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 @math -dimensional lattice differential equations with unidirectional coupling. This class of lattice equations includes some spatial discretizations for hyperbolic conservation laws with a source term as well as a subclass of monotone systems. 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.
Abstract This work proves the existence and multiplicity results of monotonic traveling wave solutions for some lattice differential equations by using the monotone iteration method. Our results include the model of cellular neural networks (CNN). In addition to the monotonic traveling wave solutions, non-monotonic and oscillating traveling wave solutions in the delay type of CNN are also obtained. The theory of a novel class of information-processing systems, called cellular neural networks, which are capable of high-speed parallel signal processing, was presented in a previous paper (see ibid., vol.35, no.10, p.1257-72, 1988). A dynamic route approach for analyzing the local dynamics of this class of neural circuits is used to steer the system trajectories into various stable equilibrium configurations which map onto binary patterns to be recognized. Some applications of cellular neural networks to such areas as image processing and pattern recognition are demonstrated, albeit with only a crude circuit. In particular, examples of cellular neural networks which can be designed to recognize the key features of Chinese characters are presented. >