Abstract
Solitons, defined as nonlinear waves which can reflect from boundaries or transmit through each other, are found in conservative, fully integrable systems. Similar phenomena, dubbed quasisolitons, have been observed also in dissipative, “excitable” systems, either at finely tuned parameters (near a bifurcation) or in systems with crossdiffusion. Here we demonstrate that quasisolitons can be robustly observed in excitable systems with excitable kinetics and with selfdiffusion only. This includes quasisolitons of fixed shape (like KdV solitons) or envelope quasisolitons (like NLS solitons). This can happen in systems with more than two components, and can be explained by effective crossdiffusion, which emerges via adiabatic elimination of a fast but diffusing component. We describe here a reduction procedure can be used for the search of complicated wave regimes in multicomponent, stiff systems by studying simplified, soft systems.
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Motivation
Reactiondiffusion systems are an important mathematical tool in studying dissipative structures. For the properties of solutions of these systems, diffusion is as important as is reaction. For instance, Turing structures are possible only if there is significant difference in the diffusion coefficients between activator and inhibitor species^{1}. Most of literature on waves and patterns is focussed on selfdiffusion, when the flux of a reacting species is defined by the gradient of that same species. However, this ignores the phenomenon of crossdiffusion, when the flux of one species depends on gradient of another species.
There is a large variety of interesting regimes in systems with nonlinear kinetics and crossdiffusion instead of or in additional to selfdiffusion^{2}. Some of these regimes present a considerable theoretical interest, since they manifest properties that are traditionally associated with very different “realms”: on one hand, these are waves that preserve stable and often unique profile, speed and amplitude, similar to nerve pulse, on the other hand, they can penetrate through each other or reflect from boundaries, such as waves in linear systems or solitons in conservative nonlinear systems. It is interesting where in nature such regimes can be observed.
Originally, these regimes have been discovered in models motivated by population dynamics, where (nonlinear) crossdiffusion appears naturally as taxis of micro or macroorganisms with respect to each other’s population density^{3,4}. There are also examples of similar models occurring in physical applications, see e.g. ref. 5. However, real crossdiffusion is actually observed in chemical solutions along with the selfdiffusion^{6}, which brings up the question of the possibility of observing the quasisolitons in excitable chemical reactions, such as BZ reaction or some of its modifications, which would be a convenient experimental object to observe quasisolitons.
This question encounters a number of obstacles due to physical constraints. Firstly, the crossdiffusion must always be nonlinear in order to preserve positivity of chemical concentrations^{7}. Then, there is a fundamental requirement that the diffusivity matrix should have all real and positive eigenvalues^{8}, which follows from Onsager reciprocal relations^{9,10} based on the Second Law of Thermodynamics. The first of these obstacles does not seem particularly essential: quasisolitons have been observed in models both with linear^{2,11,12} and nonlinear^{3,4} crossdiffusion. The second obstacle is more serious, as in all the known theoretical examples, the diffusivity matrix required for observation of quasisolitons, has zero or complex eigenvalues. Hence, as argued e.g. in [ref. 6, page 899], it would appear that there are no chances to observe these regimes in chemical excitable systems.
In the present paper, we seek to look into this question a bit deeper. We are led by the observation that the above fundamental requirement is about actual diffusivities, which would appear in reactions written in all their elementary steps, for all intermediate reacting species, whereas for studies of waves and pattern formation one typically uses reduced mathematical models, and the diffusivities in them are effective. Patternforming reactions are supposed to be supported by a source of energy, say in the form of constant supply of a reagent which is consumed in the reaction, whereas the Second Law applies to closed systems. Although this source of energy (and of “negative entropy”, in Schrödinger’s sense^{13}) is in the reaction part, in reduced models this part is actually related to the diffusion part. In reduced models, several intermediate stages and species are lumped together or “adiabatically eliminated”, say by asymptotic methods. In classical approaches, this is typically done for wellmixed reaction (see e.g. the review^{14}), whereas those intermediate species may, and often are, diffusive. Reduction in the full reactiondiffusion context, with account of different diffusivities of the eliminated intermediate reagents is not often done; however see ref. 15 for an example. That example is for an ecological model, but the mathematics involved is equally applicable for chemical reactions. That example shows, firstly, that effective diffusion coefficients occurring after the elimination are significantly different from those before (in particular, crossdiffusion terms appear where have been none originally) and, secondly, that these effective diffusion coefficients depend on the reaction kinetics linking the eliminated species with the main species — which is the place through which the remoteness from the thermodynamic equilibrium creeps right into the diffusion matrix. Another example, more specifically about chemical systems, is the complex GinzburgLandau equation, which is in fact a reactiondiffusion system where the diffusion matrix has a complex pair of eigenvalues, but it can be obtained by asymptotic reduction from a reactiondiffusion system with a diagonal diffusion matrix; see e.g. [ref. 16, Appendix B].
In this paper, we illustrate how the fastslow analysis (adiabatic elimination of fast variables) in the spatiallyextended context can be used to reconstruct full (fuller) reactiondiffusion systems with purely selfdiffusing reagents, from reduced reactiondiffusion systems with crossdiffusion. We apply this procedure to produce examples of selfdiffusion systems that manifest quasisoliton solutions. Since the fastslow analysis is an asymptotic procedure, i.e. is formally valid in certain parametric limits but has to be applied for fixed parameter values, it inherently has limited accuracy. The procedure we exploit here is valid when the spatial scale of the solutions is large compared to the diffusion length of the fast species, and the reduced equations become nonlocal when this is not fulfilled. To relax this restriction, we suggest a euristic procedure which enhances the accuracy of the reduction/reconstruction for particular wavelike solutions.
The structure of the paper is as follows. In section “Methods: Fastslow reduction for reactiondiffusion systems” we discuss the process of adiabatic elimination of a fast reacting species in the spatiallyextended context, and show how the diffusivity matrix may be affected by that elimination. In section “Results” we present a simple application of this theory: namely, by “working backwards” the adiabatic elimination, we construct a threecomponent reactiondiffusion system with selfdiffusion only, which corresponds to a twocomponent system with crossdiffusion and quasisoliton solutions of various kinds. We conclude by discussing possible implications of our findings for future research.
Methods
Fastslow reduction for reactiondiffusion systems
Our asymptotic argument is based on two assumptions. Firstly, we consider a reactiondiffusion system, which has a fast and a slow subsystem,
where , , , , , , and the parameter represents the time separation between the slow subsystem u and the fast subsystem v. We assume that the fast subsystem (2) has a unique globally stable equilibrium for v = g(u) at any fixed u of interest,
The second assumption, to be clarified and formalized later, is that the solutions of interest are sufficiently smooth in space.
Linearizing equation (2) around the stable equilibrium,
and considering the limit , we have
At this point we see that our solutions should be so smooth that ∇^{2}u is small and therefore the resulting is small, justifying the linearization made. Formally, we set , where μ is a (symbolical) small parameter. Then we can formally resolve equation (4), keeping terms up to and including , to get
Substitution of the thus found v into equation (1) leads to
where , F_{u} = ∂_{v}f_{u}(u, v)_{v=g(u)}, and
We see that the diffusion in the reduced system (5) is not only different from the full system, but is typically nonlinear even if the original diffusion was fully linear.
In the system (5) we have dropped terms as they are complicated and will not be needed in this paper in the general form. In the special case when the fast subsystem is linear and is linked to the slow system in a linear way, the situation simplifies. We then have g(u) = Gu, G = const, and
were
and
The secondorder approximation (6) contains a biharmonic operator. We can approximately replace it with a reactiondiffusion system for solutions which are approximately oscillatory in space with one dominant wavenumber k, so that
Then we have
where . If the solutions of interest, or their Laplacians, are not approximate solutions of the Helmholtz equation, this secondorder approximation is clearly only a euristic.
The direct transformation from system (1, 2) veq) to system (5) or (9), and its inverse, can be used for searching for nontrivial regimes in the full system, based on the existing experience of nontrivial regimes in the analogues of the reduced systems:

1
Do parametric search in the reduced system, which has fewer parameters and often is easier to compute, as it is less stiff;

2
Once an interesting regime is found, estimate the dominant wavenumber k for it, say as the maximum of the power spectrum of the spatial profile;

3
Use the inverse transformation to obtain parameters for the full system corresponding to the found parameters of the reduced system;

4
See what solutions the full system with these parameters will produce.
Results
Direct transform (reduction)
As a simple example, we now consider a threecomponent extension of a twocomponent system with nonlinear kinetics, where a third component is added, which has fast linear kinetics and linked to the other components in a linear way:
In terms of equations (1, 2), this means
which leads to the reduced system (9) in the form
where
and
It is easy to see that the resulting diffusivity matrix
depends on the reaction kinetics parameters and does not have to have real eigenvalues or even be positive definite (although the utility of the reduced model is of course stipulated by the wellposedness of its Cauchy problems).
Inverse transform (reconstruction)
Equation (12) allow to determine parameters of the twocomponent system (11) from the given system (10). For the study presented later, we need also to be able to do the opposite: assuming that a twocomponent system (11) is known, and is known to be a reduction of a threecomponent system (10), to reconstruct the parameters of that threecomponent system. One obvious restriction is that it is not possible to reconstruct the value of parameter since the known system (11) corresponds to the zero limit of that parameter. However, the nonuniqueness of the inverse transform is even stronger than that, and there are infinitely many ways to choose parameters of (10), which would correspond to the same (11), even disregarding the variety of . For instance, we can also arbitrarily fix values of γ, D_{w}, D_{v}, and for given parameters of (11) (and given dominant wavenumber k, if using the euristic secondorder approximation), obtain the remaining parameters of (10) via
and is still related to D_{w} via equation (13). The only restriction is that the resulting reconstructed system should be physically reasonable. This, in particular, requires that D_{u} ≥ 0. The latter will be satified if D_{v} is chosen so that either D_{v} ≤ D_{uu}D_{vv}/(D_{u} + D_{uu}) or D_{v} ≥ D_{vv}.
Quasisolitons in reduced system with FitzHughNagumo kinetic
Numerical illustrations of some nontrivial regimes obtained in the twocomponent system, and corresponding nontrivial regimes found in the reconstructed threecomponent system are presented in Figs 1 and 2. These are for the twocomponent system (11) with the FitzHighNagumo kinetics, taken in the form
and the corresponding threecomponent system (10). We used the parameter values ε = 0.01, k_{1} = 10, , D_{uu} = D_{vv} = 0.025, D_{uv} = 1, D_{vu} = −1, γ = 30, D_{v} = 0.001, D_{w} = 0.04, and a selection of values of a as shown to the left of the panels in Fig. 1. The dominant wavenumbers k for the inverse transform were obtained as the position of the maximum of the power spectrum of the signal z(x) = u(x, t) + iv(x, t) of a selected solution (u, v) of the twocomponent system, taken at a selected fixed t. Specifically, we had k = 0.864 for a = 0.07, k = 0.879 for a = 0.25 and k = 0.942 for a = 0.35. For simulations, we used the same numerical methods as those described in refs 2 and 12. Computations were done using secondorder space differencing on the uniform grid with space step Δ_{x} = 0.1. Time differencing was firstorder, explicit in the reaction terms and implicit in the crossdiffusion terms, with time step Δ_{t} = 2.5 ⋅ 10^{−4} for the twocomponent system, and fully explicit with time step Δ_{t} = 2 ⋅ 10^{−5} for the threecomponent system. The space interval was x ∈ [0, L], where L was different in different simulations, as shown in xaxis labels in Fig. 2. Initial conditions were set as u(x, 0) = u_{s}H(σ − x), v(x, 0) = 0, w(x, 0) = 0 to initiate a wave starting from the left end of the domain. Here H() is the Heaviside function, the wave seed amplitude was typically chosen as u_{s} = 2 and length as σ = 4. To simulate propagation “on an infinite line”, L = ∞, for Fig. 1, we instantanously translated the solution by δx_{2} = 20 away from the boundary each time the pulse, as measured at the level u = 0.1, approached the boundary to a distance smaller than δx_{1} = 40, and filled in the new interval of x values by extending the u, v and w variables at constant levels. The profiles in Fig. 1 are drawn in moving frames of reference; the “comoving” space coordinate is defined as x − x_{c}, where x_{c} = x_{c}(t) is the center of mass of v^{2} at the time moment t, that is, .
For a = 0.35 and a = 0.25, both 2component and 3component systems support propagation of a solitary pulse with oscillating front and monotonous back. These shapes are steady: as one can see in the top four panels of Fig. 1, in the moving frame of reference, the solutions do not depend on time. The difference is that for a = 0.35, the pulses annihilate upon collision with a bondary, — or, rather, reflect but lose their magnitude, cannot recover and eventually decay. Whereas for a = 0.25, the pulses in both systems successfully reflect from boundaries. One can see in Fig. 2(c) the reflection from the right boundary, followed by reflection from the left boundary; the process repeats after that (not shown). The behaviour in the threecomponent variant of the system is similar, although the second reflection is not shown in panel (d). We emphasize here that the established shape, speed and amplitude of propagating pulses depend only on the parameters of the model, but not on initial conditions, as long as the initial condition is such that a propagating pulse is initiated. For instance, the profiles of the solutions obtained for σ = 2, σ = 8 and σ = 12 were indistinguishable from those shown in Fig. 1. These regimes are similar to those described in ref. 3.
The behaviour for a = 0.07 is different: the shape of the pulse changes as it propagates. These are “envelope quasisolitons”, similar to those previously reported in in refs 2 and 12, and can be described as modulated highfrequency waves with the envelope in the form of a solitary wave, where the speed of the highfrequency waves (the “phase velocity”) is different from the speed of the envelope (the “group velocity”), thus the change in shape. Note that this behaviour is similar to solitons in the nonlinear Schrödinger equation (NLS)^{17}, both in terms of the varying shape and the reflection from boundary, with the difference that here the system is dissipative so here again there are unique and stable amplitude and envelope shape, with corresponding group and phase velocities, which are determined by the parameters of the system but do not depend on initial conditions, as long as a propagating wave is initiated.
Since the speed and shape of quasisolitons are typically fixed, the reflections and collisions, when they happen, are always quasielastic, if one is to use the terminology from the conservative wave theory. That is, the properties of the solitons fully recover after collisions, even though, as can be seen in Fig. 2, it make take some time. On the other hand, there are examples of quasisolitons models in which the unique shape of quasisolitons takes a very long time to establish, so at short time intervals, one may consider a oneparametric family of quasisoliton solutions: in ref. 18, these are solutions differing in their “thickness”, i.e. distance between the front and the back. The rules of collision there are more complicated, e.g. there could be “completely inelastic” collisions, where of the two colliding waves one survives and the other annihilates.
Finally, we emphasize that the reflection from boundaries shown in Fig. 2 is for homogeneous Neumann boundary condition. Replacing those with e.g. homogeneous Dirichlet boundary conditions makes reflection much less likely. In particular, it is not observed in any of the six cases shown in Figs 1 and 2, although can be observed for other parameter values, for instance for a = 0.03, and even then the reflected wave takes much longer to fully recover. In this aspect, the quasisolitons are also different from the true solitons, e.g. in the NLS. A formal way to understand this difference is to observe that NLS is symmetric with respect to inversion of the sign of its complex field. Hence one can arrange two identical but counterpropagating solitons on an infinite line so that their nonlinear superposition at a certain point remains exactly zero at all times. Then replacing the problem with the one at halfline and zero boundary condition at that point will yield a solution in the form of a soliton reflecting from the boundary with a 180° change of phase. This construction does not work for the FitzHughNagumo kinetics (14) which is not invariant with respect to the inversion (u, v) → (−u, −v), unless a = −1, but in the latter case the system is no longer excitable. A more intuitive way to explain this is: for the pulse way to propagate, the u field must exceed the threshold a, and the boundary condition u = 0 makes it much more difficult for the reflected wave to satisfy this.
Discussion
It has been traditionally believed that a definitive property of waves in excitable media is that they annihilate when collided. Although solitonlike interaction was observed in some reactiondiffusion systems with excitable kinetics, both in numerical simulations^{19,20} and in experiments^{21,22}, solitons are mostly studied in fully integrable systems (KdV, sinGordon, nonlinear Schrödinger). Perhaps the main reason of this view on excitable media was that solitonlike interactions were always limited to narrow parameter ranges close to the boundaries between excitable and oscillatory (limit cycle) regimes of the reaction kinetics. A crucial role in the change of the attitude to excitable media as a source of solitons was played by experimental and theoretical works by Vanag and Epstein^{6,23,24,25}. They have demonstrated reactiondiffusion systems with solitonlike interaction of waves, and also spontaneous formation of wave packets. At the same time, we have shown that in excitable systems with crossdiffusion, the solitonlike behaviour of waves can be quite typical, including solutions similar to group (envelope) solitons^{12}. These works resonate with Vanag and Epstein’s reports of crossdiffusion in the chemical system BZAOT^{6}. In the present work, we have demonstrated that quasisolitons, including group (envelope) quasisolitons, can observed in reactiondiffusion systems with selfdiffusion only. This has been found with the help of twocomponent systems with effective crossdiffusion, which are obtained by semirigorous adiabatric reduction of a multicomonent reactiondiffusion system with selfdiffusion only. Adiabatic elimination of fast fields, which gives rise to nontrivial dissipative terms, that, for physical reasons, cannot exist in the straightforward form, can be observed in various physical settings. For instance, in nonlinear optics, adiabatic elimination of the acoustic field gives rise to an extra term in the NLS equation for the optical field, that is similar to stimulated Raman scattering which cannot appear in that equation directly, thus dubbed “pseudostimulatedRamanscattering”^{26,27,28}. By the analogy with that result, the effective crossdiffusion we described here could be called “pseudocrossdiffusion”. For the purposes of the present communication, an important feature of the effective crossdiffusion is that the resulting diffusivity matrix is not constraint by the thermodynamic restrictions of symmetry and positive definiteness. We believe that application of such reduction, accounting for the emergence of effective crossdiffusion, may lead to finding new interesting regimes in systems that have traditionally been studied without crossdiffusion, e.g. Brusselator^{29} and Oregonator^{30,31}, which remains an interesting direction for further study.
Additional Information
How to cite this article: Biktashev, V. N. and Tsyganov, M. A. Quasisolitons in selfdiffusive excitable systems, or Why asymmetric diffusivity obeys the Second Law. Sci. Rep. 6, 30879; doi: 10.1038/srep30879 (2016).
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Acknowledgements
VNB is supported in part by EPSRC grant EP/N014391/1 (UK).
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Biktashev, V., Tsyganov, M. Quasisolitons in selfdiffusive excitable systems, or Why asymmetric diffusivity obeys the Second Law. Sci Rep 6, 30879 (2016). https://doi.org/10.1038/srep30879
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