Multidimensional Stability of Waves Travelling Through Rectangular Lattices in Rational Directions

TL;DR

This paper proves the nonlinear stability of traveling wave solutions in reaction-diffusion systems on rectangular lattices in rational directions, using advanced Green's function estimates to handle complex cases without comparison principles.

Contribution

It introduces a novel stability analysis for multidimensional lattice waves in rational directions using point-wise Green's function techniques, extending previous methods to more general systems.

Findings

Traveling waves in rational directions are nonlinearly stable.

New analytical techniques enable stability analysis without comparison principles.

Results apply to complex reaction-diffusion lattice systems.

Abstract

We consider general reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. We show that travelling wave solutions to such systems that propagate in rational directions are nonlinearly stable under small perturbations. We employ recently developed techniques involving point-wise Green's functions estimates for functional differential equations of mixed type (MFDEs), allowing our results to be applied even in situations where comparison principles are not available.