Existence and Uniqueness of Traveling Waves in a Class of Unidirectional Lattice Differential Equations

Aaron Hoffman, Franklin W. Olin College of Engineering
Benjamin Kennedy, Gettysburg College

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© 2011 American Institute of Mathematical Sciences. This article was published in Discrete and Continuous Dynamical Systems - Series A, vol. 30, iss. 1, pages 137-167 and may be found here.

Abstract

We prove the existence and uniqueness, for wave speeds sufficiently large, of monotone traveling wave solutions connecting stable to unstable spatial equilibria for a class of -dimensional lattice differential equations with unidirectional coupling. This class of lattice equations includes some cellular neural networks, monotone systems, and semi-discretizations for hyperbolic conservation laws with a source term. We obtain a variational characterization of the critical wave speed above which monotone traveling wave solutions are guaranteed to exist. We also discuss non-monotone waves, and the coexistence of monotone and non-monotone waves.