Three-Dimensional Nonlinear Lattices: From Oblique Vortices and Octupoles to Discrete Diamonds and Vortex Cubes

We construct a variety of novel localized states with distinct topological structures in the 3D discrete nonlinear Schr{\"{o}}dinger equation. The states can be created in Bose-Einstein condensates trapped in strong optical lattices, and crystals built of microresonators. These new structures, most of which have no counterparts in lower dimensions, range from purely real patterns of dipole, quadrupole and octupole types to vortex solutions, such as "diagonal" and "oblique" vortices, with axes oriented along the respective directions $(1,1,1)$ and $(1,1,0)$. Vortex "cubes" (stacks of two quasi-planar vortices with like or opposite polarities) and "diamonds" (discrete skyrmions formed by two vortices with orthogonal axes) are constructed too. We identify stability regions of these 3D solutions and compare them with their 2D counterparts, if any. An explanation for the stability/instability of most solutions is proposed. The evolution of unstable states is studied as well.Comment: 4 pages, 4 figures, submitted January 200

Dynamics and Manipulation of Matter-Wave Solitons in Optical Superlattices

We analyze the existence and stability of bright, dark, and gap matter-wave solitons in optical superlattices. Then, using these properties, we show that (time-dependent) ``dynamical superlattices'' can be used to controllably place, guide, and manipulate these solitons. In particular, we use numerical experiments to displace solitons by turning on a secondary lattice structure, transfer solitons from one location to another by shifting one superlattice substructure relative to the other, and implement solitonic ``path-following'', in which a matter wave follows the time-dependent lattice substructure into oscillatory motion.Comment: 6 pages, revtex, 6 figures, to appear in Physics Letters A; minor modifications from last versio

Families of Matter-Waves for Two-Component Bose-Einstein Condensates

We produce several families of solutions for two-component nonlinear Schr\"{o}dinger/Gross-Pitaevskii equations. These include domain walls and the first example of an antidark or gray soliton in the one component, bound to a bright or dark soliton in the other. Most of these solutions are linearly stable in their entire domain of existence. Some of them are relevant to nonlinear optics, and all to Bose-Einstein condensates (BECs). In the latter context, we demonstrate robustness of the structures in the presence of parabolic and periodic potentials (corresponding, respectively, to the magnetic trap and optical lattices in BECs).Comment: 6 pages, 4 figures, EPJD in pres

Exploring Rigidly Rotating Vortex Configurations and their Bifurcations in Atomic Bose-Einstein Condensates

In the present work, we consider the problem of a system of few vortices $N \leq 5$ as it emerges from its experimental realization in the field of atomic Bose-Einstein condensates. Starting from the corresponding equations of motion, we use a two-pronged approach in order to reveal the configuration space of the system's preferred dynamical states. On the one hand, we use a Monte-Carlo method parametrizing the vortex "particles" by means of hyperspherical coordinates and identifying the minimal energy ground states thereof for $N=2, ..., 5$ and different vortex particle angular momenta. We then complement this picture with a dynamical systems analysis of the possible rigidly rotating states. The latter reveals all the supercritical and subcritical pitchfork, as well as saddle-center bifurcations that arise exposing the full wealth of the problem even at such low dimensional cases. By corroborating the results of the two methods, it becomes fairly transparent which branch the Monte-Carlo approach selects for different values of the angular momentum which is used as a bifurcation parameter.Comment: 12 pages, 7 figures. New improved result

Hydrodynamics and two-dimensional dark lump solitons for polariton superfluids

We study a two-dimensional incoherently pumped exciton-polariton condensate described by an open-dissipative Gross-Pitaevskii equation for the polariton dynamics coupled to a rate equation for the exciton density. Adopting a hydrodynamic approach, we use multiscale expansion methods to derive several models appearing in the context of shallow water waves with viscosity. In particular, we derive a Boussinesq/Benney-Luke–type equation and its far-field expansion in terms of Kadomtsev-Petviashvili-I (KP-I) equations for right- and left-going waves. From the KP-I model, we predict the existence of vorticity-free, weakly (algebraically) localized two-dimensional dark-lump solitons. We find that, in the presence of dissipation, dark lumps exhibit a lifetime three times larger than that of planar dark solitons. Direct numerical simulations show that dark lumps do exist, and their dissipative dynamics is well captured by our analytical approximation. It is also shown that lumplike and vortexlike structures can spontaneously be formed as a result of the transverse “snaking” instability of dark soliton stripes.Europe Union project AEI/FEDER: MAT2016-79866-

Polarized States and Domain Walls in Spinor Bose-Einstein Condensates

We study spin-polarized states and their stability in anti-ferromagnetic states of spinor (F=1) quasi-one-dimensional Bose-Einstein condensates. Using analytical approximations and numerical methods, we find various types of polarized states, including: patterns of the Thomas-Fermi type; structures with a pulse-shape in one component inducing a hole in the other components; states with holes in all three components; and domain walls. A Bogoliubov-de Gennes analysis reveals that families of these states contain intervals of a weak oscillatory instability, except for the domain walls, which are always stable. The development of the instabilities is examined by means of direct numerical simulations.Comment: 7 pages, 9 figures, submitted to Phys. Rev.

Discrete surface solitons in two dimensions

We investigate fundamental localized modes in 2D lattices with an edge (surface). Interaction with the edge expands the stability area for ordinary solitons, and induces a difference between perpendicular and parallel dipoles; on the contrary, lattice vortices cannot exist too close to the border. Furthermore, we show analytically and numerically that the edge stabilizes a novel wave species, which is entirely unstable in the uniform lattice, namely, a "horseshoe" soliton, consisting of 3 sites. Unstable horseshoes transform themselves into a pair of ordinary solitons.Comment: 6 pages, 4 composite figure