No abstract

We observe a crossover from strong to weak chaos in the spatiotemporal evolution of multiple site excitations within disordered chains with cubic nonlinearity. Recent studies have shown that Anderson localization is destroyed, and the wave packet spreading is characterized by an asymptotic divergence of the second moment m2 in time (as t 1/3 ), due to weak chaos. In the present paper, we observe the existence of a qualitatively new dynamical regime of strong chaos, in which the second moment spreads even faster (as t 1/2 ), with a crossover to the asymptotic law of weak chaos at larger times. We analyze the pecularities of these spreading regimes and perform extensive numerical simulations over large times with ensemble averaging. A technique of local derivatives on logarithmic scales is developed in order to quantitatively visualize the slow crossover processes.

In the absence of nonlinearity all eigenmodes of a chain with disorder are spatially localized (Anderson localization). The width of the eigenvalue spectrum and the average eigenvalue spacing inside the localization volume set two frequency scales. An initially localized wave packet spreads in the presence of nonlinearity. Nonlinearity introduces frequency shifts, which define three different evolution outcomes: (i) localization as a transient, with subsequent subdiffusion; (ii) the absence of the transient and immediate subdiffusion; (iii) self-trapping of a part of the packet and subdiffusion of the remainder. The subdiffusive spreading is due to a finite number of packet modes being resonant. This number does not change on average and depends only on the disorder strength. Spreading is due to corresponding weak chaos inside the packet, which slowly heats the cold exterior. The second moment of the packet grows as t;{alpha}. We find alpha=1/3.

We introduce a new, simple and efficient method for determining the ordered or chaotic nature of orbits in two-dimensional (2D), four-dimensional (4D) and six-dimensional (6D) symplectic maps: the computation of the alignment indices. For a given orbit we follow the evolution in time of two different initial deviation vectors computing the norms of the difference d − (parallel alignment index) and the addition d + (antiparallel alignment index) of the two vectors. The time evolution of the smaller alignment index reflects the chaotic or ordered nature of the orbit. In 2D maps the smaller alignment index tends to zero for both ordered and chaotic orbits but with completely different time rates, which allows us to distinguish between the two cases. In 4D and 6D maps the smaller alignment index tends to zero in the case of chaotic orbits, while it tends to a positive non-zero value in the case of ordered orbits. The efficiency of the new method is also shown in a case of weak chaos and a comparison with other known methods that separate chaotic from regular orbits is presented.