Dynamical chaos in nonlinear Schr\"odinger models with subquadratic power nonlinearity

TL;DR

This paper introduces an analytical method for nonlinear Schrödinger lattices with subquadratic nonlinearity, revealing subdiffusive spreading, complex microscopic organization, and the role of degenerate states in the dynamics.

Contribution

It develops a novel iteration algorithm based on multinomial theorem and graph mapping to analyze asymptotic spreading in nonlinear Schrödinger models with random potential.

Findings

Spreading is subdiffusive with long-time trapping and Levy flights.

Degenerate states cause long-distance jumps in the lattice.

Quadratic nonlinearity has a delocalization threshold.

Abstract

We devise an analytical method to deal with a class of nonlinear Schr\"odinger lattices with random potential and subquadratic power nonlinearity. An iteration algorithm is proposed based on multinomial theorem, using Diophantine equations and a mapping procedure onto a Cayley graph. Based on this algorithm, we were 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 L\'evy 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.