Self-trapping transition in nonlinear cubic lattices

Abstract

We explore the fundamental question about the critical nonlinearity value needed to dynamically localize energy in discrete nonlinear cubic (Kerr) lattices. We focus on the effective frequency and participation ratio of the profile to determine the transition into localization in one-, two-, and three-dimensional lattices. A simple and general criterion is developed -for the case of an initially localized excitation -to define the transition region in parameter space ("dynamical tongue") from a delocalized to a localized profile. We introduce a method for computing the dynamically excited frequencies, which helps us to validate our stationary ansatz approach and the effective frequency concept. A general analytical estimate of the critical nonlinearity is obtained, with an extra parameter to be determined. We found this parameter to be almost constant for two-dimensional systems, and proved its validity by applying it successfully to two-dimensional binary lattices.

Stationary discrete solitons in a driven dissipative Bose-Hubbard chain

“…In the so-called classical limit of the Bose-Hubbard model, the operators are replaced by their c-number average, obtaining the well-known discrete nonlinear Schrödinger equation (DNLS) [6]. In this limit, discrete solitons, both theoretically and experimentally, exist in different dimensions and topologies [7][8][9][10].…”

Superdiffusive transport and energy localization in disordered granular crystals

“…We thereby separate the cases in which oscillations of the numerical data for the second moment are admissible as statistical fluctuations from the ones in which oscillations are larger than statistical fluctuations. It is also useful to compute the spectral density associated with the dynamics, as that allows one to identify which frequencies are involved in the dynamics [88]. We use the spatiotemporal displacement distribution to calculate the normalized spectral density…”

Mobility of high-power solitons in saturable nonlinear photonic lattices

“…Since those regions exhibit bistability, the appearance of an intermediate asymmetric and unstable solution is inevitable, as it was shown in Ref. [8,9] for two-dimensional systems. Therefore, in this case, the effective energy barrier will strongly depend on the intermediate solutions (IS).…”

“…We demonstrate that in order to achieve a good mobility it is necessary to increase the amount of power up to the bifurcation point where the IS disappears. A constraint method [9,10] is used to identify the ISs and describe a pseudo-potential landscape among all stationary modes. By using the system properties, we found recurrent resonant behavior in power for gaussianlike shaped pulses showing enhanced mobility.…”

“…Now, in order to go deeper in the understanding of the dynamics of 1D saturable WAs, we construct an energy landscape. By defining the center of mass X ≡ n n|u n | 2 /P and using a constraint method [9,10], we compute H versus X and P . Fig.2(c) shows our computations around the first bistable region.…”