ENEA National Laboratory, Centro Ricerche Frascati, 00044 Frascati, Rome, Italy.
Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany.
Phys Rev E. 2023 Mar;107(3-1):034203. doi: 10.1103/PhysRevE.107.034203.
We devise an analytical method to deal with a class of nonlinear Schrödinger lattices with random potential and subquadratic power nonlinearity. An iteration algorithm is proposed based on the multinomial theorem, using Diophantine equations and a mapping procedure onto a Cayley graph. Based on this algorithm, we are able to obtain several hard results pertaining to asymptotic spreading of the nonlinear field beyond a perturbation theory approach. In particular, we show that the spreading process is subdiffusive and has complex microscopic organization involving both long-time trapping phenomena on finite clusters and long-distance jumps along the lattice consistent with Lévy flights. The origin of the flights is associated with the occurrence of degenerate states in the system; the latter are found to be a characteristic of the subquadratic model. The limit of quadratic power nonlinearity is also discussed and shown to result in a delocalization border, above which the field can spread to long distances on a stochastic process and below which it is Anderson localized similarly to a linear field.
我们设计了一种分析方法来处理一类具有随机势和次二次幂非线性的非线性薛定谔格子。基于多项定理，利用丢番图方程和映射到 Cayley 图的过程，提出了一种迭代算法。基于该算法，我们能够获得一些关于非线性场在微扰理论方法之外的渐近扩展的困难结果。具体来说，我们表明扩展过程是亚扩散的，并且具有复杂的微观组织，包括有限团簇上的长时间捕获现象以及沿着格子的长距离跳跃，与 Lévy 飞行一致。飞行的起源与系统中简并态的出现有关；后者被发现是次二次模型的特征。还讨论了二次幂非线性的极限，并表明它导致了离域边界，在该边界之上，场可以在随机过程中传播到长距离，而在该边界之下，它类似于线性场被安德森局域化。
