4,398 research outputs found
Approximation of small-amplitude weakly coupled oscillators with discrete nonlinear Schrodinger equations
Small-amplitude weakly coupled oscillators of the Klein-Gordon lattices are
approximated by equations of the discrete nonlinear Schrodinger type. We show
how to justify this approximation by two methods, which have been very popular
in the recent literature. The first method relies on a priori energy estimates
and multi-scale decompositions. The second method is based on a resonant normal
form theorem. We show that although the two methods are different in the
implementation, they produce equivalent results as the end product. We also
discuss applications of the discrete nonlinear Schrodinger equation in the
context of existence and stability of breathers of the Klein--Gordon lattice
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
Numerical computation of travelling breathers in Klein-Gordon chains
We numerically study the existence of travelling breathers in Klein-Gordon
chains, which consist of one-dimensional networks of nonlinear oscillators in
an anharmonic on-site potential, linearly coupled to their nearest neighbors.
Travelling breathers are spatially localized solutions having the property of
being exactly translated by sites along the chain after a fixed propagation
time (these solutions generalize the concept of solitary waves for which
). In the case of even on-site potentials, the existence of small
amplitude travelling breathers superposed on a small oscillatory tail has been
proved recently (G. James and Y. Sire, to appear in {\sl Comm. Math. Phys.},
2004), the tail being exponentially small with respect to the central
oscillation size. In this paper we compute these solutions numerically and
continue them into the large amplitude regime for different types of even
potentials. We find that Klein-Gordon chains can support highly localized
travelling breather solutions superposed on an oscillatory tail. We provide
examples where the tail can be made very small and is difficult to detect at
the scale of central oscillations. In addition we numerically observe the
existence of these solutions in the case of non even potentials
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
Multibreather and vortex breather stability in Klein--Gordon lattices: Equivalence between two different approaches
In this work, we revisit the question of stability of multibreather
configurations, i.e., discrete breathers with multiple excited sites at the
anti-continuum limit of uncoupled oscillators. We present two methods that
yield quantitative predictions about the Floquet multipliers of the linear
stability analysis around such exponentially localized in space, time-periodic
orbits, based on the Aubry band method and the MacKay effective Hamiltonian
method and prove that their conclusions are equivalent. Subsequently, we
showcase the usefulness of the methods by a series of case examples including
one-dimensional multi-breathers, and two-dimensional vortex breathers in the
case of a lattice of linearly coupled oscillators with the Morse potential and
in that of the discrete model
Discrete Breathers
Nonlinear classical Hamiltonian lattices exhibit generic solutions in the
form of discrete breathers. These solutions are time-periodic and (typically
exponentially) localized in space. The lattices exhibit discrete translational
symmetry. Discrete breathers are not confined to certain lattice dimensions.
Necessary ingredients for their occurence are the existence of upper bounds on
the phonon spectrum (of small fluctuations around the groundstate) of the
system as well as the nonlinearity in the differential equations. We will
present existence proofs, formulate necessary existence conditions, and discuss
structural stability of discrete breathers. The following results will be also
discussed: the creation of breathers through tangent bifurcation of band edge
plane waves; dynamical stability; details of the spatial decay; numerical
methods of obtaining breathers; interaction of breathers with phonons and
electrons; movability; influence of the lattice dimension on discrete breather
properties; quantum lattices - quantum breathers. Finally we will formulate a
new conceptual aproach capable of predicting whether discrete breather exist
for a given system or not, without actually solving for the breather. We
discuss potential applications in lattice dynamics of solids (especially
molecular crystals), selective bond excitations in large molecules, dynamical
properties of coupled arrays of Josephson junctions, and localization of
electromagnetic waves in photonic crystals with nonlinear response.Comment: 62 pages, LaTeX, 14 ps figures. Physics Reports, to be published; see
also at http://www.mpipks-dresden.mpg.de/~flach/html/preprints.htm
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