The $\Gamma$-limit of traveling waves in the FitzHugh-Nagumo system

TL;DR

This paper extends the application of $ ext{Gamma}$-convergence to analyze the asymptotic behavior of traveling wave solutions in the FitzHugh-Nagumo system, a model for biological wave phenomena.

Contribution

It is the first work to apply $ ext{Gamma}$-convergence to non-stationary, traveling wave problems in the FitzHugh-Nagumo system, broadening its scope.

Findings

Established the $ ext{Gamma}$-limit for traveling waves in the FitzHugh-Nagumo system.

Extended the applicability of $ ext{Gamma}$-convergence to non-stationary problems.

Provided a variational framework for analyzing wave dynamics.

Abstract

Patterns and waves are basic and important phenomena that govern the dynamics of physical and biological systems. A common theme in investigating such systems is to identify the intrinsic factors responsible for such self-organization. The $\Gamma$-convergence is a well-known technique applicable to variational formulation in studying the concentration phenomena of stable patterns. A geometric variational functional associated with the $\Gamma$-limit of standing waves of FitzHugh-Nagumo system has recently been built. This article studies the $\Gamma$-limit of traveling waves. To the best of our knowledge, this is the first attempt to expand the scope of applicability of $\Gamma$-convergence to cover non-stationary problems.