Traveling waves of infinitely many pulses in nerve equations

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Abstract

Some nerve axon equations, in addition to admitting a traveling wave solution in the form of a solitary pulse, also admit a family of double pulse solutions. In this case, application and extension of the results of Silnikov further show the existence of an uncountable discrete family of traveling wave solutions which are trains of infinitely many pulses. All solutions in this family travel with the velocity of the solitary pulse solution. Included are nonperiodic doubly infinite trains, trains which are infinite in only one direction, and a countable family of periodic solutions. The set of integer sequences which identify the spacing between individual pulses in a solution serve as a natural index for the family. A proof and computational results are provided for the piecewise linear FitzHugh- Nagumo equations.

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