  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.
