Two-dimensional discrete solitons in rotating lattices (original) (raw)
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Alternate Solitons: Nonlinearly Managed One- and Two-Dimensional Solitons in Optical Lattices
Studies in Applied Mathematics, 2005
We consider a model of Bose-Einstein condensates which combines a stationary optical lattice (OL) and periodic change of the sign of the scattering length (SL) due to the Feshbach-resonance management. Ordinary solitons and ones of the gap type being possible, respectively, in the model with constant negative and positive SL, an issue of interest is to find solitons alternating, in the case of the low-frequency modulation, between shapes of both types, across the zero-SL point. We find such alternate solitons and identify their stability regions in the 2D and 1D models. Three types of the dynamical regimes are distinguished: stable, unstable, and semi-stable. In the latter case, the soliton sheds off a conspicuous part of its initial norm before relaxing to a stable regime. In the 2D case, the threshold (minimum number of atoms) necessary for the existence of the alternate solitons is essentially higher than its counterparts for the ordinary and gap solitons in the static model. In the 1D model, the alternate solitons are also found only above a certain threshold, while the static 1D models have no threshold. In the 1D case, stable antisymmetric alternate solitons are found too. Additionally, we consider a possibility to apply the variational approximation (VA) to the description of stationary gap solitons, in the case of constant positive SL. It predicts the solitons in the first finite bandgap very accurately, and does it reasonably well in the second gap too. In higher bands, the VA predicts a border between tightly and loosely bound solitons.
Solitons in nonlinear lattices
This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are also surveyed, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation. The solitons are considered in one, two, and three dimensions. Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of one-dimensional solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for theoretical and experimental studies alike. In both the one-dimensional and two-dimensional cases, the mechanism that creates solitons in NLs in principle is different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.
Introduction to Solitons in Photonic Lattices
Springer Series in Optical Sciences, 2009
We present a review on wave propagation in nonlinear photonic lattices: arrays of optical waveguides made of nonlinear media. Such periodic structures provide an excellent environment for the direct experimental observations and theoretical studies of universal phenomena arising from the interplay between nonlinearity and Bloch periodicity. In particular, we review one-dimensional and two-dimensional lattice solitons, spatial gap solitons, and vortex lattice solitons.
Physical Review A, 2010
We investigate solitons and nonlinear Bloch waves in Bose-Einstein condensates trapped in optical lattices. By introducing specially designed localized profiles of the spatial modulation of the attractive nonlinearity, we construct an infinite number of exact soliton solutions in terms of the Mathieu and elliptic functions, with the chemical potential belonging to the semi-infinite bandgap of the optical-lattice-induced spectrum. Starting from the exact solutions, we employ the relaxation method to construct generic families of soliton solutions in a numerical form. The stability of the solitons is investigated through the computation of the eigenvalues for small perturbations, and also by direct simulations. Finally, we demonstrate a virtually exact (in the numerical sense) composition relation between nonlinear Bloch waves and solitons.
Physics Reports, 2008
We provide an overview of recent experimental and theoretical developments in the area of optical discrete solitons. By nature, discrete solitons represent self-trapped wavepackets in nonlinear periodic structures and result from the interplay between lattice diffraction (or dispersion) and material nonlinearity. In optics, this class of self-localized states has been successfully observed in both one-and two-dimensional nonlinear waveguide arrays. In recent years such photonic lattices have been implemented or induced in a variety of material systems, including those with cubic (Kerr), quadratic, photorefractive, and liquid-crystal nonlinearities. In all cases the underlying periodicity or discreteness leads to altogether new families of optical solitons that have no counterpart whatsoever in continuous systems. We first review the linear properties of photonic lattices that are key in the understanding of discrete solitons. The physics and dynamics of the fundamental discrete and gap solitons are then analyzed along with those of many other exotic classes -e.g. twisted, vector and multi-band, cavity, spatio-temporal, random-phase, vortex, and non-local lattice solitons, just to mention a few. The possibility of all-optically routing optical discrete solitons in 2D and 3D periodic environments using soliton collisions is also presented. Finally, soliton formation in optical quasi-crystals and at the boundaries of waveguide array structures are discussed.
Communications in Nonlinear Science and Numerical Simulation, 2020
We study coupled unstaggered-staggered soliton pairs emergent from a system of two coupled discrete nonlinear Schrödinger (DNLS) equations with the self-attractive on-site self-phasemodulation nonlinearity, coupled by the repulsive cross-phase-modulation interaction, on 1D and 2D lattice domains. These mixed modes are of a "symbiotic" type, as each component in isolation may only carry ordinary unstaggered solitons. While most work on DNLS systems addressed symmetric on-site-centered fundamental solitons, these models give rise to a variety of other excited states, which may also be stable. The simplest among them are antisymmetric states in the form of discrete twisted solitons, which have no counterparts in the continuum limit. In the extension to 2D lattice domains, a natural counterpart of the twisted states are vortical solitons. We first introduce a variational approximation (VA) for the solitons, and then correct it numerically to construct exact stationary solutions, which are then used as initial conditions for simulations to check if the stationary states persist under time evolution. Twocomponent solutions obtained include (i) 1D fundamental-twisted and twisted-twisted soliton pairs, (ii) 2D fundamental-fundamental soliton pairs, and (iii) 2D vortical-vortical soliton pairs. We also highlight a variety of other transient dynamical regimes, such as breathers and amplitude death. The findings apply to modeling binary Bose-Einstein condensates, loaded in a deep lattice potential, with identical or different atomic masses of the two components, and arrays of bimodal optical waveguides.