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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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Shena J, Kominis Y, Bountis A, Kovanis V. Spatial control of localized oscillations in arrays of coupled laser dimers. Phys Rev E 2020; 102:012201. [PMID: 32794981 DOI: 10.1103/physreve.102.012201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/08/2020] [Accepted: 06/19/2020] [Indexed: 11/07/2022]
Abstract
Arrays of coupled semiconductor lasers are systems possessing radically complex dynamics that makes them useful for numerous applications in beam forming and beam shaping. In this work, we investigate the spatial controllability of oscillation amplitudes in an array of coupled photonic dimers, each consisting of two semiconductor lasers driven by differential pumping rates. We consider parameter values for which each dimer's stable phase-locked state has become unstable through a Hopf bifurcation and we show that, by assigning appropriate pumping rate values to each dimer, high-amplitude oscillations coexist with negligibly low-amplitude oscillations. The spatial profile of the amplitude of oscillations across the array can be dynamically controlled by appropriate pumping rate values in each dimer. This feature is shown to be quite robust, even for random detuning between the lasers, and suggests a mechanism for dynamically reconfigurable production of a large diversity of spatial profiles of laser amplitude oscillations.
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Affiliation(s)
- Joniald Shena
- National University of Science and Technology MISiS, 4 Leninsky Avenue, Moscow 119049, Russia
| | - Yannis Kominis
- School of Applied Mathematical and Physical Science, National Technical University of Athens, 157 80 Athens, Greece
| | - Anastasios Bountis
- Department of Mathematics, School of Science and Technology, Nazarbayev University, Astana, Republic of Kazakhstan
| | - Vassilios Kovanis
- Bradley Department of Electrical and Computer Engineering, Northern Virginia, Virginia Polytechnic Institute and University, Blacksburg, Virginia 24060, USA
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Wang L, Malomed BA, Yan Z. Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media. Phys Rev E 2019; 99:052206. [PMID: 31212420 DOI: 10.1103/physreve.99.052206] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2019] [Indexed: 11/07/2022]
Abstract
We introduce a model based on the one-dimensional nonlinear Schrödinger equation with critical (quintic) or supercritical self-focusing nonlinearity. We demonstrate that a family of solitons, which are unstable in this setting against the critical or supercritical collapse, is stabilized by pinning to an attractive defect, that may also include a parity-time (PT)-symmetric gain-loss component. The model can be realized as a planar waveguide in nonlinear optics, and in a super-Tonks-Girardeau bosonic gas. For the attractive defect with the delta-functional profile, a full family of the pinned solitons is found in an exact analytical form. In the absence of the gain-loss term, the solitons' stability is investigated in an analytical form too, by means of the Vakhitov-Kolokolov criterion; in the presence of the PT-balanced gain and loss, the stability is explored by means of numerical methods. In particular, the entire family of pinned solitons is stable in the quintic (critical) medium if the gain-loss term is absent. A stability region for the pinned solitons persists in the model with an arbitrarily high power of the self-focusing nonlinearity. A weak gain-loss component gives rise to intricate alternations of stability and instability in the system's parameter plane. Those solitons which are unstable under the action of the supercritical self-attraction are destroyed by the collapse. On the other hand, if the self-attraction-driven instability is weak and the gain-loss term is present, unstable solitons spontaneously transform into localized breathers, while the collapse does not occur. The same outcome may be caused by a combination of the critical nonlinearity with the gain and loss. Instability of the solitons is also possible when the PT-symmetric gain-loss term is added to the subcritical nonlinearity. The system with self-repulsive nonlinearity is briefly considered too, producing completely stable families of pinned localized states.
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Affiliation(s)
- Li Wang
- Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.,School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 59978, Israel
| | - Zhenya Yan
- Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.,School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
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Luz E, Lutsky V, Granot E, Malomed BA. Robust [Formula: see text] symmetry of two-dimensional fundamental and vortex solitons supported by spatially modulated nonlinearity. Sci Rep 2019; 9:4483. [PMID: 30872648 PMCID: PMC6418126 DOI: 10.1038/s41598-019-40752-x] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2018] [Accepted: 02/18/2019] [Indexed: 11/16/2022] Open
Abstract
The real spectrum of bound states produced by [Formula: see text]-symmetric Hamiltonians usually suffers breakup at a critical value of the strength of gain-loss terms, i.e., imaginary part of the complex potential. The breakup essentially impedes the use of [Formula: see text]-symmetric systems for various applications. On the other hand, it is known that the [Formula: see text] symmetry can be made unbreakable in a one-dimensional (1D) model with self-defocusing nonlinearity whose strength grows fast enough from the center to periphery. The model is nonlinearizable, i.e., it does not have a linear spectrum, while the (unbreakable) [Formula: see text] symmetry in it is defined by spectra of continuous families of nonlinear self-trapped states (solitons). Here we report results for a 2D nonlinearizable model whose [Formula: see text] symmetry remains unbroken for arbitrarily large values of the gain-loss coefficient. Further, we introduce an extended 2D model with the imaginary part of potential ~xy in the Cartesian coordinates. The latter model is not a [Formula: see text]-symmetric one, but it also supports continuous families of self-trapped states, thus suggesting an extension of the concept of the [Formula: see text] symmetry. For both models, universal analytical forms are found for nonlinearizable tails of the 2D modes, and full exact solutions are produced for particular solitons, including ones with the unbreakable [Formula: see text] symmetry, while generic soliton families are found in a numerical form. The [Formula: see text]-symmetric system gives rise to generic families of stable single- and double-peak 2D solitons (including higher-order radial states of the single-peak solitons), as well as families of stable vortex solitons with m = 1, 2, and 3. In the model with imaginary potential ~xy, families of single- and multi-peak solitons and vortices are stable if the imaginary potential is subject to spatial confinement. In an elliptically deformed version of the latter model, an exact solution is found for vortex solitons with m = 1.
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Affiliation(s)
- Eitam Luz
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, 69978 Israel
- Department of Electrical and Electronic Engineering, Ariel University, Ariel, Israel
| | - Vitaly Lutsky
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, 69978 Israel
| | - Er’el Granot
- Department of Electrical and Electronic Engineering, Ariel University, Ariel, Israel
| | - Boris A. Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv, 69978 Israel
- Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv, 69978 Israel
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Chen Z, Li Y, Malomed BA. 𝒫𝒯-symmetric and antisymmetric nonlinear states in a split potential box. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2018; 376:20170369. [PMID: 29891496 PMCID: PMC6000150 DOI: 10.1098/rsta.2017.0369] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 03/29/2018] [Indexed: 05/20/2023]
Abstract
We introduce a one-dimensional [Formula: see text]-symmetric system, which includes the cubic self-focusing, a double-well potential in the form of an infinitely deep potential box split in the middle by a delta-functional barrier of an effective height ε, and constant linear gain and loss, γ, in each half-box. The system may be readily realized in microwave photonics. Using numerical methods, we construct [Formula: see text]-symmetric and antisymmetric modes, which represent, respectively, the system's ground state and first excited state, and identify their stability. Their instability mainly leads to blowup, except for the case of ε=0, when an unstable symmetric mode transforms into a weakly oscillating breather, and an unstable antisymmetric mode relaxes into a stable symmetric one. At ε>0, the stability area is much larger for the [Formula: see text]-antisymmetric state than for its symmetric counterpart. The stability areas shrink with increase of the total power, P In the linear limit, which corresponds to [Formula: see text], the stability boundary is found in an analytical form. The stability area of the antisymmetric state originally expands with the growth of γ, and then disappears at a critical value of γThis article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.
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Affiliation(s)
- Zhaopin Chen
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
| | - Yongyao Li
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 52800, People's Republic of China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
- 1, ITMO University, St. Petersburg 197101, Russia
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Huang C, Dong L. Stable vortex solitons in a ring-shaped partially-PT-symmetric potential. OPTICS LETTERS 2016; 41:5194-5197. [PMID: 27842091 DOI: 10.1364/ol.41.005194] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We address the existence and stability of vortex solitons in a ring-shaped partially-parity-time (pPT) configuration. In sharp contrast to the reported nonlinear modes in PT- or pPT-symmetric systems, stable vortex solitons with different topological charges can be supported by the proposed pPT potential, despite the system always being beyond the symmetry-breaking point. Vortex solitons are characterized by the number of phase singularities which equals the corresponding topological charge. At higher power, unstable higher-charged vortices degenerate into stable vortices with lower charges. Robust nonlinear vortices can be easily excited by an input Gaussian beam. Our results provide, to the best of our knowledge, the first example of stable solitons in a symmetry-breaking system.
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Mertens FG, Cooper F, Arévalo E, Khare A, Saxena A, Bishop AR. Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials. Phys Rev E 2016; 94:032213. [PMID: 27739801 DOI: 10.1103/physreve.94.032213] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/29/2016] [Indexed: 11/07/2022]
Abstract
We discuss the behavior of solitary wave solutions of the nonlinear Schrödinger equation (NLSE) as they interact with complex potentials, using a four-parameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE. We compare our method with a collective coordinate approach of Kominis and give examples where the two methods give qualitatively different answers. In our variational approach, we are able to give analytic results for the small oscillation frequency of the solitary wave oscillating parameters which agree with the numerical solution of the collective coordinate equations. We also verify that instabilities set in when the slope dp(t)/dv(t) becomes negative when plotted parametrically as a function of time, where p(t) is the momentum of the solitary wave and v(t) the velocity.
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Affiliation(s)
- Franz G Mertens
- Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany
| | - Fred Cooper
- Santa Fe Institute, Santa Fe, New Mexico 87501, USA and Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - Edward Arévalo
- Pontifical Catholic University of Chile, Departamento de Física, Santiago, Región Metropolitana, Chile
| | - Avinash Khare
- Physics Department, Savitribai Phule Pune University, Pune 411007, India
| | - Avadh Saxena
- Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - A R Bishop
- Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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Kartashov YV, Konotop VV, Torner L. Topological States in Partially-PT-Symmetric Azimuthal Potentials. PHYSICAL REVIEW LETTERS 2015; 115:193902. [PMID: 26588383 DOI: 10.1103/physrevlett.115.193902] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2015] [Indexed: 06/05/2023]
Abstract
We introduce partially-parity-time (pPT)-symmetric azimuthal potentials composed from individual PT-symmetric cells located on a ring, where two azimuthal directions are nonequivalent in a sense that in such potential excitations carrying topological dislocations exhibit different dynamics for different directions of energy circulation in the initial field distribution. Such nonconservative ratchetlike structures support rich families of stable vortex solitons in cubic nonlinear media, whose properties depend on the sign of the topological charge due to the nonequivalence of azimuthal directions. In contrast, oppositely charged vortex solitons remain equivalent in similar fully-PT-symmetric potentials. The vortex solitons in the pPT- and PT-symmetric potentials are shown to feature qualitatively different internal current distributions, which are described by different discrete rotation symmetries of the intensity profiles.
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Affiliation(s)
- Yaroslav V Kartashov
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
- Institute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow Region 142190, Russia
| | - Vladimir V Konotop
- Centro de Física Teórica e Computacional and Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, Campo Grande 2, Edifício C8, Lisboa 1749-016, Portugal
| | - Lluis Torner
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
- Universitat Politecnica de Catalunya, 08034 Barcelona, Spain
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D'Ambroise J, Kevrekidis PG, Malomed BA. Staggered parity-time-symmetric ladders with cubic nonlinearity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:033207. [PMID: 25871238 DOI: 10.1103/physreve.91.033207] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/30/2014] [Indexed: 06/04/2023]
Abstract
We introduce a ladder-shaped chain with each rung carrying a parity-time- (PT-) symmetric gain-loss dimer. The polarity of the dimers is staggered along the chain, meaning alternation of gain-loss and loss-gain rungs. This structure, which can be implemented as an optical waveguide array, is the simplest one which renders the system PT-symmetric in both horizontal and vertical directions. The system is governed by a pair of linearly coupled discrete nonlinear Schrödinger equations with self-focusing or defocusing cubic onsite nonlinearity. Starting from the analytically tractable anticontinuum limit of uncoupled rungs and using the Newton's method for continuation of the solutions with the increase of the inter-rung coupling, we construct families of PT-symmetric discrete solitons and identify their stability regions. Waveforms stemming from a single excited rung and double ones are identified. Dynamics of unstable solitons is investigated too.
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Affiliation(s)
- Jennie D'Ambroise
- Department of Mathematics and Statistics, Amherst College, Amherst, Massachusetts 01002-5000, USA
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-9305, USA
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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Akhmediev N, Kartashov Y. Nonlinear Photonics 2014: introduction. OPTICS EXPRESS 2015; 23:484-491. [PMID: 25835694 DOI: 10.1364/oe.23.000484] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
International Conference "Nonlinear Photonics-2014" took place in Barcelona, Spain on July 27-31, 2014. It was a part of the "Advanced Photonics Congress" which is becoming a traditional notable event in the world of photonics. The current focus issue of Optics Express contains contributions from the participants of the Conference and the Congress. The articles in this focus issue by no means represent the total number of the congress contributions (around 400). However, it demonstrates wide range of topics covered at the event. The next conference of this series is to be held in 2016 in Australia, which is the home of many researchers working in the field of photonics in general and nonlinear photonics in particular.
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