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Malomed BA. Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings. ENTROPY (BASEL, SWITZERLAND) 2024; 26:137. [PMID: 38392392 PMCID: PMC10887582 DOI: 10.3390/e26020137] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/08/2024] [Revised: 01/26/2024] [Accepted: 01/28/2024] [Indexed: 02/24/2024]
Abstract
This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.
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Affiliation(s)
- Boris A Malomed
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica 1000000, Chile
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Kartashov YV. Light bullets in moiré lattices. OPTICS LETTERS 2022; 47:4528-4531. [PMID: 36048696 DOI: 10.1364/ol.471022] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/20/2022] [Accepted: 07/27/2022] [Indexed: 06/15/2023]
Abstract
We predict that photonic moiré lattices produced by two mutually twisted periodic sublattices in a medium with Kerr nonlinearity can support stable three-dimensional (3D) light bullets localized in both space and time. The stability of light bullets and their properties are closely connected with the properties of linear spatial eigenmodes of moiré lattices that undergo localization-delocalization transition (LDT) upon the increase of the depth of one of the sublattices forming the moiré lattice, but only for twist angles corresponding to incommensurate, aperiodic moiré structures. Above the LDT threshold, such incommensurate moiré lattices support stable light bullets without an energy threshold. In contrast, commensurate-or periodic-moiré lattices arising at Pythagorean twist angles, whose eigenmodes are delocalized Bloch waves, can support stable light bullets only above a certain energy threshold. Moiré lattices below the LDT threshold cannot support stable light bullets for our parameters. Our results illustrate that the periodicity/aperiodicity of the underlying lattice is a crucial factor in determining the stability properties of the nonlinear 3D states.
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Ruban VP. Discrete vortices on spatially nonuniform two-dimensional electric networks. Phys Rev E 2020; 102:012204. [PMID: 32794945 DOI: 10.1103/physreve.102.012204] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2019] [Accepted: 06/19/2020] [Indexed: 06/11/2023]
Abstract
Two-dimensional arrays of nonlinear electric oscillators are considered theoretically where nearest neighbors are coupled by relatively small constant but nonequal capacitors. The dynamics is approximately reduced to a weakly dissipative defocusing discrete nonlinear Schrödinger equation with translationally noninvariant linear dispersive coefficients. Behavior of quantized discrete vortices in such systems is shown to depend strongly on the spatial profile of the internode coupling as well as on the ratio between time-increasing healing length and lattice spacings. In particular, vortex clusters can be stably trapped for some initial period of time by a circular barrier in the coupling profile, but then, due to gradual dissipative broadening of vortex cores, they lose stability and suddenly start to move.
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Affiliation(s)
- Victor P Ruban
- Landau Institute for Theoretical Physics, RAS, Chernogolovka, Moscow Region, 142432 Russia
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Zhang X, Xu X, Zheng Y, Chen Z, Liu B, Huang C, Malomed BA, Li Y. Semidiscrete Quantum Droplets and Vortices. PHYSICAL REVIEW LETTERS 2019; 123:133901. [PMID: 31697515 DOI: 10.1103/physrevlett.123.133901] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2019] [Revised: 08/01/2019] [Indexed: 06/10/2023]
Abstract
We consider a binary bosonic condensate with weak mean-field (MF) residual repulsion, loaded in an array of nearly one-dimensional traps coupled by transverse hopping. With the MF force balanced by the effectively one-dimensional attraction, induced in each trap by the Lee-Hung-Yang correction (produced by quantum fluctuations around the MF state), stable on-site- and intersite-centered semidiscrete quantum droplets (QDs) emerge in the array, as fundamental ones and self-trapped vortices, with winding numbers, at least, up to five, in both tightly bound and quasicontinuum forms. The application of a relatively strong trapping potential leads to squeezing transitions, which increase the number of sites in fundamental QDs and eventually replace vortex modes by fundamental or dipole ones. The results provide the first realization of stable semidiscrete vortex QDs, including ones with multiple vorticity.
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Affiliation(s)
- Xiliang Zhang
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Xiaoxi Xu
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Yiyin Zheng
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Zhaopin Chen
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and the Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel
| | - Bin Liu
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Chunqing Huang
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Boris A Malomed
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and the Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel
| | - Yongyao Li
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and the Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel
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Ruban VP. Vortex knots on three-dimensional lattices of nonlinear oscillators coupled by space-varying links. Phys Rev E 2019; 100:012205. [PMID: 31499831 DOI: 10.1103/physreve.100.012205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2019] [Indexed: 06/10/2023]
Abstract
Quantized vortices in a complex wave field described by a defocusing nonlinear Schrödinger equation with a space-varying dispersion coefficient are studied theoretically and compared to vortices in the Gross-Pitaevskii model with external potential. A discrete variant of the equation is used to demonstrate numerically that vortex knots in three-dimensional arrays of oscillators coupled by specially tuned weak links can exist for as long times as many as tens of typical vortex turnover periods.
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Affiliation(s)
- Victor P Ruban
- Landau Institute for Theoretical Physics RAS, Chernogolovka, Moscow region, 142432 Russia
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Xue JK, Zhang AX. Superfluid fermi gas in optical lattices: self-trapping, stable, moving solitons and breathers. PHYSICAL REVIEW LETTERS 2008; 101:180401. [PMID: 18999797 DOI: 10.1103/physrevlett.101.180401] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2008] [Indexed: 05/27/2023]
Abstract
We predict the existence of self-trapping, stable, moving solitons and breathers of Fermi wave packets along the Bose-Einstein condensation (BEC)-BCS crossover in one dimension (1D), 2D, and 3D optical lattices. The dynamical phase diagrams for self-trapping, solitons, and breathers of the Fermi matter waves along the BEC-BCS crossover are presented analytically and verified numerically by directly solving a discrete nonlinear Schrödinger equation. We find that the phase diagrams vary greatly along the BEC-BCS crossover; the dynamics of Fermi wave packet are different from that of Bose wave packet.
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Affiliation(s)
- Ju-Kui Xue
- Physics and Electronics Engineering College, Northwest Normal University, Lanzhou 730070, People's Republic of China.
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Hoq QE, Carretero-González R, Kevrekidis PG, Malomed BA, Frantzeskakis DJ, Bludov YV, Konotop VV. Surface solitons in three dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:036605. [PMID: 18851178 DOI: 10.1103/physreve.78.036605] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/05/2008] [Indexed: 05/26/2023]
Abstract
We study localized modes on the surface of a three-dimensional dynamical lattice. The stability of these structures on the surface is investigated and compared to that in the bulk of the lattice. Typically, the surface makes the stability region larger, an extreme example of that being the three-site "horseshoe"-shaped structure, which is always unstable in the bulk, while at the surface it is stable near the anticontinuum limit. We also examine effects of the surface on lattice vortices. For the vortex placed parallel to the surface, the increased stability-region feature is also observed, while the vortex cannot exist in a state normal to the surface. More sophisticated localized dynamical structures, such as five-site horseshoes and pyramids, are also considered.
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Affiliation(s)
- Q E Hoq
- Department of Mathematics, Western New England College, Springfield, Massachusetts 01119, USA
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Cuevas J, Malomed BA, Kevrekidis PG. Two-dimensional discrete solitons in rotating lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:046608. [PMID: 17995128 DOI: 10.1103/physreve.76.046608] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/15/2007] [Indexed: 05/25/2023]
Abstract
We introduce a two-dimensional discrete nonlinear Schrödinger (DNLS) equation with self-attractive cubic nonlinearity in a rotating reference frame. The model applies to a Bose-Einstein condensate stirred by a rotating strong optical lattice, or light propagation in a twisted bundle of nonlinear fibers. Two types of localized states are constructed: off-axis fundamental solitons (FSs), placed at distance R from the rotation pivot, and on-axis (R=0) vortex solitons (VSs), with vorticities S=1 and 2 . At a fixed value of rotation frequency Omega , a stability interval for the FSs is found in terms of the lattice coupling constant C , 0<C<C_{cr}(R) , with monotonically decreasing C_{cr}(R) . VSs with S=1 have a stability interval, C[over ]_{cr};{(S=1)}(Omega)<C<C_{cr};{(S=1)}(Omega) , which exists for Omega below a certain critical value, Omega_{cr};{(S=1)} . This implies that the VSs with S=1 are destabilized in the weak-coupling limit by the rotation. On the contrary, VSs with S=2 , that are known to be unstable in the standard DNLS equation, with Omega=0 , are stabilized by the rotation in region 0<C<C_{cr};{(S=2)} , with C_{cr};{(S=2)} growing as a function of Omega . Quadrupole and octupole on-axis solitons are considered too, their stability regions being weakly affected by Omega not equal 0 .
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Affiliation(s)
- Jesús Cuevas
- Departamento de Física Aplicada I, Escuela Universitaria Politécnica, C/ Virgen de Africa, 7, 41011 Sevilla, Spain
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Leblond H, Malomed BA, Mihalache D. Three-dimensional vortex solitons in quasi-two-dimensional lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:026604. [PMID: 17930164 DOI: 10.1103/physreve.76.026604] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2006] [Indexed: 05/25/2023]
Abstract
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.
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Affiliation(s)
- Hervé Leblond
- Laboratoire POMA, UMR 6136, Université d'Angers, 2 Bd Lavoisier, 49000 Angers, France
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Susanto H, Kevrekidis PG, Malomed BA, Carretero-González R, Frantzeskakis DJ. Discrete surface solitons in two dimensions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:056605. [PMID: 17677184 DOI: 10.1103/physreve.75.056605] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/26/2006] [Revised: 02/28/2007] [Indexed: 05/16/2023]
Abstract
We investigate fundamental localized modes in two-dimensional lattices with an edge (surface). The interaction with the edge expands the stability area for fundamental solitons, and induces a difference between dipoles oriented perpendicular and parallel to the surface. On the contrary, lattice vortex solitons cannot exist too close to the border. We also show, analytically and numerically, that the edge supports a species of localized patterns, which exists too but is unstable in the uniform lattice, namely, a horseshoe-shaped soliton, whose "skeleton" consists of three lattice sites. Unstable horseshoes transform themselves into a pair of ordinary solitons.
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Affiliation(s)
- H Susanto
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
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Efremidis NK, Hizanidis K, Malomed BA, Di Trapani P. Three-dimensional vortex solitons in self-defocusing media. PHYSICAL REVIEW LETTERS 2007; 98:113901. [PMID: 17501055 DOI: 10.1103/physrevlett.98.113901] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2006] [Indexed: 05/15/2023]
Abstract
We demonstrate that families of vortex solitons are possible in a bidispersive three-dimensional nonlinear Schrödinger equation. These solutions can be considered as extensions of two-dimensional dark vortex solitons which, along the third dimension, remain localized due to the interplay between dispersion and nonlinearity. Such vortex solitons can be observed in optical media with normal dispersion, normal diffraction, and defocusing nonlinearity.
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Affiliation(s)
- Nikolaos K Efremidis
- Department of Applied Mathematics, University of Crete, 71409 Heraklion, Crete, Greece
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Kevrekidis PG, Carretero-González R, Frantzeskakis DJ, Malomed BA, Diakonos FK. Skyrmion-like states in two- and three-dimensional dynamical lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:026603. [PMID: 17358432 DOI: 10.1103/physreve.75.026603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2006] [Revised: 11/06/2006] [Indexed: 05/14/2023]
Abstract
We construct, in discrete two-component systems with cubic nonlinearity, stable states emulating Skyrmions of the classical field theory. In the two-dimensional case, an analog of the baby Skyrmion is built on the square lattice as a discrete vortex soliton of a complex field [whose vorticity plays the role of the Skyrmion's winding number (WN)], coupled to a radial "bubble" in a real lattice field. The most compact quasi-Skyrmion on the cubic lattice is composed of a nearly planar complex-field discrete vortex and a three-dimensional real-field bubble; unlike its continuum counterpart which must have WN=2, this stable discrete state exists with WN=1. Analogs of Skyrmions in the one-dimensional lattice are also constructed. Stability regions for all these states are found in an analytical approximation and verified numerically. The dynamics of unstable discrete Skyrmions (which leads to the onset of lattice turbulence) and their partial stabilization by external potentials are explored too.
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Affiliation(s)
- P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
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Kevrekidis PG, Frantzeskakis DJ, Carretero-González R, Malomed BA, Bishop AR. Discrete solitons and vortices on anisotropic lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:046613. [PMID: 16383560 DOI: 10.1103/physreve.72.046613] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2005] [Indexed: 05/05/2023]
Abstract
We consider the effects of anisotropy on solitons of various types in two-dimensional nonlinear lattices, using the discrete nonlinear Schrödinger equation as a paradigm model. For fundamental solitons, we develop a variational approximation that predicts that broad quasicontinuum solitons are unstable, while their strongly anisotropic counterparts are stable. By means of numerical methods, it is found that, in the general case, the fundamental solitons and simplest on-site-centered vortex solitons ("vortex crosses") feature enhanced or reduced stability areas, depending on the strength of the anisotropy. More surprising is the effect of anisotropy on the so-called "super-symmetric" intersite-centered vortices ("vortex squares"), with the topological charge equal to the square's size : we predict in an analytical form by means of the Lyapunov-Schmidt theory, and confirm by numerical results, that arbitrarily weak anisotropy results in dramatic changes in the stability and dynamics in comparison with the degenerate, in this case, isotropic, limit.
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Affiliation(s)
- P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
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Carretero-González R, Kevrekidis PG, Malomed BA, Frantzeskakis DJ. Three-dimensional nonlinear lattices: from oblique vortices and octupoles to discrete diamonds and vortex cubes. PHYSICAL REVIEW LETTERS 2005; 94:203901. [PMID: 16090247 DOI: 10.1103/physrevlett.94.203901] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2005] [Indexed: 05/03/2023]
Abstract
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).
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Affiliation(s)
- R Carretero-González
- Nonlinear Dynamical Systems Group, Department of Mathematics and Statistics, San Diego State University, San Diego California 92182-7720, USA
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Porter MA, Carretero-González R, Kevrekidis PG, Malomed BA. Nonlinear lattice dynamics of Bose-Einstein condensates. CHAOS (WOODBURY, N.Y.) 2005; 15:15115. [PMID: 15836292 DOI: 10.1063/1.1858114] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
The Fermi-Pasta-Ulam (FPU) model, which was proposed 50 years ago to examine thermalization in nonmetallic solids and develop "experimental" techniques for studying nonlinear problems, continues to yield a wealth of results in the theory and applications of nonlinear Hamiltonian systems with many degrees of freedom. Inspired by the studies of this seminal model, solitary-wave dynamics in lattice dynamical systems have proven vitally important in a diverse range of physical problems-including energy relaxation in solids, denaturation of the DNA double strand, self-trapping of light in arrays of optical waveguides, and Bose-Einstein condensates (BECs) in optical lattices. BECs, in particular, due to their widely ranging and easily manipulated dynamical apparatuses-with one to three spatial dimensions, positive-to-negative tuning of the nonlinearity, one to multiple components, and numerous experimentally accessible external trapping potentials-provide one of the most fertile grounds for the analysis of solitary waves and their interactions. In this paper, we review recent research on BECs in the presence of deep periodic potentials, which can be reduced to nonlinear chains in appropriate circumstances. These reductions, in turn, exhibit many of the remarkable nonlinear structures (including solitons, intrinsic localized modes, and vortices) that lie at the heart of the nonlinear science research seeded by the FPU paradigm.
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Affiliation(s)
- Mason A Porter
- School of Mathematics and Center for Nonlinear Science, School of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA.
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