Wavy fronts and speed bifurcation in excitable systems with cross diffusion

Cross-diffusion waves by cellular automata modeling: Pattern formation in porous media

Porous earth materials exhibit large-scale deformation patterns, such as deformation bands, which emerge from complex small-scale interactions. This paper introduces a cross-diffusion framework designed to capture these multiscale, multiphysics phenomena, inspired by the study of multi-species chemical systems. A microphysics-enriched continuum approach is developed to accurately predict the formation and evolution of these patterns. Utilizing a cellular automata algorithm for discretizing the porous network structure, the proposed framework achieves substantial computational efficiency in simulating the pattern formation process. This study focuses particularly on a dynamic regime termed "cross-diffusion wave," an instability in porous media where cross-diffusion plays a significant role-a phenomenon experimentally observed in materials like dry snow. The findings demonstrate that external thermodynamic forces can initiate pattern formation in cross-coupled dynamic systems, with the propagation speed of deformation bands primarily governed by cross-diffusion and a specific cross-reaction coefficient. Owing to the universality of thermodynamic force-flux relationships, this study sheds light on a generic framework for pattern formation in cross-coupled multi-constituent reactive systems.

Transport-driven chemical oscillations: a review

Chemical oscillators attract transversal interest not only as useful models for understanding and controlling (bio)chemical complexity far from the equilibrium, but also as a promising means to design smart materials and power synthetic functional behaviors. We review and classify oscillatory phenomena in systems where a periodic variation in the concentration of the constitutive chemical species is induced by transport instabilities either triggered by simple reactions or without any reactive process at play. These phenomena, where the origin of the dynamical complexity is shifted from chemical to physical nonlinearities, can facilitate a variety of processes commonly encountered in chemistry and chemical engineering. We present an excursus through the main examples, discussing phenomenology, properties and modeling of different mechanisms that can lead to these kinds of oscillations. In particular, we reproduce the relevant results reported in the pertinent literature and, in parallel, propose new kinds of proof-of-concept systems substantiated by preliminary studies which can inspire new research lines. In the landscape of physically driven chemical oscillations, we devote particular attention to transport phenomena, actively or passively combined to (reactive or nonreactive) chemical species, which provide multiple pathways towards spontaneous oscillatory instabilities. Though with different specificities, the great part of these systems can be reduced to a common theoretical description. We finally overview possible perspectives in the study of physically driven oscillatory instabilities, showing how the related control can impact fundamental and applied open problems, ranging from origin of life studies to the optimization of processes with environmental relevance.

Solitary pulses and periodic wave trains in a bistable FitzHugh-Nagumo model with cross diffusion and cross advection

We describe analytically, and simulate numerically, traveling waves with oscillatory tails in a bistable, piecewise-linear reaction-diffusion-advection system of the FitzHugh-Nagumo type with linear cross-diffusion and cross-advection terms of opposite signs. We explore the dynamics of two wave types, namely, solitary pulses and their infinite sequences, i.e., periodic wave trains. The effects of cross diffusion and cross advection on wave profiles and speed of propagation are analyzed. For pulses, in the speed diagram splitting of a curve into several branches occurs, corresponding to different waves (wave branching). For wave trains, in the dispersion relation diagram there are oscillatory curves and the discontinuous curve of an isola with two branches. The corresponding wave trains have symmetric or asymmetric profiles. Numerical simulations show that for large values of the period there exist two wave trains, which come closer and closer together and are subject to fusion into one when the value of the period is decreasing. Other types of waves are also briefly discussed.

Nonlinear waves in a quintic FitzHugh-Nagumo model with cross diffusion: Fronts, pulses, and wave trains

We study a tristable piecewise-linear reaction-diffusion system, which approximates a quintic FitzHugh-Nagumo model, with linear cross-diffusion terms of opposite signs. Basic nonlinear waves with oscillatory tails, namely, fronts, pulses, and wave trains, are described. The analytical construction of these waves is based on the results for the bistable case [Zemskov et al., Phys. Rev. E 77, 036219 (2008) and Phys. Rev. E 95, 012203 (2017) for fronts and for pulses and wave trains, respectively]. In addition, these constructions allow us to describe novel waves that are specific to the tristable system. Most interesting is the pulse solution with a zigzag-shaped profile, the bright-dark pulse, in analogy with optical solitons of similar shapes. Numerical simulations indicate that this wave can be stable in the system with asymmetric thresholds; there are no stable bright-dark pulses when the thresholds are symmetric. In the latter case, the pulse splits up into a tristable front and a bistable one that propagate with different speeds. This phenomenon is related to a specific feature of the wave behavior in the tristable system, the multiwave regime of propagation, i.e., the coexistence of several waves with different profile shapes and propagation speeds at the same values of the model parameters.

Oscillatory multipulsons: Dissipative soliton trains in bistable reaction-diffusion systems with cross diffusion of attractive-repulsive type

One-dimensional localized sequences of bound (coupled) traveling pulses, wave trains with a finite number of pulses, are described in a piecewise-linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross-diffusion terms of opposite signs. The simplest case of two bound pulses, the paired-pulse waves (pulse pairs), is solved analytically. The solutions contain oscillatory tails in the wave profiles so that the pulse pairs consist of a double-peak core and wavy edges. Several pulse pairs with different profile shapes and propagation speeds can appear for the same parameter values of the model when the cross diffusion is dominant. The more general case of many bound pulses, multipulse waves, is studied numerically. It is shown that, dependent on the values of the cross-diffusion coefficients, the multipulse waves upon collision can pass through one another with unchanged size and shape, exhibiting soliton behavior. Moreover, multipulse collisions with the system boundaries can generate a rich variety of wave transformations: the transition from the multipulse waves to pulse-front waves and further to simple fronts or to annihilation as well the transition to solitary pulses or to multipulse waves with lower numbers of pulses. Analytical and numerical results for the pulse pairs agree well with each other.

Multifront regime of a piecewise-linear FitzHugh-Nagumo model with cross diffusion

Oscillatory reaction-diffusion fronts are described analytically in a piecewise-linear approximation of the FitzHugh-Nagumo equations with linear cross-diffusion terms, which correspond to a pursuit-evasion situation. Fundamental dynamical regimes of front propagation into a stable and into an unstable state are studied, and the shape of the waves for both regimes is explored in detail. We find that oscillations in the wave profile may either be negligible due to rapid attenuation or noticeable if the damping is slow or vanishes. In the first case, we find fronts that display a monotonic profile of the kink type, whereas in the second case the oscillations give rise to fronts with wavy tails. Further, the oscillations may be damped with exponential decay or undamped so that a saw-shaped pattern forms. Finally, we observe an unexpected feature in the behavior of both types of the oscillatory waves: the coexistence of several fronts with different profile shapes and propagation speeds for the same parameter values of the model, i.e., a multifront regime of wave propagation.

Oscillatory pulse-front waves in a reaction-diffusion system with cross diffusion

We explore traveling waves with oscillatory tails in a bistable piecewise linear reaction-diffusion system of the FitzHugh-Nagumo type with linear cross diffusion. These waves differ fundamentally from the standard simple fronts of the kink type. In contrast to kinks, the waves studied here have a complex shape profile with a front-back-front (a pulse-front) pattern. The characteristic feature of such pulse-front waves is a hybrid type of the speed diagram, which on the one hand reflects the typical dynamical behavior of the fronts in the FitzHugh-Nagumo model, related to the nonequilibrium Ising-Bloch bifurcation, and on the other hand exhibits also the solitary pulse scenario where several waves appear simultaneously with different speeds of propagation. We describe analytically the wave profiles and heteroclinic trajectories in the phase plane and discuss their morphology and transformation. The phenomena of wave formation and propagation are also studied by numerical simulations of the model partial differential equations. These simulations support the view that the pulse-front waves are constructed of fronts and pulses.

Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion

We study waves with exponentially decaying oscillatory tails in a reaction-diffusion system with linear cross diffusion. To be specific, we consider a piecewise linear approximation of the FitzHugh-Nagumo model, also known as the Bonhoeffer-van der Pol model. We focus on two types of traveling waves, namely solitary pulses that correspond to a homoclinic solution, and sequences of pulses or wave trains, i.e., a periodic solution. The effect of cross diffusion on wave profiles and speed of propagation is analyzed. We find the intriguing result that both pulses and wave trains occur in the bistable cross-diffusive FitzHugh-Nagumo system, whereas only fronts exist in the standard bistable system without cross diffusion.

Cross-diffusion-driven hydrodynamic instabilities in a double-layer system: General classification and nonlinear simulations

Cross diffusion, whereby a flux of a given species entrains the diffusive transport of another species, can trigger buoyancy-driven hydrodynamic instabilities at the interface of initially stable stratifications. Starting from a simple three-component case, we introduce a theoretical framework to classify cross-diffusion-induced hydrodynamic phenomena in two-layer stratifications under the action of the gravitational field. A cross-diffusion-convection (CDC) model is derived by coupling the fickian diffusion formalism to Stokes equations. In order to isolate the effect of cross-diffusion in the convective destabilization of a double-layer system, we impose a starting concentration jump of one species in the bottom layer while the other one is homogeneously distributed over the spatial domain. This initial configuration avoids the concurrence of classic Rayleigh-Taylor or differential-diffusion convective instabilities, and it also allows us to activate selectively the cross-diffusion feedback by which the heterogeneously distributed species influences the diffusive transport of the other species. We identify two types of hydrodynamic modes [the negative cross-diffusion-driven convection (NCC) and the positive cross-diffusion-driven convection (PCC)], corresponding to the sign of this operational cross-diffusion term. By studying the space-time density profiles along the gravitational axis we obtain analytical conditions for the onset of convection in terms of two important parameters only: the operational cross-diffusivity and the buoyancy ratio, giving the relative contribution of the two species to the global density. The general classification of the NCC and PCC scenarios in such parameter space is supported by numerical simulations of the fully nonlinear CDC problem. The resulting convective patterns compare favorably with recent experimental results found in microemulsion systems.

Wavy fronts in a hyperbolic FitzHugh-Nagumo system and the effects of cross diffusion

We study a hyperbolic version of the FitzHugh-Nagumo (also known as the Bonhoeffer-van der Pol) reaction-diffusion system. To be able to obtain analytical results, we employ a piecewise linear approximation of the nonlinear kinetic term. The hyperbolic version is compared with the standard parabolic FitzHugh-Nagumo system. We completely describe the dynamics of wavefronts and discuss the properties of the speed equation. The nonequilibrium Ising-Bloch bifurcation of traveling fronts is found to occur in the hyperbolic case as well as in the parabolic system. Waves in the hyperbolic case typically propagate with lower speeds, in absolute value, than waves in the parabolic one. We find the interesting feature that the hyperbolic and parabolic front trajectories coincide in the phase plane for the FitzHugh-Nagumo model with a diagonal diffusion matrix, which is the case of self-diffusion, and differ for the system with cross diffusion.

Cross-diffusion-induced convective patterns in microemulsion systems

Cross-diffusion phenomena are experimentally and theoretically shown to be able to induce convective fingering around an initially stable stratification of two microemulsions with different compositions.

Self-sustained peristaltic waves: explicit asymptotic solutions

A simple nonlinear model for the coupled problem of fluid flow and contractile wall deformation is proposed to describe peristalsis. In the context of the model the ability of a transporting system to perform autonomous peristaltic pumping is interpreted as the ability to propagate sustained waves of wall deformation. Piecewise-linear approximations of nonlinear functions are used to analytically demonstrate the existence of traveling-wave solutions. Explicit formulas are derived which relate the speed of self-sustained peristaltic waves to the rheological properties of the transporting vessel and the transported fluid. The results may contribute to the development of diagnostic and therapeutic procedures for cases of peristaltic motility disorders.