Abstract
Anomalous diffusion is simulated in this paper by studying the transport of discrete solitons in a lattice with evolving disorder. We find a Richardson-type diffusion for the small solitons and a regime of transient diffusion for larger solitons within the ensemble-averaged description. As a comparison, the time-averaged observables present a ballistic scaling for both cases. However, distribution of these observables changes remarkably with the soliton size. Our results suggest violation of ergodicity for the solitons' diffusive processes, which are expected to shed light on further understanding of the discreteness-disorder-nonlinearity interaction.
Highlights
Discrete solitons (DSs), the self-trapped wave packets balancing lattice dispersion and nonlinearity, have been observed in a variety of physical systems such as optics, Bose-Einstein condensates, and crystals [1,2,3,4,5]
One of the theoretical efforts is dedicated to the understanding of DS mobility in ordered lattices [6,7,8,9], where some prototype forms of discrete nonlinear Schrödinger (DNLS) models have been considered by using the concept of the Peierls-Nabarro (PN) potentials [10,11]
We look at the Richardson-type diffusion of DSs in the sense of single particle tracking, for which a typical time-averaged mean squared displacement over an ANOMALOUS DIFFUSION OF DISCRETE SOLITONS
Summary
Discrete solitons (DSs), the self-trapped wave packets balancing lattice dispersion and nonlinearity, have been observed in a variety of physical systems such as optics, Bose-Einstein condensates, and crystals [1,2,3,4,5]. For the wave-packet spreading (width expansion) described by a linear Schrödinger equation with evolving disorder, the optical experiment [18] as well as theoretical study [19] suggest a transport faster than ballistic (we call it hyperdiffusion), while the diffusion rate may depend on the correlation properties of the relevant random potential [20,21]. For the highly localized DSs, we might be interested in their particlelike nature, considering random walks of the DS’s center-of-mass In such a vein, former works have studied for continuous systems the Brownian motion of solitons [24,25], transport of nonlocal solitons [26,27,28], as well as the Anderson localization of solitons [29,30]. Due to addition of the discreteness, the transport of DSs would present more sophisticated and interesting visions as revealed hereafter
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