50 research outputs found
Fractional dynamics of coupled oscillators with long-range interaction
We consider one-dimensional chain of coupled linear and nonlinear oscillators
with long-range power-wise interaction. The corresponding term in dynamical
equations is proportional to . It is shown that the
equation of motion in the infrared limit can be transformed into the medium
equation with the Riesz fractional derivative of order , when
. We consider few models of coupled oscillators and show how their
synchronization can appear as a result of bifurcation, and how the
corresponding solutions depend on . The presence of fractional
derivative leads also to the occurrence of localized structures. Particular
solutions for fractional time-dependent complex Ginzburg-Landau (or nonlinear
Schrodinger) equation are derived. These solutions are interpreted as
synchronized states and localized structures of the oscillatory medium.Comment: 34 pages, 18 figure
Electromagnetic Fields on Fractals
Fractals are measurable metric sets with non-integer Hausdorff dimensions. If
electric and magnetic fields are defined on fractal and do not exist outside of
fractal in Euclidean space, then we can use the fractional generalization of
the integral Maxwell equations. The fractional integrals are considered as
approximations of integrals on fractals. We prove that fractal can be described
as a specific medium.Comment: 15 pages, LaTe
Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum Transformations
Simple derivation is presented of the four families of infinitely many shape
invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi
polynomials. Darboux-Crum transformations are applied to connect the well-known
shape invariant Hamiltonians of the radial oscillator and the
Darboux-P\"oschl-Teller potential to the shape invariant potentials of
Odake-Sasaki. Dutta and Roy derived the two lowest members of the exceptional
Laguerre polynomials by this method. The method is expanded to its full
generality and many other ramifications, including the aspects of generalised
Bochner problem and the bispectral property of the exceptional orthogonal
polynomials, are discussed.Comment: LaTeX2e with amsmath, amssymb, amscd 26 pages, no figure
Kolmogorov-Sinai entropy in field line diffusion by anisotropic magnetic turbulence
The Kolmogorov-Sinai (KS) entropy in turbulent diffusion of magnetic field
lines is analyzed on the basis of a numerical simulation model and theoretical
investigations. In the parameter range of strongly anisotropic magnetic
turbulence the KS entropy is shown to deviate considerably from the earlier
predicted scaling relations [Rev. Mod. Phys. {\bf 64}, 961 (1992)]. In
particular, a slowing down logarithmic behavior versus the so-called Kubo
number (, where is the ratio of the rms magnetic fluctuation field to the magnetic field
strength, and and are the correlation lengths in respective
dimensions) is found instead of a power-law dependence. These discrepancies are
explained from general principles of Hamiltonian dynamics. We discuss the
implication of Hamiltonian properties in governing the paradigmatic
"percolation" transport, characterized by , associating it with the
concept of pseudochaos (random non-chaotic dynamics with zero Lyapunov
exponents). Applications of this study pertain to both fusion and astrophysical
plasma and by mathematical analogy to problems outside the plasma physics.
This research article is dedicated to the memory of Professor George M.
ZaslavskyComment: 15 pages, 2 figures. Accepted for publication on Plasma Physics and
Controlled Fusio
Fractional Derivative as Fractional Power of Derivative
Definitions of fractional derivatives as fractional powers of derivative
operators are suggested. The Taylor series and Fourier series are used to
define fractional power of self-adjoint derivative operator. The Fourier
integrals and Weyl quantization procedure are applied to derive the definition
of fractional derivative operator. Fractional generalization of concept of
stability is considered.Comment: 20 pages, LaTe
Hamiltonian formulation of systems with linear velocities within Riemann-Liouville fractional derivatives
The link between the treatments of constrained systems with fractional
derivatives by using both Hamiltonian and Lagrangian formulations is studied.
It is shown that both treatments for systems with linear velocities are
equivalent.Comment: 10 page
Nonholonomic Constraints with Fractional Derivatives
We consider the fractional generalization of nonholonomic constraints defined
by equations with fractional derivatives and provide some examples. The
corresponding equations of motion are derived using variational principle.Comment: 18 page
Continuous Limit of Discrete Systems with Long-Range Interaction
Discrete systems with long-range interactions are considered. Continuous
medium models as continuous limit of discrete chain system are defined.
Long-range interactions of chain elements that give the fractional equations
for the medium model are discussed. The chain equations of motion with
long-range interaction are mapped into the continuum equation with the Riesz
fractional derivative. We formulate the consistent definition of continuous
limit for the systems with long-range interactions. In this paper, we consider
a wide class of long-range interactions that give fractional medium equations
in the continuous limit. The power-law interaction is a special case of this
class.Comment: 23 pages, LaTe