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

Kei Nishi, Yasumasa Nishiura, Takashi Teramoto

Research output: Contribution to journalArticle

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.

Original languageEnglish
Pages (from-to)183-207
Number of pages25
JournalPhysica D: Nonlinear Phenomena
Volume398
DOIs
Publication statusPublished - 2019 Nov

Keywords

  • Bifurcation theory
  • Localized patterns
  • Reaction–diffusion system
  • Reduced equations

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

Fingerprint Dive into the research topics of 'Reduction approach to the dynamics of interacting front solutions in a bistable reaction–diffusion system and its application to heterogeneous media'. Together they form a unique fingerprint.

  • Cite this