Toward nonlinear stability of sources via a modified Burgers equation

TL;DR

This paper investigates the nonlinear stability of source solutions in reaction-diffusion systems by analyzing a modified Burgers equation model, revealing how outward transport affects stability and wave organization.

Contribution

It introduces a modified Burgers equation as a simplified model to analyze the nonlinear stability of sources with outward group velocities in reaction-diffusion systems.

Findings

Demonstrates how outward transport influences stability analysis.

Establishes nonlinear stability results for the modified Burgers model.

Provides insights into wave number selection and source dynamics.

Abstract

Coherent structures are solutions to reaction-diffusion systems that are time-periodic in an appropriate moving frame and spatially asymptotic at $x=\pm\infty$ to spatially periodic travelling waves. This paper is concerned with sources which are coherent structures for which the group velocities in the far field point away from the core. Sources actively select wave numbers and therefore often organize the overall dynamics in a spatially extended system. Determining their nonlinear stability properties is challenging as localized perturbations may lead to a non-localized response even on the linear level due to the outward transport. Using a modified Burgers equation as a model problem that captures some of the essential features of coherent structures, we show how this phenomenon can be analysed and nonlinear stability be established in this simpler context.