Traveling pulses in a coupled FitzHugh–Nagumo equation

Abstract

For a coupled FitzHugh–Nagumo (FHN) equation with a parameter a∈[0,1], when a∈(0,1∕2) the existence of traveling pulses of this equation was proved in 2013 by Holzer, Doelman and Kaper. Here we adopt a new approach to obtain the existence of the traveling pulse of the same equation for a∈[0,1∕2), which includes the degenerate case a=0, where the origin on the critical set loses normal hyperbolicity while it is for a∈(0,1). Also we show that the pulse does not exhibit an oscillatory tail at the homoclinic orbit when the time t→±∞, whereas the classical FHN equation could have an oscillatory tail of the traveling pulse depending on the choice of the parameters of the system. Finally we present an explanation on why traveling pulses cannot exist when a∈[1∕2,1].