Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2605.12605

Stability of localized solutions to lattice dynamical systems

Bocheng Ruan·Jack M. Hughes·Jason J. Bramburger
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

nlin.PS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Bocheng Ruan, Jack M. Hughes, Jason J. Bramburger. 2026-05-12. Stability of localized solutions to lattice dynamical systems. https://arxiv.org/abs/2605.12605

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Duck hunting with quantum mechanics

We bridge two sides of singular perturbation theory: the classical theory of slow-fast systems and the semi-classical approach to quantum mechanical systems. For a specific but physically important class of dynamical systems, we show that purely classical and exotic objects, so-called canard solutions, are shadows of instantons in the corresponding quantum system. We demonstrate that canard solutions exist in a domain of parameter space whose boundaries are determined by an instanton action. We illustrate our statements analytically for the relevant example, the overdamped Josephson junction, and confirm them numerically. For the Josephson junction, the canard window is the exponentially narrow gap between consecutive Shapiro steps.

nlin.PS↗
arXiv2026

Rolls and Snaking in a Swift-Hohenberg Equation with Non-smooth Nonlinearity

We study rolls and homoclinic snaking in a variation of the one-dimensional Swift-Hohenberg equation, whose standard forms are prototypical order-parameter models for pattern formation in the sciences. Motivated by classes of differential equation models that involve continuous non-smooth low order nonlinear terms, we replace the standard quadratic-cubic nonlinearity by $ν|u|^α-u^3$, $α\in [1,2]$ with $ν> 0$. In the vicinity of zero, for $α<2$ this nonlinearity falls outside the scope of classical Taylor expansion and bifurcation analysis. Our partially analytical and partially numerical results highlight that the non-smooth term modifies the criticality of pattern-forming bifurcations and alters the associated branches of solutions. In particular, $α\in(1,2)$ implies subcriticality of roll bifurcations for any $ν>0$. At $α=1$ differentiability is lost, which has a strong impact on the bifurcations of sign-changing rolls including the disappearance of homoclinic snaking. Homoclinic snaking thus emerges non-smoothly as $α$ increases from $α=1$, and persists when retaining an additional, e.g., quadratic term.

nlin.PS↗