Tarasov Vasily E, Zaslavsky George M
Skobeltsyn Institute of Nuclear Physics, Moscow State University, Moscow 119992, Russia.
Chaos. 2006 Jun;16(2):023110. doi: 10.1063/1.2197167.
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, when 0<alpha<2. We consider a few models of coupled oscillators and show how their synchronization can appear as a result of bifurcation, and how the corresponding solutions depend on alpha. The presence of a fractional derivative also leads 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.
我们考虑一个具有长程幂次相互作用的线性和非线性耦合振子的一维链。动力学方程中的相应项与1//n - m/α + 1成正比。结果表明，当0 < α < 2时，红外极限下的运动方程可以转化为具有α阶里斯分数导数的介质方程。我们考虑了几个耦合振子模型，并展示了它们的同步如何作为分岔的结果出现，以及相应的解如何依赖于α。分数导数的存在还导致了局域结构的出现。推导了分数时间相关的复金兹堡 - 朗道（或非线性薛定谔）方程的特殊解。这些解被解释为振荡介质的同步态和局域结构。
