Existence of wavefronts and impulses to FitzHugh–Nagumo equations

Consideration is given to the FitzHugh–Nagumo equations <f>ut=uxx-f(u)-v,vt=#x03B5;(u-γv),</f> where <f>f(u)=u(u-1)(u-a)</f>, <f>0<a<1/2</f>, <f>#x03B5;>0,γ>9/((1-2a)(2-a))</f>. It is shown that there exist some <f>co>0</f> so that the system has a heteroclinic travelling wave solution with wave speed <f>co</f> for any <f>#x03B5;>0</f>, and there exists <f>coo∈(0,co)</f>, such that (1.2) has a homoclinic solution with wave speed <f>coo</f> if and only if <f>#x03B5;γ2<1</f>. [Copyright &y& Elsevier]