Stability of localized solutions to lattice dynamical systems

TL;DR

This paper introduces a theoretical framework for analyzing the spectral stability of localized solutions in lattice dynamical systems, using Evans functions to count unstable eigenvalues in multi-dimensional settings.

Contribution

It develops a general approach leveraging front and back solutions and Evans functions to analyze stability of localized states in high-dimensional lattices.

Findings

The Evans function factorizes asymptotically for well-separated localization regions.

The framework applies to multi-pulse and oscillatory localized solutions.

Numerical examples demonstrate bifurcation structures and eigenvalue spectra.

Abstract

Localized patterns are spatially confined structures that arise in lattice dynamical systems and play an important role in physics, biology, and materials science. While their existence and bifurcation structure are well-understood, the stability of these solutions remains largely unexplored, particularly in discrete and high-dimensional settings. In this work, we develop a general theoretical framework to analyze the spectral stability of localized steady states in one-dimensional and multi-dimensional rectangular lattices. Our approach leverages the properties of front and back solutions, combined with a discrete Evans function formulation, to characterize the spectrum of localized solutions. We prove that, for well-separated regions of localization, the Evans function asymptotically factorizes into contributions from the underlying fronts and backs, allowing explicit counting of unstable eigenvalues. This framework applies to solutions with single or multiple plateaus, including oscillatory and multi-pulse configurations. We illustrate the results on a real-valued cubic-quintic Ginzburg-Landau lattice, a prototypical Nagumo-type system, and provide numerical demonstrations of bifurcation structures and eigenvalue spectra.