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The dynamics of pulse solutions in a bistable reaction-diffusion system arestudied analytically by reducing partial differential equations (PDEs) tofinite-dimensional ordinary differential equations (ODEs). For the reduction,we apply the multiple-scales method to the mixed ODE-PDE system obtained bytaking a singular limit of the PDEs. The reduced equations describe theinterface motion of a pulse solution formed by two interacting front solutions.This motion is in qualitatively good agreement with that observed for theoriginal PDE system. Furthermore, it is found that the reduction not onlyfacilitates the analytical study of the pulse solution, especially thespecification of the onset of local bifurcations, but also allows us toelucidate the global bifurcation structure behind the pulse behavior. As anapplication, the pulse dynamics in a heterogeneous bump-type medium areexplored numerically and analytically. The reduced ODEs clarify the transitionmechanisms between four pulse behaviors that occur at different parametervalues.

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