Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · nlin/0604075

q-breathers in finite two- and three-dimensional nonlinear acoustic lattices

M. V. Ivanchenko·O. I. Kanakov·K. G. Mishagin·S. Flach
View original source ↗Read source document ↗Export metadata ↓

Abstract

Nonlinear interaction between normal modes dramatically affects energy equipartition, heat conduction and other fundamental processes in extended systems. In their celebrated experiment Fermi, Pasta and Ulam (FPU, 1955) observed that in simple one-dimensional nonlinear atomic chains the energy must not always be equally shared among the modes. Recently, it was shown that exact and stable time-periodic orbits, coined $q$-breathers (QBs), localize the mode energy in normal mode space in an exponential way, and account for many aspects of the FPU problem. Here we take the problem into more physically important cases of two- and three-dimensional acoustic lattices to find existence and principally different features of QBs. By use of perturbation theory and numerical calculations we obtain that the localization and stability of QBs is enhanced with increasing system size in higher lattice dimensions opposite to their one-dimensional analogues.

Explore related subjects

nlin.PSnlin.SI

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

M. V. Ivanchenko, O. I. Kanakov, K. G. Mishagin, S. Flach. 2006-04-28. q-breathers in finite two- and three-dimensional nonlinear acoustic lattices. https://doi.org/10.1103/physrevlett.97.025505

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

KEEP EXPLORING

Related papers

arXiv2026

Routes to chaos in a mass-conserving two-species reaction-diffusion model

Mass-conserving reaction-diffusion systems with two species correspond to a seemingly simple case where pattern formation occurs under the influence of a conservation law. Here, we first revisit their linear stability behavior and point out that generically two instabilities can occur: a stationary large-scale mass-conserving (Cahn-Hilliard) instability and an instability that combines features of a stationary large-scale non-mass-conserving (Allen-Cahn) instability and a oscillatory large-scale mass-conserving (conserved-Hopf) instability. We term it an Allen-Cahn-Hopf instability. Second, we investigate the nonlinear dynamics for a specific model related to the formation of cell polarization where only a Cahn-Hilliard instability can occur, i.e., all primary bifurcations are stationary. We analyze how secondary and further bifurcations subsequently give rise to various oscillatory states. The emerging rich spectrum of spatiotemporal behavior includes several period-doubling cascades related to different forms of spatial and temporal symmetry breaking. Beside regular states, three types of low-dimensional spatiotemporal chaos occur and involve transitions like fusion and an outer crises. Our results demonstrate the importance of nonlinear interactions in the dynamics of mass-conserving reaction-diffusion systems, and show that even a simple two-species system with primary bifurcations of Cahn-Hilliard type can show complex spatiotemporal behavior.

nlin.PS↗
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↗