Existence and Stability of Multiple Impulse Solutions of a Nerve Equation

A piecewise linear partial differential equation which models the propagation of a nerve impulse along an axon is studied. For the equation parameters used there are two solutions which represent solitary impulses traveling with fixed form, one with a slower velocity cₛ₁ than the velocity cf1 of the other. We show that near these velocities there exist multiple impulse solutions, and exhibit four such solutions each of which has two pulses. The local stability of these solutions is analyzed. The two solutions whose velocities are near cₛ₁ demonstrate instabilities which indicate growth or decay of the separate pulses. For one of the multiple impulse solutions with velocity near cₛ₁ and for one of those with velocity near cf1, the second pulse deomonstrates an instability which indicates a change in the spacing between the two pulses. The other solution with velocity near cf1 is stable.