Abstract

 

In the study of FitzHugh-Nagumo equations, we look for a standing pulse with profile staying close to a trivial background state except in one localized spatial region where the change is substantial. This amounts to seeking a homoclinic orbit for a corresponding Hamiltonian system and we utilize a variational formulation which involves a nonlocal term. Such a functional is referred to as Helmholtz free energy in modeling microphase separation in diblock copolymers, while its global minimizer does not exist in our setting of dealing with standing pulse. The homoclinic orbit obtained here is a local minimizer extracted from a suitable topological class of admissible functions. In contrast with the known results for positive standing pulses in literature,  new technique has been attempted in seek of standing pulse solution with sign change.