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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations

Pages: 137 - 167, Volume 30, Issue 1, May 2011      doi:10.3934/dcds.2011.30.137

 
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Aaron Hoffman - Franklin W. Olin College of Engineering, 1000 Olin Way, Needham, MA 02492-1200, United States (email)
Benjamin Kennedy - Gettysburg College, Department of Mathematics, 300 North Washington St., Gettysburg, PA 17325-1400, United States (email)

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 $N$-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.

Keywords:  Lattice differential equation, traveling wave, monostable.
Mathematics Subject Classification:  Primary: 37L60, 34K10; Secondary: 34C19, 34K19

Received: October 2009;      Revised: November 2010;      Available Online: February 2011.

 References