1,010 research outputs found
Compact high order finite difference schemes for linear Schrödinger problems on non-uniform meshes
In the present paper a general technique is developed for construction
of compact high-order finite difference schemes to approximate Schrödinger
problems on nonuniform meshes. Conservation of the finite difference schemes
is investigated. Discrete transparent boundary conditions are constructed for
the given high-order finite difference scheme. The same technique is applied
to construct compact high-order approximations of the Robin and Szeftel type
boundary conditions. Results of computational experiments are presente
Compact high order finite difference schemes for linear Schrödinger problems on non-uniform meshes
In the present paper a general technique is developed for construction
of compact high-order finite difference schemes to approximate Schrödinger
problems on nonuniform meshes. Conservation of the finite difference schemes
is investigated. Discrete transparent boundary conditions are constructed for
the given high-order finite difference scheme. The same technique is applied
to construct compact high-order approximations of the Robin and Szeftel type
boundary conditions. Results of computational experiments are presente
Nonlinear Phenomena of Ultracold Atomic Gases in Optical Lattices: Emergence of Novel Features in Extended States
The system of a cold atomic gas in an optical lattice is governed by two
factors: nonlinearity originating from the interparticle interaction, and the
periodicity of the system set by the lattice. The high level of controllability
associated with such an arrangement allows for the study of the competition and
interplay between these two, and gives rise to a whole range of interesting and
rich nonlinear effects. This review covers the basic idea and overview of such
nonlinear phenomena, especially those corresponding to extended states. This
includes "swallowtail" loop structures of the energy band, Bloch states with
multiple periodicity, and those in "nonlinear lattices", i.e., systems with the
nonlinear interaction term itself being a periodic function in space.Comment: 39 pages, 21 figures; review article to be published in a Special
Issue of Entropy on "Non-Linear Lattice
Multipole-mode surface solitons
We discover multipole-mode solitons supported by the surface between two
distinct periodic lattices imprinted in Kerr-type nonlinear media. Such
solitons are possible because the refractive index modulation at both sides of
the interface glues together their out-of-phase individual constituents.
Remarkably, we find that the new type of solitons may feature highly asymmetric
shapes and yet they are stable over wide domains of their existence, a rare
property to be attributed to their surface nature.Comment: 14 pages, 3 figures, to appear in Optics Letter
Systems of Points with Coulomb Interactions
Large ensembles of points with Coulomb interactions arise in various settings
of condensed matter physics, classical and quantum mechanics, statistical
mechanics, random matrices and even approximation theory, and give rise to a
variety of questions pertaining to calculus of variations, Partial Differential
Equations and probability. We will review these as well as "the mean-field
limit" results that allow to derive effective models and equations describing
the system at the macroscopic scale. We then explain how to analyze the next
order beyond the mean-field limit, giving information on the system at the
microscopic level. In the setting of statistical mechanics, this allows for
instance to observe the effect of the temperature and to connect with
crystallization questions.Comment: 30 pages, to appear as Proceedings of the ICM201
Homogenized description of defect modes in periodic structures with localized defects
A spatially localized initial condition for an energy-conserving wave
equation with periodic coefficients disperses (spatially spreads) and decays in
amplitude as time advances. This dispersion is associated with the continuous
spectrum of the underlying differential operator and the absence of discrete
eigenvalues. The introduction of spatially localized perturbations in a
periodic medium leads to defect modes, states in which energy remains trapped
and spatially localized. In this paper we study weak, localized perturbations
of one-dimensional periodic Schr\"odinger operators. Such perturbations give
rise to such defect modes, and are associated with the emergence of discrete
eigenvalues from the continuous spectrum. Since these isolated eigenvalues are
located near a spectral band edge, there is strong scale-separation between the
medium period and the localization length of the defect mode. Bound states
therefore have a multi-scale structure: a "carrier Bloch wave" times a "wave
envelope", which is governed by a homogenized Schr\"odinger operator with
associated effective mass, depending on the spectral band edge which is the
site of the bifurcation. Our analysis is based on a reformulation of the
eigenvalue problem in Bloch quasi-momentum space, using the Gelfand-Bloch
transform and a Lyapunov-Schmidt reduction to a closed equation for the
near-band-edge frequency components of the bound state. A rescaling of the
latter equation yields the homogenized effective equation for the wave
envelope, and approximations to bifurcating eigenvalues and eigenfunctions.Comment: The title differs from version 1. To appear in Communications in
Mathematical Science
(Invited) Two-color soliton meta-atoms and molecules
We present a detailed overview of the physics of two-color soliton molecules in nonlinear waveguides, i.e. bound states of localized optical pulses which are held together due to an incoherent interaction mechanism. The mutual confinement, or trapping, of the subpulses, which leads to a stable propagation of the pulse compound, is enabled by the nonlinear Kerr effect. Special attention is paid to the description of the binding mechanism in terms of attractive potential wells, induced by the refractive index changes of the subpulses, exerted on one another through cross-phase modulation. Specifically, we discuss nonlinear-photonics meta atoms, given by pulse compounds consisting of a strong trapping pulse and a weak trapped pulse, for which trapped states of low intensity are determined by a Schrödinger-type eigenproblem. We discuss the rich dynamical behavior of such meta-atoms, demonstrating that an increase of the group-velocity mismatch of both subpulses leads to an ionization-like trapping-to-escape transition. We further demonstrate that if both constituent pulses are of similar amplitude, molecule-like bound-states are formed. We show that -periodic amplitude variations permit a coupling of these pulse compound to dispersive waves, resulting in the resonant emission of Kushi-comb-like multi-frequency radiation
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