Transition from circular to spiral waves and from Mexican hat to upside-down Mexican hat-solutions: The cases of local and nonlocal λ−ω reaction-diffusion-convection fractal systems with variable coefficients

Abstract

Nonlinear partial differential equations admitting traveling wave solutions play an important role in the description and analysis of real-life physical processes and nonlinear phenomena. In this study, we prove that the excitable $\lambda -\omega$reaction-diffusion-convection system introduced by Kopell and Howard can exhibit, in fractal dimensions, a large variety of spatial patterns. We have considered two independent models: a local reaction-diffusion-convection model characterized by variable coefficients that are subject to particular power laws and a nonlocal reaction-diffusion model characterized by symmetric kernels and a variable diffusion coefficient. Each model is characterized by a number of motivating properties and features. In the 1st model, the amplitude is governed by a 2nd-order differential equation, whereas in the 2nd-model, the amplitude is governed by a 4th-order differential equation, which is, under some conditions, comparable to the Swift-Hohenberg equation with variable coefficients that arise in the study of pattern formation, which belongs to the family of extended Fisher-Kolmogorov stationary equations used to study pattern-forming systems in biological and chemical systems. We report the emergence of superstructures that are suppressed for fractal dimensions much less than unity. These superstructures include superspiral waves characterized by a circular symmetry detected in various oscillatory media and the emergence of reflection of waves that take place in non-uniform reaction-diffusion systems, besides the emergence of micro-spiral waves that emerge at the cellular level. A transition from spiral waves to perfectly rotating waves is observed, besides a transition from Mexican hat shaped solutions to upside-down Mexican hat shaped solutions. The domain size has a very strong impact on the rotational frequency of spiral and circular waves. These new phenomena associated with configuration patterns through a reaction-diffusion-convection system with different scales and characterized by variable coefficients can be applied for modeling a wide class of reaction-diffusion-convection problems. Supplementary properties have been obtained and discussed accordingly.