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

TL;DR

This paper establishes the existence and uniqueness of monotone traveling wave solutions in certain unidirectional lattice differential equations for large wave speeds, including variational characterizations and coexistence phenomena.

Contribution

It introduces a variational approach to determine critical wave speeds and analyzes both monotone and non-monotone wave solutions in lattice differential equations.

Findings

Existence and uniqueness of monotone traveling waves for large speeds

Variational characterization of critical wave speed

Coexistence of monotone and non-monotone waves

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 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.