Existence of standing pulse solutions to a skew-gradient system

TL;DR

This paper proves the existence of standing pulse solutions in a nonlinear skew-gradient reaction-diffusion system, expanding understanding of localized patterns beyond linear inhibitor models.

Contribution

It introduces a variational method to establish standing pulse solutions with nonlinear reaction terms, addressing a gap in existing research.

Findings

Existence of standing pulse solutions with sign change proven.

Qualitative properties of these solutions analyzed.

Extension to nonlinear inhibitor effects in skew-gradient systems.

Abstract

Reaction-diffusion systems have been primary tools for studying pattern formation. A skew-gradient system is well known to encompass a class of activator-inhibitor type reaction-diffusion systems that exhibit localized patterns such as fronts and pulses. While there is a substantial literature for the case of a linear inhibitor equation, the study of nonlinear inhibitor effect is still limited. To fill this research gap, we investigate standing pulse solutions to a skew-gradient system in which both activator and inhibitor reaction terms inherit nonlinear structures. Using a variational approach that involves several nonlocal terms, we establish the existence of standing pulse solutions with a sign change. In addition, we explore some qualitative properties of the standing pulse solutions.