arXiv subjects

George M. Zaslavsky

Publications and source records attributed to George M. Zaslavsky.

At least 19 recordsLinked to original sources

arXiv2011

Fokker-Planck Equation with Fractional Coordinate Derivatives

Using the generalized Kolmogorov-Feller equation with long-range interaction, we obtain kinetic equations with fractional derivatives with respect to coordinates. The method of successive approximations with the averaging with respect to fast variable is used. The main assumption is that the correlator of probability densities of particles to make a step has a power-law dependence. As a result, we obtain Fokker-Planck equation with fractional coordinate derivative of order $1<α<2$.

physics.class-ph↗

arXiv2011

Fractional dynamics of systems with long-range interaction

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range power wise interaction defined by a term proportional to 1/|n-m|^{α+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 number of ordinary differential equations of motion of the particles into a partial differential equation with the Riesz fractional derivative of order α, when 0<α<2. Few models of coupled oscillators are considered and their synchronized states and localized structures are discussed in details. Particularly, we discuss some solutions of time-dependent fractional Ginzburg-Landau (or nonlinear Schrodinger) equation.

nlin.CD↗

arXiv2011

Fractional Equations of Kicked Systems and Discrete Maps

Starting from kicked equations of motion with derivatives of non-integer orders, we obtain "fractional" discrete maps. These maps are generalizations of well-known universal, standard, dissipative, kicked damped rotator maps. The main property of the suggested fractional maps is a long-term memory. The memory effects in the fractional discrete maps mean that their present state evolution depends on all past states with special forms of weights. These forms are represented by combinations of power-law functions.

nlin.CD↗

arXiv2007

Fractional Generalization of Kac Integral

Generalization of the Kac integral and Kac method for paths measure based on the Levy distribution has been used to derive fractional diffusion equation. Application to nonlinear fractional Ginzburg-Landau equation is discussed.

math-ph↗

arXiv2007

Conservation Laws and Hamilton's Equations for Systems with Long-Range Interaction and Memory

Using the fact that extremum of variation of generalized action can lead to the fractional dynamics in the case of systems with long-range interaction and long-term memory function, we consider two different applications of the action principle: generalized Noether's theorem and Hamiltonian type equations. In the first case, we derive conservation laws in the form of continuity equations that consist of fractional time-space derivatives. Among applications of these results, we consider a chain of coupled oscillators with a power-wise memory function and power-wise interaction between oscillators. In the second case, we consider an example of fractional differential action 1-form and find the corresponding Hamiltonian type equations from the closed condition of the form.

math-ph↗

arXiv2007

Fractional Dynamics of Systems with Long-Range Space Interaction and Temporal Memory

Field equations with time and coordinates derivatives of noninteger order are derived from stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a fractional generalization of the Ginzburg-Landau and nonlinear Schrodinger equations. As another example, dynamical equations for particles chain with power-law interaction and memory are considered in the continuous limit. The obtained fractional equations can be applied to complex media with/without random parameters or processes.

math-ph↗

arXiv2006

Transition to Chaos in Discrete Nonlinear Schrodinger Equation with Long-Range Interaction

Discrete nonlinear Schrodinger 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 $α$DNLS. It exhibits competition between the nonlinearity and a level of correlation between interacting far-distanced oscillators, that is defined by the value of $α$. We consider transition to chaos in this system as a function of $α$ and nonlinearity. It is shown that decreasing of $α$ with respect to nonlinearity stabilize the system. Connection of the model to the fractional genezalization of the NLS (called FNLS) in the long-wave approximation is also discussed and some of the results obtained for $α$DNLS can be correspondingly extended to the FNLS.

math-ph↗

arXiv2006

Coupled oscillators with power-law interaction and their fractional dynamics analogues

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 types of simulations: using the chain of oscillators and using the continuous medium equation with the fractional derivative.

math-ph↗

arXiv2006

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.

math-ph↗

arXiv2006

Evidence of fractional transport in point-vortex flow

Advection properties of passive particles in flows generated by point vortices are considered. Transport properties are anomalous with characteristic transport exponent $μ\sim 1.5$. This behavior is linked back to the presence of coherent fractal structures within the flow. A fractional kinetic analysis allows to link the characteristic transport exponent $μ$ to the trapping time exponent $γ=1+μ$. The quantitative agreement is found for different systems of vortices investigated and a clear signature is obtained of the fractional nature of transport in these flows.

nlin.CD↗

arXiv2006

Chaotic Jets

The problem of characterizing the origin of the non-Gaussian properties of transport resulting from Hamiltonian dynamics is addressed. For this purpose the notion of chaotic jet is revisited and leads to the definition of a diagnostic able to capture some singular properties of the dynamics. This diagnostic is applied successfully to the problem of advection of passive tracers in a flow generated by point vortices. We present and discuss this diagnostic as a result of which clues on the origin of anomalous transport in these systems emerge.

nlin.CD↗

arXiv2006

Anomalous transport in Charney-Hasegawa-Mima flows

Transport properties of particles evolving in a system governed by the Charney-Hasegawa-Mima equation are investigated. Transport is found to be anomalous with a non linear evolution of the second moments with time. The origin of this anomaly is traced back to the presence of chaotic jets within the flow. All characteristic transport exponents have a similar value around $μ=1.75$, which is also the one found for simple point vortex flows in the literature, indicating some kind of universality. Moreover the law $γ=μ+1$ linking the trapping time exponent within jets to the transport exponent is confirmed and an accumulation towards zero of the spectrum of finite time Lyapunov exponent is observed. The localization of a jet is performed, and its structure is analyzed. It is clearly shown that despite a regular coarse grained picture of the jet, motion within the jet appears as chaotic but chaos is bounded on successive small scales.

nlin.CD↗

arXiv2005

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 $1/|n-m|^{α+1}$. 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 $α$, when $0<α<2$. 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 $α$. 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.

nlin.PS↗

arXiv2005

Fractional Ginzburg-Landau equation for fractal media

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 the fractional Ginzburg-Landau equation or nonlinear Schrodinger equation with fractional derivatives are presented. The Agrawal variational principle and its generalization have been applied.

physics.class-ph↗

arXiv2005

Dynamics with Low-Level Fractionality

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 $α<2$ is dissipation. In systems with many spacially coupled elements (oscillators) the fractional derivative, along the space coordinate, corresponds to a long range interaction. We discuss a method of constructing a solution using an expansion in $ε=n-α$ with small $ε$ and positive integer $n$. The method is applied to the fractional linear and nonlinear oscillators and to fractional Ginzburg-Landau or parabolic equations.

physics.class-ph↗

arXiv2003

Ray dynamics in a long-range acoustic propagation experiment

A ray-based wavefield description is employed in the interpretation of measurements made during the November 1994 Acoustic Engineering Test (the AET experiment). In this experiment phase-coded pulse-like signals with 75 Hz center frequency and 37.5 Hz bandwidth were transmitted near the sound channel axis in the eastern North Pacific Ocean. The resulting acoustic signals were recorded on a moored vertical receiving array at a range of 3252 km. In our analysis both mesoscale and internal-wave-induced sound speed perturbations are taken into account. Much of this analysis exploits results that relate to the subject of ray chaos; these results follow from the Hamiltonian structure of the ray equations. We present evidence that all of the important features of the measured AET wavefields, including their stability, are consistent with a ray-based wavefield description in which ray trajectories are predominantly chaotic.

nlin.CD↗

arXiv2002

Jets, Stickiness and Anomalous Transport

Dynamical and statistical properties of the vortex and passive particle advection in chaotic flows generated by four and sixteen point vortices are investigated. General transport properties of these flows are found anomalous and exhibit a superdiffusive behavior with typical second moment exponent (μ\sim 1.75). The origin of this anomaly is traced back to the presence of coherent structures within the flow, the vortex cores and the region far from where vortices are located. In the vicinity of these regions stickiness is observed and the motion of tracers is quasi-ballistic. The chaotic nature of the underlying flow dictates the choice for thorough analysis of transport properties. Passive tracer motion is analyzed by measuring the mutual relative evolution of two nearby tracers. Some tracers travel in each other vicinity for relatively large times. This is related to an hidden order for the tracers which we call jets. Jets are localized and found in sticky regions. Their structure is analyzed and found to be formed of a nested sets of jets within jets. The analysis of the jet trapping time statistics shows a quantitative agreement with the observed transport exponent.

nlin.CD↗

arXiv2001

Ray dynamics in ocean acoustics

Recent results relating to ray dynamics in ocean acoustics are reviewed. Attention is focussed on long-range propagation in deep ocean environments. For this class of problems, the ray equations may be simplified by making use of a one-way formulation in which the range variable appears as the independent (time-like) variable. Topics discussed include integrable and non-integrable ray systems, action-angle variables, nonlinear resonances and the KAM theorem, ray chaos, Lyapunov exponents, predictability, nondegeneracy violation, ray intensity statistics, semiclassical breakdown, wave chaos, and the connection between ray chaos and mode coupling. The Hamiltonian structure of the ray equations plays an important role in all of these topics.

nlin.CD↗