11,734 research outputs found
Nonlinear Lattice Dynamics of Bose-Einstein Condensates
The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine
thermalization in non-metallic solids and develop ``experimental'' techniques
for studying nonlinear problems, continues to yield a wealth of results in the
theory and applications of nonlinear Hamiltonian systems with many degrees of
freedom. Inspired by the studies of this seminal model, solitary-wave dynamics
in lattice dynamical systems have proven vitally important in a diverse range
of physical problems--including energy relaxation in solids, denaturation of
the DNA double strand, self-trapping of light in arrays of optical waveguides,
and Bose-Einstein condensates (BECs) in optical lattices. BECS, in particular,
due to their widely ranging and easily manipulated dynamical apparatuses--with
one to three spatial dimensions, positive-to-negative tuning of the
nonlinearity, one to multiple components, and numerous experimentally
accessible external trapping potentials--provide one of the most fertile
grounds for the analysis of solitary waves and their interactions. In this
paper, we review recent research on BECs in the presence of deep periodic
potentials, which can be reduced to nonlinear chains in appropriate
circumstances. These reductions, in turn, exhibit many of the remarkable
nonlinear structures (including solitons, intrinsic localized modes, and
vortices) that lie at the heart of the nonlinear science research seeded by the
FPU paradigm.Comment: 10 pages, revtex, two-columns, 3 figs, accepted fpr publication in
Chaos's focus issue on the 50th anniversary of the publication of the
Fermi-Pasta-Ulam problem; minor clarifications (and a couple corrected typos)
from previous versio
Local Causal States and Discrete Coherent Structures
Coherent structures form spontaneously in nonlinear spatiotemporal systems
and are found at all spatial scales in natural phenomena from laboratory
hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary
climate dynamics. Phenomenologically, they appear as key components that
organize the macroscopic behaviors in such systems. Despite a century of
effort, they have eluded rigorous analysis and empirical prediction, with
progress being made only recently. As a step in this, we present a formal
theory of coherent structures in fully-discrete dynamical field theories. It
builds on the notion of structure introduced by computational mechanics,
generalizing it to a local spatiotemporal setting. The analysis' main tool
employs the \localstates, which are used to uncover a system's hidden
spatiotemporal symmetries and which identify coherent structures as
spatially-localized deviations from those symmetries. The approach is
behavior-driven in the sense that it does not rely on directly analyzing
spatiotemporal equations of motion, rather it considers only the spatiotemporal
fields a system generates. As such, it offers an unsupervised approach to
discover and describe coherent structures. We illustrate the approach by
analyzing coherent structures generated by elementary cellular automata,
comparing the results with an earlier, dynamic-invariant-set approach that
decomposes fields into domains, particles, and particle interactions.Comment: 27 pages, 10 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/dcs.ht
Multi-component gap solitons in spinor Bose-Einstein condensates
We model the nonlinear behaviour of spin-1 Bose-Einstein condensates (BECs)
with repulsive spin-independent interactions and either ferromagnetic or
anti-ferromagnetic (polar) spin-dependent interactions, loaded into a
one-dimensional optical lattice potential. We show that both types of BECs
exhibit dynamical instabilities and may form spatially localized
multi-component structures. The localized states of the spinor matter waves
take the form of vector gap solitons and self-trapped waves that exist only
within gaps of the linear Bloch-wave band-gap spectrum. Of special interest are
the nonlinear localized states that do not exhibit a common spatial density
profile shared by all condensate components, and consequently cannot be
described by the single mode approximation (SMA), frequently employed within
the framework of the mean-field treatment. We show that the non-SMA states can
exhibits Josephson-like internal oscillations and self-magnetisation, i.e.
intrinsic precession of the local spin. Finally, we demonstrate that
non-stationary states of a spinor BEC in a lattice exhibit coherent undamped
spin-mixing dynamics, and that their controlled conversion into a stationary
state can be achieved by the application of an external magnetic field.Comment: 12 pages, 13 figure
Dynamic and Static Excitations of a Classical Discrete Anisotropic Heisenberg Ferromagnetic Spin Chain
Using Jacobi elliptic function addition formulas and summation identities we
obtain several static and moving periodic soliton solutions of a classical
anisotropic, discrete Heisenberg spin chain with and without an external
magnetic field. We predict the dispersion relations of these nonlinear
excitations and contrast them with that of magnons and relate these findings to
the materials realized by a discrete spin chain. As limiting cases, we discuss
different forms of domain wall structures and their properties.Comment: Accepted for publication in Physica
Three-dimensional matter-wave vortices in optical lattices
We predict the existence of spatially localized nontrivial vortex states of a
Bose-Einstein condensate with repulsive atomic interaction confined by a
three-dimensional optical lattice. Such vortex-like structures include planar
vortices, their co- and counter-rotating bound states, and distinctly
three-dimensional non-planar vortex states. We demonstrate numerically that
many of these vortex structures are remarkably robust, and therefore can be
generated and observed in experiment.Comment: 4 pages, 6 figure
Cluster coherent potential approximation for electronic structure of disordered alloys
We extend the single-site coherent potential approximation (CPA) to include
the effects of non-local disorder correlations (alloy short-range order) on the
electronic structure of random alloy systems. This is achieved by mapping the
original Anderson disorder problem to that of a selfconsistently embedded
cluster. This cluster problem is then solved using the equations of motion
technique. The CPA is recovered for cluster size , and the disorder
averaged density-of-states (DOS) is always positive definite. Various new
features, compared to those observed in CPA, and related to repeated scattering
on pairs of sites, reflecting the effect of SRO are clearly visible in the DOS.
It is explicitly shown that the cluster-CPA method always yields
positive-definite DOS. Anderson localization effects have been investigated
within this approach. In general, we find that Anderson localization sets in
before band splitting occurs, and that increasing partial order drives a
continuous transition from an Anderson insulator to an incoherent metal.Comment: 7 pages, 6 figures. submitted to PR
Transverse Patterns in Nonlinear Optical Resonators
The book is devoted to the formation and dynamics of localized structures
(vortices, solitons) and extended patterns (stripes, hexagons, tilted waves) in
nonlinear optical resonators such as lasers, optical parametric oscillators,
and photorefractive oscillators. The theoretical analysis is performed by
deriving order parameter equations, and also through numerical integration of
microscopic models of the systems under investigation. Experimental
observations, and possible technological implementations of transverse optical
patterns are also discussed. A comparison with patterns found in other
nonlinear systems, i.e. chemical, biological, and hydrodynamical systems, is
given. This article contains the table of contents and the introductory chapter
of the book.Comment: 37 pages, 14 figures. Table of contents and introductory chapter of
the boo
Local dynamical lattice instabilities: Prerequisites for resonant pairing superconductivity
Fluctuating local diamagnetic pairs of electrons, embedded in a Fermi sea,
are candidates for non-phonon-mediated superconductors without the stringent
conditions on Tc which arise in phonon-mediated BCS classical low-Tc
superconductors. The local accumulations of charge, from which such diamagnetic
fluctuations originate, are irrevocably coupled to local dynamical lattice
instabilities and form composite charge-lattice excitations of the system. For
a superconducting phase to be realized, such excitations must be itinerant
spatially phase-coherent modes. This can be achieved by resonant pair tunneling
in and out of polaronic cation-ligand sites. Materials in which
superconductivity driven by such local lattice instability can be expected,
have a Tc which is controlled by the phase stiffness rather than the amplitude
of the diamagnetic pair fluctuations. Above Tc, a pseudogap phase will be
maintained up to a T*, where this pairing amplitude disappears. We discuss the
characteristic local charge and lattice properties which characterize this
pseudogap phase and which form the prerequisites for establishing a
phase-coherent macroscopic superconducting state.Comment: 15 pages, 13 figure
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