140 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
Bifurcations of discrete breathers in a diatomic Fermi-Pasta-Ulam chain
Discrete breathers are time-periodic, spatially localized solutions of the
equations of motion for a system of classical degrees of freedom interacting on
a lattice. Such solutions are investigated for a diatomic Fermi-Pasta-Ulam
chain, i. e., a chain of alternate heavy and light masses coupled by anharmonic
forces. For hard interaction potentials, discrete breathers in this model are
known to exist either as ``optic breathers'' with frequencies above the optic
band, or as ``acoustic breathers'' with frequencies in the gap between the
acoustic and the optic band. In this paper, bifurcations between different
types of discrete breathers are found numerically, with the mass ratio m and
the breather frequency omega as bifurcation parameters. We identify a period
tripling bifurcation around optic breathers, which leads to new breather
solutions with frequencies in the gap, and a second local bifurcation around
acoustic breathers. These results provide new breather solutions of the FPU
system which interpolate between the classical acoustic and optic modes. The
two bifurcation lines originate from a particular ``corner'' in parameter space
(omega,m). As parameters lie near this corner, we prove by means of a center
manifold reduction that small amplitude solutions can be described by a
four-dimensional reversible map. This allows us to derive formally a continuum
limit differential equation which characterizes at leading order the
numerically observed bifurcations.Comment: 30 pages, 10 figure
The Anti-FPU Problem
We present a detailed analysis of the modulational instability of the
zone-boundary mode for one and higher-dimensional Fermi-Pasta-Ulam (FPU)
lattices. Following this instability, a process of relaxation to equipartition
takes place, which we have called the Anti-FPU problem because the energy is
initially fed into the highest frequency part of the spectrum, at variance with
the original FPU problem (low frequency excitations of the lattice). This
process leads to the formation of chaotic breathers in both one and two
dimensions. Finally, the system relaxes to energy equipartition on time scales
which increase as the energy density is decreased. We show that breathers
formed when cooling the lattice at the edges, starting from a random initial
state, bear strong qualitative similarities with chaotic breathers
Breathers and Thermal Relaxation in Fermi-Pasta-Ulam Arrays
Breather stability and longevity in thermally relaxing nonlinear arrays
depend sensitively on their interactions with other excitations. We review the
relaxation of breathers in Fermi-Pasta-Ulam arrays, with a specific focus on
the different relaxation channels and their dependence on the interparticle
interactions, dimensionality, initial condition, and system parameters
Classical and quantum nonlinear localized excitations in discrete systems
Pre-pint tomado de ArxivDiscrete breathers, or intrinsic localized modes, are spatially localized, time–periodic, nonlinear
excitations that can exist and propagate in systems of coupled dynamical units. Recently, some
experiments show the sighting of a form of discrete breather that exist at the atomic scale in a
magnetic solid. Other observations of breathers refer to systems such as Josephson–junction arrays,
photonic crystals and optical-switching waveguide arrays. All these observations underscore their
importance in physical phenomena at all scales. The authors review some of their latest theoretical
contributions in the field of classical and quantum breathers, with possible applications to these
widely different physical systems and to many other such as DNA, proteins, quantum dots, quantum
computing, etc
Breathers in oscillator chains with Hertzian interactions
We prove nonexistence of breathers (spatially localized and time-periodic
oscillations) for a class of Fermi-Pasta-Ulam lattices representing an
uncompressed chain of beads interacting via Hertz's contact forces. We then
consider the setting in which an additional on-site potential is present,
motivated by the Newton's cradle under the effect of gravity. Using both direct
numerical computations and a simplified asymptotic model of the oscillator
chain, the so-called discrete p-Schr\"odinger (DpS) equation, we show the
existence of discrete breathers and study their spectral properties and
mobility. Due to the fully nonlinear character of Hertzian interactions,
breathers are found to be much more localized than in classical nonlinear
lattices and their motion occurs with less dispersion. In addition, we study
numerically the excitation of a traveling breather after an impact at one end
of a semi-infinite chain. This case is well described by the DpS equation when
local oscillations are faster than binary collisions, a situation occuring e.g.
in chains of stiff cantilevers decorated by spherical beads. When a hard
anharmonic part is added to the local potential, a new type of traveling
breather emerges, showing spontaneous direction-reversing in a spatially
homogeneous system. Finally, the interaction of a moving breather with a point
defect is also considered in the cradle system. Almost total breather
reflections are observed at sufficiently high defect sizes, suggesting
potential applications of such systems as shock wave reflectors
Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation
We study nonlinear waves in Newton's cradle, a classical mechanical system
consisting of a chain of beads attached to linear pendula and interacting
nonlinearly via Hertz's contact forces. We formally derive a spatially discrete
modulation equation, for small amplitude nonlinear waves consisting of slow
modulations of time-periodic linear oscillations. The fully-nonlinear and
unilateral interactions between beads yield a nonstandard modulation equation
that we call the discrete p-Schroedinger (DpS) equation. It consists of a
spatial discretization of a generalized Schroedinger equation with p-Laplacian,
with fractional p>2 depending on the exponent of Hertz's contact force. We show
that the DpS equation admits explicit periodic travelling wave solutions, and
numerically find a plethora of standing wave solutions given by the orbits of a
discrete map, in particular spatially localized breather solutions. Using a
modified Lyapunov-Schmidt technique, we prove the existence of exact periodic
travelling waves in the chain of beads, close to the small amplitude modulated
waves given by the DpS equation. Using numerical simulations, we show that the
DpS equation captures several other important features of the dynamics in the
weakly nonlinear regime, namely modulational instabilities, the existence of
static and travelling breathers, and repulsive or attractive interactions of
these localized structures
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