Milovanov Alexander V, Iomin Alexander
ENEA National Laboratory, Centro Ricerche Frascati, I-00044 Frascati, Rome, Italy Space Research Institute, Russian Academy of Sciences, 117997 Moscow, Russia and Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany.
Department of Physics and Solid State Institute, Technion, Haifa, 32000, Israel and Max-Planck-Institut für Physik komplexer Systeme, 01187 Dresden, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Jun;89(6):062921. doi: 10.1103/PhysRevE.89.062921. Epub 2014 Jun 26.
We study the phenomena of Anderson localization in the presence of nonlinear interaction on a lattice. A class of nonlinear Schrödinger models with arbitrary power nonlinearity is analyzed. We conceive the various regimes of behavior, depending on the topology of resonance overlap in phase space, ranging from a fully developed chaos involving Lévy flights to pseudochaotic dynamics at the onset of delocalization. It is demonstrated that the quadratic nonlinearity plays a dynamically very distinguished role in that it is the only type of power nonlinearity permitting an abrupt localization-delocalization transition with unlimited spreading already at the delocalization border. We describe this localization-delocalization transition as a percolation transition on the infinite Cayley tree (Bethe lattice). It is found in the vicinity of the criticality that the spreading of the wave field is subdiffusive in the limit t→+∞. The second moment of the associated probability distribution grows with time as a power law ∝ t^{α}, with the exponent α=1/3 exactly. Also we find for superquadratic nonlinearity that the analog pseudochaotic regime at the edge of chaos is self-controlling in that it has feedback on the topology of the structure on which the transport processes concentrate. Then the system automatically (without tuning of parameters) develops its percolation point. We classify this type of behavior in terms of self-organized criticality dynamics in Hilbert space. For subquadratic nonlinearities, the behavior is shown to be sensitive to the details of definition of the nonlinear term. A transport model is proposed based on modified nonlinearity, using the idea of "stripes" propagating the wave process to large distances. Theoretical investigations, presented here, are the basis for consistency analysis of the different localization-delocalization patterns in systems with many coupled degrees of freedom in association with the asymptotic properties of the transport.
我们研究了晶格上存在非线性相互作用时的安德森局域化现象。分析了一类具有任意幂次非线性的非线性薛定谔模型。我们根据相空间中共振重叠的拓扑结构，设想了各种行为状态，范围从涉及列维飞行的完全发展的混沌到离域化开始时的伪混沌动力学。结果表明，二次非线性在动力学上起着非常独特的作用，因为它是唯一一种允许在离域化边界处就出现具有无限扩展的突然局域化 - 离域化转变的幂次非线性类型。我们将这种局域化 - 离域化转变描述为无限凯莱树（贝塞晶格）上的渗流转变。发现在临界附近，波场的扩展在极限(t→ +∞)时是亚扩散的。相关概率分布的二阶矩随时间按幂律(\propto t^{α})增长，其中指数(α = 1/3)。此外，我们还发现对于超二次非线性，混沌边缘的类似伪混沌状态是自我控制的，因为它对传输过程集中的结构拓扑有反馈作用。然后系统自动（无需参数调整）发展其渗流点。我们根据希尔伯特空间中的自组织临界动力学对这种行为类型进行分类。对于次二次非线性，行为显示对非线性项定义的细节敏感。基于修改后的非线性提出了一种传输模型，利用“条纹”将波过程传播到远距离的思想。这里给出的理论研究是对具有许多耦合自由度的系统中不同局域化 - 离域化模式与传输渐近性质相关的一致性分析的基础。
