Three-Dimensional Solitary Waves and Vortices in a Discrete Nonlinear Schrödinger Lattice (original) (raw)
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Three-dimensional vortex solitons in quasi-two-dimensional lattices
Physical Review E, 2007
We consider the three-dimensional (3D) Gross-Pitaevskii or nonlinear Schrödinger equation with a quasi-2D square-lattice potential (which corresponds to the optical lattice trapping a self-attractive Bose-Einstein condensate, or, in some approximation, to a photonic-crystal fiber, in terms of nonlinear optics). Stable 3D solitons, with embedded vorticity S=1 and 2, are found by means of the variational approximation and in a numerical form. They are built, basically, as sets of four fundamental solitons forming a rhombus, with phase shifts piS2 between adjacent sites, and an empty site in the middle. The results demonstrate two species of stable 3D solitons, which were not studied before, viz., localized vortices ("spinning light bullets," in terms of optics) with S>1 , and vortex solitons (with any S not equal 0 ) supported by a lattice in the 3D space. Typical scenarios of instability development (collapse or decay) of unstable localized vortices are identified too.
Stable higher-order vortices and quasivortices in the discrete nonlinear Schrödinger equation
Physical Review E, 2004
Vortex solitons with the topological charge S = 3, and "quasivortex" (multipole) solitons, which exist instead of the vortices with S = 2 and 4, are constructed on a square lattice in the discrete nonlinear Schrödinger equation (true vortices with S = 2 were known before, but they are unstable). For each type of solitary wave, its stability interval is found, in terms of the intersite coupling constant. The interval shrinks with increase of S. At couplings above a critical value, oscillatory instabilities set in, resulting in breakup of the vortex or quasivortex into lattice solitons with a lower vorticity. Such localized states may be observed in optical guiding structures, and in Bose-Einstein condensates loaded into optical lattices.
Persistence and stability of discrete vortices in nonlinear Schrödinger lattices
Physica D: Nonlinear Phenomena, 2005
We study discrete vortices in the two-dimensional nonlinear Schrödinger lattice with small coupling between lattice nodes. The discrete vortices in the anti-continuum limit of zero coupling represent a finite set of excited nodes on a closed discrete contour with a non-zero charge. Using the Lyapunov-Schmidt reductions, we analyze continuation and termination of the discrete vortices for small coupling between lattice nodes. An example of a square discrete contour is considered that includes the vortex cell (also known as the off-site vortex). We classify families of symmetric and asymmetric discrete vortices that bifurcate from the anti-continuum limit. We predict analytically and confirm numerically the number of unstable eigenvalues associated with each family of such discrete vortices.
Physical Review Letters, 2005
We construct a variety of novel localized topological structures in the 3D discrete nonlinear Schrödinger 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 multipole patterns and diagonal vortices to vortex ''cubes'' (stack of two quasiplanar vortices) and ''diamonds'' (formed by two orthogonal vortices).
Physical Review E, 2008
We consider a prototypical dynamical lattice model, namely, the discrete nonlinear Schroedinger equation on nonsquare lattice geometries. We present a systematic classification of the solutions that arise in principal six-lattice-site and three-lattice-site contours in the form of both discrete multipole solitons and discrete vortices. Additionally to identifying the possible states, we analytically track their linear stability both qualitatively and quantitatively. We find that among the six-site configurations, the hexapole of alternating phases, as well as the vortex of topological charge S=2 have intervals of stability; among three-site states, only the vortex of topological charge S=1 may be stable in the case of focusing nonlinearity. These conclusions are confirmed both for hexagonal and for honeycomb lattices by means of detailed numerical bifurcation analysis of the stationary states from the anticontinuum limit, and by direct simulations to monitor the dynamical instabilities, when the latter arise. The dynamics reveal a wealth of nonlinear behavior resulting not only in single-site solitary wave forms, but also in robust multisite breathing structures.
Solitary vortices and gap solitons in rotating optical lattices
Physical Review A, 2009
We report on results of a systematic analysis of two-dimensional solitons and localized vortices in models including a rotating periodic potential and the cubic nonlinearity, with the latter being both self-attractive and self-repulsive. The models apply to Bose-Einstein condensates stirred by rotating optical lattices and to twisted photonic-crystal fibers, or bundled arrays of waveguides, in nonlinear optics. In the case of the attractive nonlinearity, we construct compound states in the form of vortices, quadrupoles, and supervortices, all trapped in the slowly rotating lattice, and identify their stability limits ͑fundamental solitons in this setting were studied previously͒. In rapidly rotating potentials, vortices decouple from the lattice in the azimuthal direction and assume an annular shape. In the model with the repulsive nonlinearity, which was not previously explored in this setting, gap solitons and vortices are found in both cases of the slow and rapid rotations. It is again concluded that the increase in the rotation frequency leads to the transition from fully trapped corotating vortices to ring-shaped ones. We also study "crater-shaped" vortices in the attraction model, which, unlike their compound counterparts, are trapped, essentially, in one cell of the lattice. Previously, only unstable vortices of this type were reported. We demonstrate that they have a certain stability region. Solitons and vortices are found here in the numerical form, and, in parallel, by means of the variational approximation. ¡ £ ¥ £¨© © © © © ¡ £ ¥ £ © © FIG. 5. An example of a stable vortex chain carrying topological charge S = 5, built in a slowly rotating lattice. Parameters are A =1, ⍀ = 0.05, and N = 40. The meaning of the panels is explained in the text.
Vortex and dipole solitons in complex two-dimensional nonlinear lattices
Physical Review A, 2012
Using computational methods, it is found that the two-dimensional nonlinear Schrödinger (NLS) equation with a quasicrystal lattice potential admits multiple dipole and vortex solitons. The linear and the nonlinear stability of these solitons is investigated using direct simulations of the NLS equation and its linearized equation. It is shown that certain multiple vortex structures on quasicrystal lattices can be linearly unstable but nonlinearly stable. These results have application to investigations of localized structures in nonlinear optics and Bose-Einstein condensates.
Observation of discrete vortex solitons in optically induced photonic lattices
Physical review letters, 2004
We report on the first experimental observation of discrete vortex solitons in two-dimensional optically induced photonic lattices. We demonstrate strong stabilization of an optical vortex by the lattice in a self-focusing nonlinear medium and study the generation of the discrete vortices from a broad class of singular beams.