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We consider a one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction. The corresponding term in dynamical equations is proportional to 1//n-m/alpha+1. 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 alpha,… (More)

The notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order. This type of equation appears in the description of chaotic dynamics, wave propagation in fractal media, and field theory. For the fractional linear oscillator the physical meaning of the derivative of order ao2 is dissipation.… (More)

We derive the fractional generalization of the Ginzburg–Landau equation from the variational Euler–Lagrange equation for fractal media. To describe fractal media we use the fractional integrals considered as approximations of integrals on fractals. Some simple solutions of the Ginzburg–Landau equation for fractal media are considered and different forms of… (More)

Chaotic ray dynamics in deep sea propagation models is considered using the approaches developed in the theory of dynamical chaos. It has been demonstrated that the mechanism of emergence of ray chaos due to overlapping of nonlinear ray-medium resonances should play an important role in long range sound propagation. Analytical estimations, supported by… (More)

The one-dimensional chain of coupled oscillators with long-range power-law interaction is considered. The equation of motion in the infrared limit are mapped onto the continuum equation with the Riesz fractional derivative of order α, when 0 < α < 2. The evolution of soliton-like and breather-like structures are obtained numerically and compared for both… (More)

- H Weitzner, G M Zaslavsky
- 2002

We present two observations related to the application of linear (LFE) and nonlinear fractional equations (NFE). First, we give the comparison and estimates of the role of the fractional derivative term to the normal diffusion term in a LFE. The transition of the solution from normal to anomalous transport is demonstrated and the dominant role of the power… (More)

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction defined by a term proportional to 1/jn À mj a+1. Continuous medium equation for this system can be obtained in the so-called infrared limit when the wave number tends to zero. We construct a transform operator that maps the system of large… (More)

Islands are divided according to their phase space structure—resonant islands and tangle islands are considered. It is proved that in the near-integrable limit these correspond to two distinct sets, hence that in general their definitions are not trivially equivalent. It is demonstrated and proved that accelerator modes of the standard map and of the web… (More)

Discrete nonlinear Schrödinger equation (DNLS) describes a chain of oscillators with nearest neighbor interactions and a specific nonlinear term. We consider its modification with long-range interaction through a potential proportional to 1/l 1+α with fractional α < 2 and l as a distance between oscillators. This model is called αDNLS. It exhibits… (More)