Reduction approach to the dynamics of interacting front solutions in a bistable reaction-diffusion system and its application to heterogeneous media

TL;DR

This paper develops a reduction method from PDEs to ODEs to analyze pulse solutions in bistable reaction-diffusion systems, revealing bifurcation structures and pulse behaviors, with applications to heterogeneous media.

Contribution

It introduces a multiple-scales reduction technique for PDEs to ODEs, enabling analytical and numerical study of pulse dynamics and bifurcations in heterogeneous media.

Findings

Reduced equations accurately describe pulse interface motion

Bifurcation analysis reveals global structure of pulse behaviors

Transition mechanisms between pulse states are clarified

Abstract

The dynamics of pulse solutions in a bistable reaction-diffusion system are studied analytically by reducing partial differential equations (PDEs) to finite-dimensional ordinary differential equations (ODEs). For the reduction, we apply the multiple-scales method to the mixed ODE-PDE system obtained by taking a singular limit of the PDEs. The reduced equations describe the interface motion of a pulse solution formed by two interacting front solutions. This motion is in qualitatively good agreement with that observed for the original PDE system. Furthermore, it is found that the reduction not only facilitates the analytical study of the pulse solution, especially the specification of the onset of local bifurcations, but also allows us to elucidate the global bifurcation structure behind the pulse behavior. As an application, the pulse dynamics in a heterogeneous bump-type medium are explored numerically and analytically. The reduced ODEs clarify the transition mechanisms between four pulse behaviors that occur at different parameter values.