invariant solutions with physical interest are possible and spiral waves are among them. Aλ−ωmodel that has been extensively studied was proposed by Smoes and Dreitlein [26] with the aim of finding the local dissipative structures observed in the Belousov– Zhabotinskii reaction. For this model, some analytic solutions were obtained for the case of one spatial variable [26,27]. These results were later obtained using Lie group theory as special cases of more general solutions by Steeb and Strampp [28]. Previously, numerical solutions were obtained for the case of two spatial variables [29], showing some classes of dissipative structures. With the use of Lie group theory for the case of two spatial variables we demonstrate the existence of different types of rotating spiral waves . We can prove analytically the regularity at the origin for some of them. A similar method has been used by Greenberg [15] with a different system. We have also studied numerically the regularity of other solutions. The regularity of the solutions implies that there is no need to add additional kinetics to produce the waves in the core of the spiral. Also, the potentiality of the method suggests the application to many models of reaction-diffusion systems. 2. Reaction–diffusion models We consider models described by systems of partial differential equations (SPDE) of the form ∂u1 ∂t =D∇2u1+f(u 1,u 2) ∂u2 ∂t =D∇2u2+g(u1,u 2) (1) where u1=u1(x,y,t),u 2=u 2(x,y,t) represent, for example, relative concentrations, i.e. deviations with respect to mean values, of two chemical reactants or relative populations of two biological species, and can take negative values. They diffuse through the plane (x, y) and react with kinetics given by the nonlinear functions f(u 1,u 2)and g(u1,u 2).D represents the diffusion coefficient, which can be made equal to 1 after a suitable rescaling of (x, y). The Dreitlein–Smoes model, which is analysed in some detail in this paper, is obtained as a consequence of a particular kinetics of some chemical reactants. Spiral waves in reaction-diffusion systems 4261 In a recent paper [25] the authors show that system (1) is a λ−ωsystem if, and only if, it is invariant under a Lie group of transformations [21], with characteristics Q1=a1u1x+a2u1y+a3u1t+a4(xu1y−yu1x)+a5u2 Q2=a1u2x+a2u2y+a3u2t+a4(xu2y−yu2x)−a5u1 (2) where the set {ai}5 i=1represents arbitrary constants. The λ−ωreaction-diffusion systems with two reactants are described by systems (1) with fand gof the form f(u 1,u 2)=λ(z)u1−w(z)u2;g(u1,u 2)=ω(z)u1+λ(z)u2, where z2=u2 1+u2 2. Usually, λ(z) is supposed to be a positive function of zfor 0 ⩽z<z 0and negative for z>z 0 . Also, ω(z) is supposed to be a positive function of z, in order to assure that the model, without diffusion, has a limit cycle with amplitude z0and frequency w(z0). Thus, λ−ωsystems have been proposed as models for chemical or biological systems which exhibit oscillating behaviour in homogeneous situations. Each mono-parametric subgroup is associated with a set of determined constants {ai}5 i=1. Invariant solutions with respect to different subgroups of the full group exhibit a great variety of patterns, and spiral wave patterns are among them. In this paper we are interested in the question about the existence of regular rotating spiral waves. These waves are invariant with respect to rotations with phase shift and are also time-periodic. Then, we must look for the characteristics associated with these groups. It is easy to see that they correspond to the cases a1=a2=a3=0, and a1=a2=a4=0. We denote these groups by G45 and G35, respectively. Their characteristics are given by Q1 45 =a4(xu1y−yu1x)+a5u2Q2 45 =a4(xu2y−yu2x)−a5u1 Q1 35 =a3u1t+a5u2Q2 35 =a3u2t−a5u1.(3) It is clearly appropriate to use polar coordinates. That is, we introduce the set of variables (r, θ) and (z, φ), where x=rcos(θ) y =rsin(θ) u1=zcos(φ) u2=zsin(φ). Thus, zrepresents the amplitude and φthe phase of the wave. This change of variables gives rise to considerable simplification of the characteristics. We easily obtain Qz 45 =a4zθQφ 45 =a4φθ−a5 Qz 35 =a3ztQφ 35 =a3φt−a5.(4) These characteristics are determinated except for a multiplicative constant. Then, it is possible to choose a5=1, and in these new coordinates, system (1) reads zrr +zr r+zθθ r2−zφ2 r+φ2 θ r2+zλ(z) −zt=0 φrr +φr r+φθθ r2+2φr zr z+2φθ r2 zθ z+ω(z) −φt=0. (5) The characteristics associated to a group vanish for solutions which are invariant with respect to this group. Thus, invariant solutions with respect to G45 must have the form u1=z(r,t)cosθ a4+β(r,t)u2=z(r,t)sinθ a4+β(r,t).(6) 4262 JFRArchilla et al Since the concentrations uimust be continuous at all points in space, they must satisfy ui(r, θ) =ui(r, θ +2π). Hence, the possible values for a4are of the form 1/n, with nan integer, and solutions (6) take the form u1=z(r,t)cos(nθ +β(r,t)) u2=z(r,t)sin(nθ +β(r, t)). (7) These solutions are invariant with respect to phase rotations of amplitude 2π/n. If, in addition, they are invariant with respect to the group G35, solutions take the form u1=z(r)cos(nθ +t +β(r)) u2=z(r)sin(nθ +t +β(r)) (8) with =1/a3. Thus, Lie point theory of transformations permits us to establish in a somewhat straightforward way whether the model admits spiral waves solutions. There is no need to previously assume the existence of these type of solutions, as many authors do when they are looking for solutions of λ−ωsystems. Of course there still remains the problem of studying the solutions of (9) with physical content. Substitution of (8) into (5) yields zrr +zr r+zλ(z) −β2 r−n2 r2=0(9a) β rr +βr r+2βr zr z+ω(z) −=0.(9b) The wavefronts corresponding to equations (8) are not defined as curves with u1and u2constant, because the function z(r) could mask the shape of the spirals. They are better defined as curves of constant phase, which are steadily rotating waves with angular frequency ω0=1/na3. In any case, both definitions coincide for certain phases; for example, we may consider the wavefronts, where u2=0, with phase 2πm: xm=rcos−t na3−1 nβ(r) +2πm n ym=rsin−t na3−1 nβ(r) +2πm n(10) with m=0,1,...,n−1. Equation (9b) suggests the convenience of first considering the case where β(r) is constant. 2.1. β(r) constant In this case the function ω(z) must also be a constant of value =1/a3. The wavefronts are straight lines. This shape may not be seen as a spiral in the conventional sense, but spiral waves are often defined as rotating, time-periodic, spatial structures [30], and this shape is usually included among spirals [16]. An interesting model of λ−ωtype, with ω=−2 constant is proposed by Smoes and Dreitlein [26] with λ(z) =k−z2, where kis a real parameter. From (9b) we observe that a3=−1 2. Hence ω0=−2 n. The reduced equation is zrr +zr r+zk−n2 r2−z2=0.(11) This equation is invariant with respect to the transformation Z=√Z r=r √k=k. (12) Spiral waves in reaction-diffusion systems 4263 Figure 1. Plot of z(r), in the case β(r) constant, obtained by means of numerical integration of equation (13) with αc=0.5831746. Thus, the solutions for any value of k>0 may be obtained from the solutions for k=1. Hence, when β(r) is constant it is sufficient to study the equation zrr +zr r+z1−n2 r2−z2=0.(13) The solutions are rotating spirals with phase 8=nt +t +β. In the appendix we investigate the existence of one-armed spiral waves (n=1), such that z(r) is analytical in r=0, and bounded for r→∞. We are able to demonstrate analytically the existence of these types of solutions. The following boundary conditions must be satisfied by solutions of equation (13) which are regular at the origin lim r→0z(r) =0 lim r→∞z(r) =1.(14) For the case β(r) constant, we have found numerically the value αc=0.5831746. In figure 1 we plot z(r) obtained by means of the numerical integration of equation (13). The associated solution u1=z(r)cos(θ −2t) with t=0 is represented in figure 2, and we can see that the wavefronts are straight lines. 2.2. β(r) non-constant The case β(r) non-constant is clearly more interesting from a physical point of view. A monotonous function β(r) will produce the shape of conventional spiral waves observed in many physical systems, as, for example, the Belousov–Zhabotinskii reaction. However, it is not easy to find the appropriate λ(z) and β(z), which allow bounded solutions for both z(r) and βr(r). It is easy to see that a necessary condition is that w(z(∞)) =, and that if w(z) =constant, as in the Dreitlein–Smoes model, there are no possible bounded solutions for β(r) non-constant [30]. However, divergent solutions for r→∞should not be disregarded. Many systems are proposed to model local dissipative structures and are not necessarily supposed to be valid when r→∞. That is the reason why we have done some numerical integration for ω(z) ==−2. This integration also poses some problems. First, equation (9a) is singular in r=0, where the values z(0)=0 and βr(0)=0 are 4264 JFRArchilla et al Figure 2. Spatial pattern of the concentration u1, in the case β(r) constant, obtained by means of numerical integration of equation (13) with αc=0.5831746. The horizontal and the vertical axes are from −5√2to+5 √ 2. (This figure can be viewed in colour in the electronic version of the article; see http://www.iop.org/EJ/welcome) known. The procedure has been chosen to substitute a power series for z(r) and βr(r) in a neighbourhood of r=0 in equations (9), and to find the relations among the first coefficients up to order 3, using them to specify the initial conditions in function of α=zr(0). Second, as suggested by some numerical integration and the demonstration in the appendix for the case β(r) constant, there are two different types of solutions. Divergent solutions for α greater than certain value αc, and oscillating solutions for smaller values of α. We look for solutions that are well behaved for relatively large r, although not for all r. Thus, we look for initial conditions very near the critical value αc. The results shown in figure 3 are for β(0)=0, βr(0)=1, k=1 and −ω=−0.1. We have numerically found the value αc=0.579889. The associated solutions u1,u2are obtained through equations (8). Figure 4 represents u1(x, y) with t=0. It is possible to find some systems with ω(z) non-constant in the literature, which allow bounded solutions for all r, as, for example, Greenberg [15], for λ(z) =1−zand ω(z) =1+ω1(z −1), or Hagan [16], for λ(z) =1−z2and ω(z) =qz2. It is not the aim of this paper to find the general conditions for ω(z) and λ(z) which lead to bounded solutions, and this problem remains open. 3. Conclusions The main aim of this article is to demonstrate the relation of certain Lie point symmetries with the existence of rotating spiral waves in reaction-diffusion systems. Application of Lie group theory to the study of nonlinear reaction-diffusion models may help to establish if these kinds of structures are a consequence of only the kinetics processes and of molecular diffusion, or if additional kinematics mechanisms must be present to produce them. Invariant solutions of subgroups of the full symmetry group, i.e. partially invariant solutions [31], as Spiral waves in reaction-diffusion systems 4265 Figure 3. Plot of z(r) and β(r), obtained by means of numerical integration of system (9) with β(0)=0, βr(0)=1, k=1, −ω=−0.1 and αc=0.579889. Figure 4. Spatial pattern of the concentration u1, obtained by means of numerical integration of system (9) with β(0)=0, βr(0)=1, k=1, −ω=−0.1 and αc=0.579889. The horizontal and the vertical axes are from −5√2to+5 √ 2. (This figure can be viewed in colour in the electronic version of the article; see http://www.iop.org/EJ/welcome) the spiral waves studied in this paper, are usually of great interest, not only because of the richness of patterns exhibited but also because, as they have a lower degree of symmetry than the system, they are probably the emerging solutions in a spontaneous symmetry breaking process [32]. Appendix A In this appendix we briefly sketch, without technical details, the method used for obtaining the characteristics of λ−ωsystems. A complete reference can be found in [21]. 4266 JFRArchilla et al Group of transformations Let Gbe a local Lie Group, x=(x1,x 2,...,x n)the set of independent variables, and u=(u1,u 2,...,u m)the set of dependent variables, in a space of functions u=u(x).A local Lie group of transformations in the space (x, u) is given by the set of equations x=X(x, u, ) u=U(x,u,) (A1) where is a continuous parameter of a local group, being =0 the value of the parameter for the identity element. The expression local means that the group properties are valid at least in some neighbourhood of =0. If the functions Xand Udepend not only on xand ubut also on some derivatives, the transformations (A1) have no geometrical interpretation, and must be seen as transformations in the space of functions u(x). In this case they are called generalized transformations. Infinitesimals For every transformation (A1) there is an infinitesimal transformation given by δx =ξ(x, u) δu =η(x, u) (A2) with small enough; ξ=(ξ1,ξ2,...,ξn)and η=(η1,η 2,...,η m)are called the infinitesimals of the transformation and are given by ξ=∂X ∂ =0 η=∂U ∂ =0 .(A3) Characteristics The characteristic of the transformation group is defined as Q=η−ξiui. An equivalent transformation [21] to (A1) that leaves the xvariables invariant is given infinitesimally by δu =Q(x, u, {ui}) where Q=∂U ∂ =0 .(A4) This is a generalized transformation which has an equivalent geometrical transformation. The expression {ui}represents the set of derivatives ∂uα/∂xiwith α=1,2,...,m and i=1,2,...,n. We represent by {uI}, where I=(i1,i 2,...,i n)is a multi-index, the set of derivatives, given explicitly by the expressions {uI}→ ∂ |I|u α ∂xi1 1∂xi2 2...∂xi n n α=1,2,...,m |I|= n X j=1 i j>0. The infinitesimal transformation for uIis given by δuI=(DIQ) where DIis the total derivative operator DI=∂ xI+uI ∂ ∂u +X J uJ,I ∂ ∂uJ|J|>0 with ∂ ∂xI=∂|I| ∂xi1 1∂xi2 2...∂xi n n .