Single and Multiple Pulse Waves for the FitzHugh–Nagumo

The FitzHugh–Nagumo equations describe the propagation of impulses along a nerve axon. We study traveling wave solutions, which satisfy a set of ordinary differential equations with a cubic nonlinearity $f( v ) = v ( 1 - v )( v - a )$. Our emphasis is on solutions which tend to zero at ± ∞; an improved proof of a previous result, that there are at least two such solutions, is presented. Further, by allowing the middle root $v = a$ of the nonlinearity to be a small parameter, better estimates of the parameter range where such solutions exist are given. In view of recent work of Evans, Fenichel, Feroe and Šil’nikov, our results imply that there are a great many more bounded traveling waves than was known previously. In particular, there is an infinite family of periodic waves all traveling with the same speed. A reasonable transversality hypothesis introduced by Evans, Fenichel and Feroe implies also that there are multiple-pulse waves at speeds close to that of a single pulse.