First and second order semi-strong interaction in reaction-diffusion systems

TL;DR

This paper develops a unified asymptotic framework to analyze the interaction laws of pulses and fronts in reaction-diffusion systems with scale separation, introducing first and second order semi-strong interactions.

Contribution

It introduces a general concept of first order semi-strong interaction, distinct from the previously studied second order, and derives laws of motion for pulses and fronts in a model-independent way.

Findings

First order semi-strong interaction is gradient-like under certain conditions.

Derived interaction laws for pulses and fronts in reaction-diffusion systems.

Applicable to models like Gray-Scott and Schnakenberg.

Abstract

Spatial scale separation often leads to sharp interfaces that can be fully localized pulses or transition layer fronts connecting different states. This paper concerns the asymptotic interaction laws of pulses and fronts in the so-called semi-strong regime of strongly differing diffusion lengths for reaction-diffusion systems in one space dimension. An asymptotic expansion and matching approach is applied in a model independent common framework. First order semi-strong interaction is introduced as a general interface interaction type. It is distinct from the semi-strong interaction studied over the past decade, which is referred to as `second order' here. Laws of motion are derived for pulses as well as fronts in abstract systems with attention to the effect of symmetries. First order interaction for pulses is shown to be gradient-like under conditions that are numerically checked for a class of equations including the Gray-Scott and Schnakenberg models.