Classification of wave regimes in excitable systems with linear cross diffusion

Abstract

We consider principal properties of various wave regimes in two selected excitable systems with linear crossdiffusion in one spatial dimension observed at different parameter values. This includes fixed-shape propagating waves, envelope waves, multi-envelope waves, and intermediate regimes appearing as waves propagating fixedshape most of the time but undergoing restructuring from time to time. Depending on parameters, most of these regimes can be with and without the "quasi-soliton" property of reflection of boundaries and penetration through each other. We also present some examples of behaviour of envelope quasi-solitons in two spatial dimensions.

Quasisolitons in self-diffusive excitable systems, or Why asymmetric diffusivity obeys the Second Law

“…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 [12,2], and can be described as modulated high-frequency 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.…”

“…There is a large variety of interesting regimes in systems with nonlinear kinetics and cross-diffusion instead of or in additional to self-diffusion [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.…”

Solitary waves in the excitable Burridge–Knopoff model

“…It would be also relevant to analyse pulse dynamics near boundaries or defects and to study the interaction between pulses. For example, transitions from pulse annihilation to crossing have been studied in different continuum excitable models such as reactiondiffusion or cross-diffusion systems [56,[85][86][87]. Considering the BK model, it would be interesting to analyse the possible effects of spatial discreteness and nonsmoothness (in particular for multivalued laws) in such collision processes.…”

Cross-Diffusion Waves as a trigger for multiscale, multiphysics Instabilities: Application to earthquakes

“…When adding a cross-diffusion term autowaves can turn into standing waves (Berenstein and Beta, 2012) having completely different properties to classical standing wave solitons. This difference defines a new class of dissipative waves which, according to Tsyganov and Biktashev (2014), presents an entirely different "world" to the waves encountered in integrable conservative systems such as the elastic-wave phenomenon. This phenomenon will be discussed further in the section on cross-diffusion waveforms where we will discuss the most important property for nucleation of earthquakes.…”