Branching of double pulse solutions from single pulse solutions in nerve axon equations (Q1112999)

Considered is a generalized FitzHugh nerve equation: \[ (1)\quad u_ t=Du_{xx}+f(u;\mu),\quad u(x,t)=(u_ 1,...,u_ m)^ t,\quad f(u;\mu)=(f_ 1,...,f_ m)^ t,\quad D=diag(1,0,...,0), \] where \(\mu\) is a parameter expressing a physiological condition and \(f(0;\mu)=0\) for all \(\mu\). It is assumed that (1) has a single pulse solution \(u(x,t)=u^ 1(z;\mu),\) \(z=x+C^ 1(\mu)t\) in a neighbourhood of \(\mu =\mu_ 0\). The author establishes a necessary and sufficient condition for the existence of double pulse solutions \(u^ 2=u^ 2(z;\mu),\) \(z=x+C^ 2(\mu)t\) branching from the single pulse solution at \(\mu =\mu_ 0\) and studies the direction of the branching of the curve \(C=C^ 2(\mu)\).