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A system of three point vortices in an unbounded plane has a special family of self-similarly contracting or expanding solutions: during the motion, vortex triangle remains similar to the original one, while its area decreases (grows) at a constant rate. A contracting con guration brings three vortices to a single point in a nite time; this phenomenon is… (More)

- Vasily E Tarasov, George M Zaslavsky
- Chaos
- 2006

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)

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/l1+α with fractional α < 2 and l as a distance between oscillators. This model is called αDNLS. It exhibits… (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)

- Michael G Brown, John A Colosi, Steven Tomsovic, Anatoly L Virovlyansky, Michael A Wolfson, George M Zaslavsky
- The Journal of the Acoustical Society of America
- 2003

Recent results relating to ray dynamics in ocean acoustics are reviewed. Attention is focused 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 (timelike) variable. Topics discussed include… (More)

- V Afraimovich, G M Zaslavsky
- Chaos
- 2003

New notions of the complexity function C(epsilon;t,s) and entropy function S(epsilon;t,s) are introduced to describe systems with nonzero or zero Lyapunov exponents or systems that exhibit strong intermittent behavior with "flights," trappings, weak mixing, etc. The important part of the new notions is the first appearance of epsilon-separation of initially… (More)

- X Leoncini, L Kuznetsov, G M Zaslavsky
- Physical review. E, Statistical, nonlinear, and…
- 2001

Dynamical and statistical properties of tracer advection are studied in a family of flows produced by three point-vortices of different signs. Tracer dynamics is analyzed by numerical construction of Poincaré sections, and is found to be strongly chaotic: advection pattern in the region around the center of vorticity is dominated by a well developed… (More)

- G M Zaslavsky, M Edelman, V E Tarasov
- Chaos
- 2007

We consider a chain of nonlinear oscillators with long-range interaction of the type 1l(1+alpha), where l is a distance between oscillators and 0<alpha<2. In the continuous limit, the system's dynamics is described by a fractional generalization of the Ginzburg-Landau equation with complex coefficients. Such a system has a new parameter alpha that is… (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)

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)