9,080 research outputs found
Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model
General boundary conditions ("branes") for the Poisson sigma model are
studied. They turn out to be labeled by coisotropic submanifolds of the given
Poisson manifold. The role played by these boundary conditions both at the
classical and at the perturbative quantum level is discussed. It turns out to
be related at the classical level to the category of Poisson manifolds with
dual pairs as morphisms and at the perturbative quantum level to the category
of associative algebras (deforming algebras of functions on Poisson manifolds)
with bimodules as morphisms. Possibly singular Poisson manifolds arising from
reduction enter naturally into the picture and, in particular, the construction
yields (under certain assumptions) their deformation quantization.Comment: 21 pages, 2 figures; minor corrections, references updated; final
versio
Excitation Thresholds for Nonlinear Localized Modes on Lattices
Breathers are spatially localized and time periodic solutions of extended
Hamiltonian dynamical systems. In this paper we study excitation thresholds for
(nonlinearly dynamically stable) ground state breather or standing wave
solutions for networks of coupled nonlinear oscillators and wave equations of
nonlinear Schr\"odinger (NLS) type. Excitation thresholds are rigorously
characterized by variational methods. The excitation threshold is related to
the optimal (best) constant in a class of discr ete interpolation inequalities
related to the Hamiltonian energy. We establish a precise connection among ,
the dimensionality of the lattice, , the degree of the nonlinearity
and the existence of an excitation threshold for discrete nonlinear
Schr\"odinger systems (DNLS).
We prove that if , then ground state standing waves exist if
and only if the total power is larger than some strictly positive threshold,
. This proves a conjecture of Flach, Kaldko& MacKay in
the context of DNLS. We also discuss upper and lower bounds for excitation
thresholds for ground states of coupled systems of NLS equations, which arise
in the modeling of pulse propagation in coupled arrays of optical fibers.Comment: To appear in Nonlinearit
On the notion of phase in mechanics
The notion of phase plays an esential role in both classical and quantum
mechanics.But what is a phase? We show that if we define the notion of phase in
phase (!) space one can very easily and naturally recover the Heisenberg-Weyl
formalism; this is achieved using the properties of the Poincare-Cartan
invariant, and without making any quantum assumption
Charged-Surface Instability Development in Liquid Helium; Exact Solutions
The nonlinear dynamics of charged-surface instability development was
investigated for liquid helium far above the critical point. It is found that,
if the surface charge completely screens the field above the surface, the
equations of three-dimensional (3D) potential motion of a fluid are reduced to
the well-known equations describing the 3D Laplacian growth process. The
integrability of these equations in 2D geometry allows the analytic description
of the free-surface evolution up to the formation of cuspidal singularities at
the surface.Comment: latex, 5 pages, no figure
Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons
We present a unified approach for qualitative and quantitative analysis of
stability and instability dynamics of positive bright solitons in
multi-dimensional focusing nonlinear media with a potential (lattice), which
can be periodic, periodic with defects, quasiperiodic, single waveguide, etc.
We show that when the soliton is unstable, the type of instability dynamic that
develops depends on which of two stability conditions is violated.
Specifically, violation of the slope condition leads to an amplitude
instability, whereas violation of the spectral condition leads to a drift
instability. We also present a quantitative approach that allows to predict the
stability and instability strength
Poisson Geometry in Constrained Systems
Constrained Hamiltonian systems fall into the realm of presymplectic
geometry. We show, however, that also Poisson geometry is of use in this
context.
For the case that the constraints form a closed algebra, there are two
natural Poisson manifolds associated to the system, forming a symplectic dual
pair with respect to the original, unconstrained phase space. We provide
sufficient conditions so that the reduced phase space of the constrained system
may be identified with a symplectic leaf in one of those. In the second class
case the original constrained system may be reformulated equivalently as an
abelian first class system in an extended phase space by these methods.
Inspired by the relation of the Dirac bracket of a general second class
constrained system to the original unconstrained phase space, we address the
question of whether a regular Poisson manifold permits a leafwise symplectic
embedding into a symplectic manifold. Necessary and sufficient for this is the
vanishing of the characteristic form-class of the Poisson tensor, a certain
element of the third relative cohomology.Comment: 41 pages, more detailed abstract in paper; v2: minor corrections and
an additional referenc
Theory of Nonlinear Dispersive Waves and Selection of the Ground State
A theory of time dependent nonlinear dispersive equations of the Schroedinger
/ Gross-Pitaevskii and Hartree type is developed. The short, intermediate and
large time behavior is found, by deriving nonlinear Master equations (NLME),
governing the evolution of the mode powers, and by a novel multi-time scale
analysis of these equations. The scattering theory is developed and coherent
resonance phenomena and associated lifetimes are derived. Applications include
BEC large time dynamics and nonlinear optical systems. The theory reveals a
nonlinear transition phenomenon, ``selection of the ground state'', and NLME
predicts the decay of excited state, with half its energy transferred to the
ground state and half to radiation modes. Our results predict the recent
experimental observations of Mandelik et. al. in nonlinear optical waveguides
Fidelity Decay as an Efficient Indicator of Quantum Chaos
Recent work has connected the type of fidelity decay in perturbed quantum
models to the presence of chaos in the associated classical models. We
demonstrate that a system's rate of fidelity decay under repeated perturbations
may be measured efficiently on a quantum information processor, and analyze the
conditions under which this indicator is a reliable probe of quantum chaos and
related statistical properties of the unperturbed system. The type and rate of
the decay are not dependent on the eigenvalue statistics of the unperturbed
system, but depend on the system's eigenvector statistics in the eigenbasis of
the perturbation operator. For random eigenvector statistics the decay is
exponential with a rate fixed precisely by the variance of the perturbation's
energy spectrum. Hence, even classically regular models can exhibit an
exponential fidelity decay under generic quantum perturbations. These results
clarify which perturbations can distinguish classically regular and chaotic
quantum systems.Comment: 4 pages, 3 figures, LaTeX; published version (revised introduction
and discussion
Extended phase space for a spinning particle
Extended phase space of an elementary (relativistic) system is introduced in
the spirit of the Souriau's definition of the `space of motions' for such
system. Our formulation is generally applicable to any homogeneous space-time
(e.g. de Sitter) and also to Poisson actions. Calculations concerning the
Minkowski case for non-zero spin particles show an intriguing alternative: we
should either accept two-dimensional trajectories or (Poisson) noncommuting
space-time coordinates.Comment: 12 pages, late
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