Journal ArticleDOI
The discrete Frenkel-Kontorova model and its extensions: I. Exact results for the ground-states
Serge Aubry,P.Y. Le Daeron +1 more
TLDR
A rigorous study of the ground states of one-dimensional models generalizing the discrete Frenkel-Kontorova model has been presented in this article, where the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic, quasi-periodic and chaotic orbits.About:
This article is published in Physica D: Nonlinear Phenomena.The article was published on 1983-09-01. It has received 607 citations till now. The article focuses on the topics: Boundary value problem.read more
Citations
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Journal ArticleDOI
Discrete Solitons in Optics
Falk Lederer,George I. Stegeman,Demetri N. Christodoulides,Gaetano Assanto,M. Segev,Yaron Silberberg +5 more
TL;DR: In this article, the authors provide an overview of recent experimental and theoretical developments in the area of optical discrete solitons, which represent self-trapped wavepackets in nonlinear periodic structures and result from the interplay between lattice diffraction (or dispersion) and material nonlinearity.
Journal ArticleDOI
Symplectic maps, variational principles, and transport
TL;DR: In this article, a Lagrangian variational formulation of twist maps is proposed to compute the flux escaping from regions bounded by partial barriers formed from minimizing orbits, which form a scaffold in the phase space and constrain the motion of remaining orbits.
Journal ArticleDOI
Action minimizing invariant measures for positive definite Lagrangian systems.
TL;DR: In this paper, a generalization of the notion of minimal measure to periodic positive definite Lagrangian systems in more degrees of freedom has been proposed, where the minimal measures can be expressed as Lipschitz sections of the tangent bundle.
Journal ArticleDOI
An analytical study of transport, mixing and chaos in an unsteady vortical flow
TL;DR: In this paper, the authors examined the transport properties of a particular two-dimensional, inviscid incompressible flow using dynamical systems techniques and derived an analytical estimate of the flux rate into and out of the vortex neighbourhood.
Journal ArticleDOI
Wave transmission in nonlinear lattices
D. Hennig,G.P. Tsironis +1 more
TL;DR: In this paper, wave transmission properties in one dimensional nonlinear lattices are discussed and the results from the theory of dynamical systems are used to investigate various aspects of wave transmission and wave localization.
References
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Book
Real and complex analysis
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
Journal ArticleDOI
Quantum theory of solids
Rudolf Peierls,Louis D. Roberts +1 more
TL;DR: In this article, the interaction of light with non-conducting crystals has been studied in the context of crystal lattices and its applications in general theory and applications, such as semi-conductivity and superconductivity.
Book
The Hopf Bifurcation and Its Applications
TL;DR: The Hopf bifurcation refers to the development of periodic orbits ("self-oscillations") from a stable fixed point, as a parameter crosses a critical value as mentioned in this paper.