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Directed Graphs and Traveling Waves.

Abstract : The equation studied here has been considered as a model for a variety of physical phenomena including population genetics and nerve conduction. Of primary interest is the limiting behavior of solutions of this equation. One expects the solution eventually to look like a traveling wave solution; that is, one which moves with constant shape and velocity. This paper determines all of the traveling wave solutions of the equation, showing there are situations when the number of traveling wave solutions can be infinite.