1
|
Malomed BA. Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings. Entropy (Basel) 2024; 26:137. [PMID: 38392392 PMCID: PMC10887582 DOI: 10.3390/e26020137] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [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.
Collapse
Affiliation(s)
- Boris A Malomed
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica 1000000, Chile
| |
Collapse
|
2
|
Malomed BA. Basic fractional nonlinear-wave models and solitons. Chaos 2024; 34:022102. [PMID: 38341765 DOI: 10.1063/5.0190039] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/02/2023] [Accepted: 01/09/2024] [Indexed: 02/13/2024]
Abstract
This review article provides a concise summary of one- and two-dimensional models for the propagation of linear and nonlinear waves in fractional media. The basic models, which originate from Laskin's fractional quantum mechanics and more experimentally relevant setups emulating fractional diffraction in optics, are based on the Riesz definition of fractional derivatives, which are characterized by the respective Lévy indices. Basic species of one-dimensional solitons, produced by the fractional models which include cubic or quadratic nonlinear terms, are outlined too. In particular, it is demonstrated that the variational approximation is relevant in many cases. A summary of the recently demonstrated experimental realization of the fractional group-velocity dispersion in fiber lasers is also presented.
Collapse
Affiliation(s)
- Boris A Malomed
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| |
Collapse
|
3
|
Sun WR, Malomed BA, Li JH. Rogue waves and instability arising from long-wave-short-wave resonance beyond the integrable regime. Phys Rev E 2024; 109:024209. [PMID: 38491594 DOI: 10.1103/physreve.109.024209] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2023] [Accepted: 01/19/2024] [Indexed: 03/18/2024]
Abstract
We consider instability and localized patterns arising from the long-wave-short-wave resonance in the nonintegrable regime numerically. We study the stability and instability of elliptic-function periodic waves with respect to subharmonic perturbations, whose period is a multiple of the period of the elliptic waves. We thus find the modulational instability (MI) of the corresponding dnoidal waves. Upon varying parameters of dnoidal waves, spectrally unstable ones can be transformed into stable states via the Hamiltonian Hopf bifurcation. For snoidal waves, we find a transition of the dominant instability scenario between the MI and the instability with a bubblelike spectrum. For cnoidal waves, we produce three variants of the MI. Evolution of the unstable states is also considered, leading to formation of rogue waves on top of the elliptic-wave and continuous-wave backgrounds.
Collapse
Affiliation(s)
- Wen-Rong Sun
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
| | - Boris A Malomed
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - Jin-Hua Li
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
| |
Collapse
|
4
|
Kumar S, Yadav A, Malomed BA. Bimetal Thin Film, Semiconductors, and 2D Nanomaterials in SPR Biosensors: An Approach to Enhanced Urine Glucose Sensing. IEEE Trans Nanobioscience 2024; PP:1-1. [PMID: 38224505 DOI: 10.1109/tnb.2024.3354571] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/17/2024]
Abstract
This work introduces a systematic approach for the development of Kretschmann configuration-based biosensors designed for non-invasive urine glucose detection. The methodology encompasses the utilization of various semiconductors, including Silicon (Si), Germanium (Ge), Gallium Nitride (GaN), Aluminum Nitride (AlN), and Indium Nitride (InN), in combination with a bimetallic layer (comprising Au and Ag films of equal thickness) to enhance the biosensor sensitivity. Additionally, 2D nanomaterials, such as Black Phosphorus and Graphene, are integrated into the semiconductor layers to enhance performance further. These configurations are meticulously optimized through the application of the transfer matrix method (TMM), and the sensing parameters are assessed using the angular modulation method. Among the semiconductors, AlN and GaN exhibit superior results. On these substrates, Graphene and Black phosphorous (BP) layers are applied, resulting in four final structures (thicknesses in nm): BK7/Au(26)/Ag(26)/Si(6)/BP(0.53)/Biosample, BK7/Au(26)/Ag(26)/AlN(14)/BP(0.53)/Biosample, BK7/Au(26)/Ag(26)/GaN(12)/BP(0.53)/Biosample, and BK7/Au(26)/Ag(26)/GaN(12)/Graphene(0.34)/Biosample. These biosensors achieve Sensitivity(° /RIU) and Figure of Merit (FoM) (1/RIU) of 380, 360, 440, 400, and 58.5, 90, 90.65, and 82.4, respectively. Subsequently, these high-performing sensors undergo testing with actual urine glucose samples. Among them, two biosensors, BK7/Au(26)/Ag(26)/AlN(14)/BP (0.53)/Biosample and BK7/Au(26)/Ag(26)/GaN(14)/Graphene(0.34)/Biosample, exhibit outstanding performance, with sensitivities (° /RIU) and FoM (1/RIU) of 394.44 & 294.44, and 112.6 & 92.01 respectively. A comparison is also made with relevant previously published work, revealing improved performance in glucose detection.
Collapse
|
5
|
The Tai LX, Longobucco M, Viet Hung N, Pałuba B, Trippenbach M, Malomed BA, Astrauskas I, Pugžlys A, Baltuška A, Buczyński R, Bugár I. Analysis of high-contrast all-optical dual-wavelength switching in asymmetric dual-core fibers. Opt Lett 2024; 49:149-152. [PMID: 38134174 DOI: 10.1364/ol.500292] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/13/2023] [Accepted: 11/17/2023] [Indexed: 12/24/2023]
Abstract
We systematically present experimental and theoretical results for the dual-wavelength switching of 1560 nm, 75 fs signal pulses (SPs) driven by 1030 nm, and 270 fs control pulses (CPs) in a dual-core fiber (DCF). We demonstrate a switching contrast of 31.9 dB, corresponding to a propagation distance of 14 mm, achieved by launching temporally synchronized SP-CP pairs into the fast core of the DCF with moderate inter-core asymmetry. Our analysis employs a system of three coupled propagation equations to identify the compensation of the asymmetry by nonlinearity as the physical mechanism behind the efficient switching performance.
Collapse
|
6
|
Xu X, Zhao F, Huang J, He H, Zhang L, Chen Z, Nie Z, Malomed BA, Li Y. Semidiscrete optical vortex droplets in quasi-phase-matched photonic crystals. Opt Express 2023; 31:38343-38354. [PMID: 38017942 DOI: 10.1364/oe.506130] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/21/2023] [Accepted: 10/18/2023] [Indexed: 11/30/2023]
Abstract
What we believe is a new scheme for producing semidiscrete self-trapped vortices ("swirling photon droplets") in photonic crystals with competing quadratic (χ(2)) and self-defocusing cubic (χ(3)) nonlinearities is proposed. The photonic crystal is designed with a striped structure, in the form of spatially periodic modulation of the χ(2) susceptibility, which is imposed by the quasi-phase-matching technique. Unlike previous realizations of semidiscrete optical modes in composite media, built as combinations of continuous and arrayed discrete waveguides, the semidiscrete vortex "droplets" are produced here in the fully continuous medium. This work reveals that the system supports two types of semidiscrete vortex droplets, viz., onsite- and intersite-centered ones, which feature, respectively, odd and even numbers of stripes, N. Stability areas for the states with different values of N are identified in the system's parameter space. Some stability areas overlap with each other, giving rise to the multistability of states with different N. The coexisting states are mutually degenerate, featuring equal values of the Hamiltonian and propagation constant. An experimental scheme to realize the droplets is outlined, suggesting new possibilities for the long-distance transmission of nontrivial vortex beams in nonlinear media.
Collapse
|
7
|
Zhao Y, Hu HJ, Zhou QQ, Qiu ZC, Xue L, Xu SL, Zhou Q, Malomed BA. Three-dimensional solitons in Rydberg-dressed cold atomic gases with spin-orbit coupling. Sci Rep 2023; 13:18079. [PMID: 37872222 PMCID: PMC10593778 DOI: 10.1038/s41598-023-44745-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2023] [Accepted: 10/11/2023] [Indexed: 10/25/2023] Open
Abstract
We present numerical results for three-dimensional (3D) solitons with symmetries of the semi-vortex (SV) and mixed-mode (MM) types, which can be created in spinor Bose-Einstein condensates of Rydberg atoms under the action of the spin-orbit coupling (SOC). By means of systematic numerical computations, we demonstrate that the interplay of SOC and long-range spherically symmetric Rydberg interactions stabilize the 3D solitons, improving their resistance to collapse. We find how the stability range depends on the strengths of the SOC and Rydberg interactions and the soft-core atomic radius.
Collapse
Affiliation(s)
- Yuan Zhao
- Laboratory of Optoelectronic Information and Intelligent Control, Hubei University of Science and Technology, Xianning, 437100, China
- School of Biomedical Engineering and Medical Imaging, Xianning Medical College, Hubei University of Science and Technology, Xianning, 437100, China
| | - Heng-Jie Hu
- Laboratory of Optoelectronic Information and Intelligent Control, Hubei University of Science and Technology, Xianning, 437100, China
- School of Electronic and Information Engineering, Hubei University of Science and Technology, Xianning, 437100, China
| | - Qian-Qian Zhou
- Laboratory of Optoelectronic Information and Intelligent Control, Hubei University of Science and Technology, Xianning, 437100, China
- School of Electronic and Information Engineering, Hubei University of Science and Technology, Xianning, 437100, China
| | - Zhang-Cai Qiu
- Laboratory of Optoelectronic Information and Intelligent Control, Hubei University of Science and Technology, Xianning, 437100, China
- School of Electronic and Information Engineering, Hubei University of Science and Technology, Xianning, 437100, China
| | - Li Xue
- Laboratory of Optoelectronic Information and Intelligent Control, Hubei University of Science and Technology, Xianning, 437100, China
- School of Electronic and Information Engineering, Hubei University of Science and Technology, Xianning, 437100, China
| | - Si-Liu Xu
- Laboratory of Optoelectronic Information and Intelligent Control, Hubei University of Science and Technology, Xianning, 437100, China.
- School of Biomedical Engineering and Medical Imaging, Xianning Medical College, Hubei University of Science and Technology, Xianning, 437100, China.
| | - Qin Zhou
- Research Center of Nonlinear Science, School of Mathematical and Physical Sciences, Wuhan Textile University, Wuhan, 430200, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Ramat Aviv, P.O.B. 39040, Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| |
Collapse
|
8
|
Liu B, Cai X, Qin X, Jiang X, Xie J, Malomed BA, Li Y. Ring-shaped quantum droplets with hidden vorticity in a radially periodic potential. Phys Rev E 2023; 108:044210. [PMID: 37978625 DOI: 10.1103/physreve.108.044210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/20/2023] [Accepted: 09/20/2023] [Indexed: 11/19/2023]
Abstract
We study the stability and characteristics of two-dimensional circular quantum droplets (QDs) with embedded hidden vorticity (HV), i.e., opposite angular momenta in two components, formed by binary Bose-Einstein condensates (BECs) trapped in a radially periodic potential. The system is modeled by the Gross-Pitaevskii equations with the Lee-Huang-Yang terms, which represent the higher-order self-repulsion induced by quantum fluctuations around the mean-field state, and a potential which is a periodic function of the radial coordinate. Ring-shaped QDs with high winding numbers (WNs) of the HV type, which are trapped in particular circular troughs of the radial potential, are produced by means of the imaginary-time-integration method. Effects of the depth and period of the potential on these QD states are studied. The trapping capacity of individual circular troughs is identified. Stable compound states in the form of nested multiring patterns are constructed too, including ones with WNs of opposite signs. The stably coexisting ring-shaped QDs with different WNs can be used for the design of BEC-based data-storage schemes.
Collapse
Affiliation(s)
- Bin Liu
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
- Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology, Foshan University, Foshan 528000, China
| | - Xiaoyan Cai
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Xizhou Qin
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
- Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology, Foshan University, Foshan 528000, China
| | - Xunda Jiang
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
- Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology, Foshan University, Foshan 528000, China
| | - Jianing Xie
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
- Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology, Foshan University, Foshan 528000, 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 69978, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - Yongyao Li
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
- Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology, Foshan University, Foshan 528000, China
| |
Collapse
|
9
|
Dong L, Fan M, Malomed BA. Stable higher-charge vortex solitons in the cubic-quintic medium with a ring potential. Opt Lett 2023; 48:4817-4820. [PMID: 37707910 DOI: 10.1364/ol.500054] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2023] [Accepted: 08/19/2023] [Indexed: 09/15/2023]
Abstract
We put forward a model for trapping stable optical vortex solitons (VSs) with high topological charges m. The cubic-quintic nonlinear medium with an imprinted ring-shaped modulation of the refractive index is shown to support two branches of VSs, which are controlled by the radius, width, and depth of the modulation profile. While the lower-branch VSs are unstable in their nearly whole existence domain, the upper branch is completely stable. Vortex solitons with m ≤ 12 obey the anti-Vakhitov-Kolokolov stability criterion. The results suggest possibilities for the creation of stable narrow optical VSs with a low power, carrying higher vorticities.
Collapse
|
10
|
Biasi A, Evnin O, Malomed BA. Obstruction to ergodicity in nonlinear Schrödinger equations with resonant potentials. Phys Rev E 2023; 108:034204. [PMID: 37849119 DOI: 10.1103/physreve.108.034204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/29/2023] [Accepted: 08/11/2023] [Indexed: 10/19/2023]
Abstract
We identify a class of trapping potentials in cubic nonlinear Schrödinger equations (NLSEs) that make them nonintegrable, but prevent the emergence of power spectra associated with ergodicity. The potentials are characterized by equidistant energy spectra (e.g., the harmonic-oscillator trap), which give rise to a large number of resonances enhancing the nonlinearity. In a broad range of dynamical solutions, spanning the regimes in which the nonlinearity may be either weak or strong in comparison with the linear part of the NLSE, the power spectra are shaped as narrow (quasidiscrete), evenly spaced spikes, unlike generic truly continuous (ergodic) spectra. We develop an analytical explanation for the emergence of these spectral features in the case of weak nonlinearity. In the strongly nonlinear regime, the presence of such structures is tracked numerically by performing simulations with random initial conditions. Some potentials that prevent ergodicity in this manner are of direct relevance to Bose-Einstein condensates: they naturally appear in 1D, 2D, and 3D Gross-Pitaevskii equations (GPEs), the quintic version of these equations, and a two-component GPE system.
Collapse
Affiliation(s)
- Anxo Biasi
- Laboratoire de Physique de l'Ecole Normale Supérieure ENS Université PSL, CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
| | - Oleg Evnin
- Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
- Theoretische Natuurkunde, Vrije Universiteit Brussel and International Solvay Institutes, Brussels 1050, Belgium
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| |
Collapse
|
11
|
Liu L, Sun WR, Malomed BA. Formation of Rogue Waves and Modulational Instability with Zero-Wavenumber Gain in Multicomponent Systems with Coherent Coupling. Phys Rev Lett 2023; 131:093801. [PMID: 37721812 DOI: 10.1103/physrevlett.131.093801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/26/2022] [Accepted: 06/20/2023] [Indexed: 09/20/2023]
Abstract
It is known that rogue waves (RWs) are generated by the modulational instability (MI) of the baseband type. Starting with the Bers-Kaup-Reiman system for three-wave resonant interactions, we identify a specific RW-building mechanism based on MI which includes zero wavenumber in the gain band. An essential finding is that this mechanism works solely under a linear relation between the MI gain and a vanishingly small wavenumber of the modulational perturbation. The same mechanism leads to the creation of RWs by MI in other multicomponent systems-in particular, in the massive Thirring model.
Collapse
Affiliation(s)
- Lei Liu
- College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
| | - Wen-Rong Sun
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, 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 69978, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| |
Collapse
|
12
|
Kumar S, Li P, Zeng L, He J, Malomed BA. A solvable model for symmetry-breaking phase transitions. Sci Rep 2023; 13:13768. [PMID: 37612417 PMCID: PMC10447515 DOI: 10.1038/s41598-023-40704-6] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2023] [Accepted: 08/16/2023] [Indexed: 08/25/2023] Open
Abstract
Analytically solvable models are benchmarks in studies of phase transitions and pattern-forming bifurcations. Such models are known for phase transitions of the second kind in uniform media, but not for localized states (solitons), as integrable equations which produce solitons do not admit intrinsic transitions in them. We introduce a solvable model for symmetry-breaking phase transitions of both the first and second kinds (alias sub- and supercritical bifurcations) for solitons pinned to a combined linear-nonlinear double-well potential, represented by a symmetric pair of delta-functions. Both self-focusing and defocusing signs of the nonlinearity are considered. In the former case, exact solutions are produced for symmetric and asymmetric solitons. The solutions explicitly demonstrate a switch between the symmetry-breaking transitions of the first and second kinds (i.e., sub- and supercritical bifurcations, respectively). In the self-defocusing model, the solution demonstrates the transition of the second kind which breaks antisymmetry of the first excited state.
Collapse
Affiliation(s)
- Shatrughna Kumar
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv, 69978, Israel
| | - Pengfei Li
- Department of Physics, Taiyuan Normal University, Jinzhong, 030619, China
| | - Liangwei Zeng
- Department of Basic Course, Guangzhou Maritime University, Guangzhou, 510725, China
| | - Jingsong He
- Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong, 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, 69978, Israel.
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile.
| |
Collapse
|
13
|
Kumar S, Li P, Malomed BA. One-dimensional Townes solitons in dual-core systems with localized coupling. Phys Rev E 2023; 108:024202. [PMID: 37723768 DOI: 10.1103/physreve.108.024202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/02/2023] [Accepted: 07/06/2023] [Indexed: 09/20/2023]
Abstract
The recent creation of Townes solitons (TSs) in binary Bose-Einstein condensates and experimental demonstration of spontaneous symmetry breaking (SSB) in solitons propagating in dual-core optical fibers has drawn renewed interest in the TS and SSB phenomenology in these and other settings. In particular, stabilization of TSs, which are always unstable in free space, is a relevant problem with various ramifications. We introduce a system which admits exact solutions for both TSs and SSB of solitons. It is based on a dual-core waveguide with quintic self-focusing and fused (localized) coupling between the cores. The respective system of coupled nonlinear Schrödinger equations gives rise to exact solutions for full families of symmetric and asymmetric solitons, which are produced by the supercritical SSB bifurcation (i.e., the symmetry-breaking phase transition of the second kind). Stability boundaries of asymmetric solitons are identified by dint of numerical methods. Unstable solitons spontaneously transform into robust moderately asymmetric breathers or strongly asymmetric states with small intrinsic oscillations. The setup can be used in the design of photonic devices operating in coupling and switching regimes.
Collapse
Affiliation(s)
- Shatrughna Kumar
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel
| | - Pengfei Li
- Department of Physics, Taiyuan Normal University, Jinzhong, 030619, China
- Institute of Computational and Applied Physics, Taiyuan Normal University, Jinzhong, 030619, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| |
Collapse
|
14
|
Zhao H, Malomed BA, Smalyukh II. Topological solitonic macromolecules. Nat Commun 2023; 14:4581. [PMID: 37516736 PMCID: PMC10387112 DOI: 10.1038/s41467-023-40335-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/27/2023] [Accepted: 07/19/2023] [Indexed: 07/31/2023] Open
Abstract
Being ubiquitous, solitons have particle-like properties, exhibiting behaviour often associated with atoms. Bound solitons emulate dynamics of molecules, though solitonic analogues of polymeric materials have not been considered yet. Here we experimentally create and model soliton polymers, which we call "polyskyrmionomers", built of atom-like individual solitons characterized by the topological invariant representing the skyrmion number. With the help of nonlinear optical imaging and numerical modelling based on minimizing the free energy, we reveal how topological point defects bind the solitonic quasi-atoms into polyskyrmionomers, featuring linear, branched, and other macromolecule-resembling architectures, as well as allowing for encoding data by spatial distributions of the skyrmion number. Application of oscillating electric fields activates diverse modes of locomotion and internal vibrations of these self-assembled soliton structures, which depend on symmetry of the solitonic macromolecules. Our findings suggest new designs of soliton meta matter, with a potential for the use in fundamental research and technology.
Collapse
Affiliation(s)
- Hanqing Zhao
- Department of Physics, University of Colorado, Boulder, CO, 80309, USA
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - Ivan I Smalyukh
- Department of Physics, University of Colorado, Boulder, CO, 80309, USA.
- Materials Science and Engineering Program, University of Colorado, Boulder, CO, 80309, USA.
- International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM²), Hiroshima University, Higashihiroshima, Hiroshima, 739-8526, Japan.
- Renewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder, CO, 80309, USA.
| |
Collapse
|
15
|
Strunin DV, Malomed BA. Symmetry-breaking transitions in quiescent and moving solitons in fractional couplers. Phys Rev E 2023; 107:064203. [PMID: 37464702 DOI: 10.1103/physreve.107.064203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/12/2023] [Accepted: 05/08/2023] [Indexed: 07/20/2023]
Abstract
We consider phase transitions, in the form of spontaneous symmetry breaking (SSB) bifurcations of solitons, in dual-core couplers with fractional diffraction and cubic self-focusing acting in each core, characterized by Lévy index α. The system represents linearly coupled optical waveguides with the fractional paraxial diffraction or group-velocity dispersion (the latter system was used in a recent experiment [Nat. Commun. 14, 222 (2023)10.1038/s41467-023-35892-8], which demonstrated the first observation of the wave propagation in an effectively fractional setup). By dint of numerical computations and variational approximation, we identify the SSB in the fractional coupler as the bifurcation of the subcritical type (i.e., the symmetry-breaking phase transition of the first kind), whose subcriticality becomes stronger with the increase of fractionality 2-α, in comparison with very weak subcriticality in the case of the nonfractional diffraction, α=2. In the Cauchy limit of α→1, it carries over into the extreme subcritical bifurcation, manifesting backward-going branches of asymmetric solitons which never turn forward. The analysis of the SSB bifurcation is extended for moving (tilted) solitons, which is a nontrivial problem because the fractional diffraction does not admit Galilean invariance. Collisions between moving solitons are studied too, featuring a two-soliton symmetry-breaking effect and merger of the solitons.
Collapse
Affiliation(s)
- Dmitry V Strunin
- School of Mathematics, Physics and Computing, University of Southern Queensland, Toowoomba, Queensland 4350, Australia
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| |
Collapse
|
16
|
Cui Y, Zhang Y, Huang L, Zhang A, Liu Z, Kuang C, Tao C, Chen D, Liu X, Malomed BA. Dichromatic "Breather Molecules" in a Mode-Locked Fiber Laser. Phys Rev Lett 2023; 130:153801. [PMID: 37115872 DOI: 10.1103/physrevlett.130.153801] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/05/2022] [Accepted: 03/20/2023] [Indexed: 06/19/2023]
Abstract
Bound states of solitons ("molecules") occur in various settings, playing an important role in the operation of fiber lasers, optical emulation, encoding, and communications. Soliton interactions are generally related to breathing dynamics in nonlinear dissipative systems, and maintain potential applications in spectroscopy. In the present work, dichromatic breather molecules (DBMs) are created in a synchronized mode-locked fiber laser. Real-time delay-shifting interference spectra are measured to display the temporal evolution of the DBMs, that cannot be observed by means of the usual real-time spectroscopy. As a result, robust out-of-phase vibrations are found as a typical intrinsic mode of DBMs. The same bound states are produced numerically in the framework of a model combining equations for the population inversion in the mode-locked laser and cross-phase-modulation-coupled complex Ginzburg-Landau equations for amplitudes of the optical fields in the fiber segments of the laser cavity. The results demonstrate that the Q-switching instability induces the onset of breathing oscillations. The findings offer new possibilities for the design of various regimes of the operation of ultrafast lasers.
Collapse
Affiliation(s)
- Yudong Cui
- State Key Laboratory of Modern Optical Instrumentation, College of Optical Science and Engineering, Zhejiang University, Hangzhou 310027, China
- ZJU-Hangzhou Global Scientific and Technological Innovation Center, No. 733 Jianshe San Road, Xiaoshan District, Hangzhou, Zhejiang, China, 311200
- Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Yusheng Zhang
- Hangzhou Institute of Advanced Studies, Zhejiang Normal University, Hangzhou 311231, China
| | - Lin Huang
- Ceyear Technologies Co., Ltd., Qingdao 266555, China
| | - Aiguo Zhang
- Ceyear Technologies Co., Ltd., Qingdao 266555, China
| | - Zhiming Liu
- Ceyear Technologies Co., Ltd., Qingdao 266555, China
| | - Cuifang Kuang
- State Key Laboratory of Modern Optical Instrumentation, College of Optical Science and Engineering, Zhejiang University, Hangzhou 310027, China
- ZJU-Hangzhou Global Scientific and Technological Innovation Center, No. 733 Jianshe San Road, Xiaoshan District, Hangzhou, Zhejiang, China, 311200
| | - Chenning Tao
- Hangzhou Institute of Advanced Studies, Zhejiang Normal University, Hangzhou 311231, China
| | - Daru Chen
- Hangzhou Institute of Advanced Studies, Zhejiang Normal University, Hangzhou 311231, China
| | - Xu Liu
- State Key Laboratory of Modern Optical Instrumentation, College of Optical Science and Engineering, Zhejiang University, Hangzhou 310027, China
- ZJU-Hangzhou Global Scientific and Technological Innovation Center, No. 733 Jianshe San Road, Xiaoshan District, Hangzhou, Zhejiang, China, 311200
| | - Boris A Malomed
- Department of Physical Electronics, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica 1000000, Chile
| |
Collapse
|
17
|
Zhao F, Xu X, He H, Zhang L, Zhou Y, Chen Z, Malomed BA, Li Y. Vortex Solitons in Quasi-Phase-Matched Photonic Crystals. Phys Rev Lett 2023; 130:157203. [PMID: 37115876 DOI: 10.1103/physrevlett.130.157203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/26/2022] [Revised: 02/02/2023] [Accepted: 03/20/2023] [Indexed: 06/19/2023]
Abstract
We report solutions for stable compound solitons in a three-dimensional quasi-phase-matched photonic crystal with the quadratic (χ^{(2)}) nonlinearity. The photonic crystal is introduced with a checkerboard structure, which can be realized by means of the available technology. The solitons are built as four-peak vortex modes of two types, rhombuses and squares (intersite- and onsite-centered self-trapped states, respectively). Their stability areas are identified in the system's parametric space (rhombuses occupy an essentially broader stability domain), while all bright vortex solitons are subject to strong azimuthal instability in uniform χ^{(2)} media. Possibilities for experimental realization of the solitons are outlined.
Collapse
Affiliation(s)
- Feiyan Zhao
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Xiaoxi Xu
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Hexiang He
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Li Zhang
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Yangui Zhou
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Zhaopin Chen
- Physics Department and Solid-State Institute, Technion, Haifa 32000, Israel
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - Yongyao Li
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
- Guangdong-Hong Kong-Macao Joint Laboratory for Intelligent Micro-Nano Optoelectronic Technology, Foshan University, Foshan 528000, China
| |
Collapse
|
18
|
Song J, Yan Z, Malomed BA. Formations and dynamics of two-dimensional spinning asymmetric quantum droplets controlled by a PT-symmetric potential. Chaos 2023; 33:033141. [PMID: 37003809 DOI: 10.1063/5.0138420] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/12/2022] [Accepted: 03/03/2023] [Indexed: 06/19/2023]
Abstract
In this paper, vortex solitons are produced for a variety of 2D spinning quantum droplets (QDs) in a PT-symmetric potential, modeled by the amended Gross-Pitaevskii equation with Lee-Huang-Yang corrections. In particular, exact QD states are obtained under certain parameter constraints, providing a guide to finding the respective generic family. In a parameter region of the unbroken PT symmetry, different families of QDs originating from the linear modes are obtained in the form of multipolar and vortex droplets at low and high values of the norm, respectively, and their stability is investigated. In the spinning regime, QDs become asymmetric above a critical rotation frequency, most of them being stable. The effect of the PT-symmetric potential on the spinning and nonspinning QDs is explored by varying the strength of the gain-loss distribution. Generally, spinning QDs trapped in the PT-symmetric potential exhibit asymmetry due to the energy flow affected by the interplay of the gain-loss distribution and rotation. Finally, interactions between spinning or nonspinning QDs are explored, exhibiting elastic collisions under certain conditions.
Collapse
Affiliation(s)
- Jin Song
- KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
| | - Zhenya Yan
- KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
| |
Collapse
|
19
|
Liu S, Zhang Y, Malomed BA, Karimi E. Experimental realisations of the fractional Schrödinger equation in the temporal domain. Nat Commun 2023; 14:222. [PMID: 36641454 PMCID: PMC9840624 DOI: 10.1038/s41467-023-35892-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2022] [Accepted: 01/06/2023] [Indexed: 01/15/2023] Open
Abstract
The fractional Schrödinger equation (FSE)-a natural extension of the standard Schrödinger equation-is the basis of fractional quantum mechanics. It can be obtained by replacing the kinetic-energy operator with a fractional derivative. Here, we report the experimental realisation of an optical FSE for femtosecond laser pulses in the temporal domain. Programmable holograms and the single-shot measurement technique are respectively used to emulate a Lévy waveguide and to reconstruct the amplitude and phase of the pulses. Varying the Lévy index of the FSE and the initial pulse, the temporal dynamics is observed in diverse forms, including solitary, splitting and merging pulses, double Airy modes, and "rain-like" multi-pulse patterns. Furthermore, the transmission of input pulses carrying a fractional phase exhibits a "fractional-phase protection" effect through a regular (non-fractional) material. The experimentally generated fractional time-domain pulses offer the potential for designing optical signal-processing schemes.
Collapse
Affiliation(s)
- Shilong Liu
- grid.28046.380000 0001 2182 2255Department of Physics, University of Ottawa, 25 Templeton, Ottawa, ON K1N 6N5 Canada ,grid.13402.340000 0004 1759 700XState Key Laboratory of Modern Optical Instrumentation, College of Optical Science and Engineering, Zhejiang University, 310027 Hangzhou, Zhejiang China
| | - Yingwen Zhang
- grid.28046.380000 0001 2182 2255Department of Physics, University of Ottawa, 25 Templeton, Ottawa, ON K1N 6N5 Canada ,grid.24433.320000 0004 0449 7958National Research Council, 100 Sussex Dr, Ottawa, ON K1A 0R6 Canada
| | - Boris A. Malomed
- grid.12136.370000 0004 1937 0546Department of Physical Electronics, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv, 69978 Israel ,grid.412182.c0000 0001 2179 0636Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - Ebrahim Karimi
- grid.28046.380000 0001 2182 2255Department of Physics, University of Ottawa, 25 Templeton, Ottawa, ON K1N 6N5 Canada ,grid.24433.320000 0004 0449 7958National Research Council, 100 Sussex Dr, Ottawa, ON K1A 0R6 Canada
| |
Collapse
|
20
|
Kumar S, Li P, Malomed BA. Domain walls in fractional media. Phys Rev E 2022; 106:054207. [PMID: 36559477 DOI: 10.1103/physreve.106.054207] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2022] [Accepted: 10/17/2022] [Indexed: 06/17/2023]
Abstract
Currently, much interest is drawn to the analysis of optical and matter-wave modes supported by the fractional diffraction in nonlinear media. We predict a new type of such states in the form of domain walls (DWs) in the two-component system of immiscible fields. Numerical study of the underlying system of fractional nonlinear Schrödinger equations demonstrates the existence and stability of DWs at all values of the respective Lévy index (α<2), which determines the fractional diffraction, and at all values of the XPM/SPM ratio β in the two-component system above the immiscibility threshold. The same conclusion is obtained for DWs in the system which includes the linear coupling, alongside the XPM interaction between the immiscible components. Analytical results are produced for the scaling of the DW's width. The DW solutions are essentially simplified in the special case of β=3, as well as close to the immiscibility threshold. In addition to symmetric DWs, asymmetric ones are constructed too, in the system with unequal diffraction coefficients and/or different Lévy indices of the two components.
Collapse
Affiliation(s)
- Shatrughna Kumar
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel
| | - Pengfei Li
- Department of Physics, Taiyuan Normal University, Jinzhong 030619, China
- Institute of Computational and Applied Physics, Taiyuan Normal University, Jinzhong 030619, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O.B. 39040, Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| |
Collapse
|
21
|
Malomed BA, Kevrekidis PG, Koukouloyannis V, Whitaker N, Frantzeskakis DJ. Interactions of solitary waves in the Adlam-Allen model. Phys Rev E 2022; 106:034209. [PMID: 36266868 DOI: 10.1103/physreve.106.034209] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/30/2021] [Accepted: 08/31/2022] [Indexed: 06/16/2023]
Abstract
We study the interactions of two or more solitary waves in the Adlam-Allen model describing the evolution of a (cold) plasma of positive and negative charges, in the presence of electric and transverse magnetic fields. In order to show that the interactions feature an exponentially repulsive nature, we elaborate two distinct approaches: (a) using energetic considerations and the Hamiltonian structure of the model, and (b) using the so-called Manton method. We compare these findings with results of direct simulations, and we identify adjustments necessary to achieve a quantitative match between them. Additional connections are made, such as with solitons of the Korteweg-de Vries equation. New challenges are identified in connection to this model and its solitary waves.
Collapse
Affiliation(s)
- Boris A Malomed
- Department of Interdisciplinary Studies, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - Panayotis G Kevrekidis
- Department of Mathematics Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - Vassilis Koukouloyannis
- Department of Mathematics, Laboratory of Applied Mathematics and Mathematical Modelling, University of the Aegean, Karlovasi, 83200 Samos, Greece
| | - Nathaniel Whitaker
- Department of Mathematics Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - Dimitrios J Frantzeskakis
- Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
| |
Collapse
|
22
|
Luo Z, Liu Y, Li Y, Batle J, Malomed BA. Stability limits for modes held in alternating trapping-expulsive potentials. Phys Rev E 2022; 106:014201. [PMID: 35974589 DOI: 10.1103/physreve.106.014201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2022] [Accepted: 06/09/2022] [Indexed: 06/15/2023]
Abstract
We elaborate a scheme of trapping-expulsion management (TEM), in the form of the quadratic potential periodically switching between confinement and expulsion, as a means of stabilization of two-dimensional dynamical states against the backdrop of the critical collapse driven by the cubic self-attraction with strength g. The TEM scheme may be implemented, as spatially or temporally periodic modulations, in optics or BEC, respectively. The consideration is carried out by dint of numerical simulations and variational approximation (VA). In terms of the VA, the dynamics amounts to a nonlinear Ermakov equation, which, in turn, is tantamount to a linear Mathieu equation. Stability boundaries are found as functions of g and parameters of the periodic modulation of the trapping potential. Below the usual collapse threshold, which is known, in the numerical form, as g<g_{c}^{(num)}≈5.85 (in the standard notation), the stability is limited by the onset of the parametric resonance. This stability limit, including the setup with the self-repulsive sign of the cubic term (g<0), is accurately predicted by the VA. At g>g_{c}^{(num)}, the collapse threshold is found with the help of full numerical simulations. The relative increase of g_{c} above g_{c}^{(num)} is ≈1.5%. It is a meaningful result, even if its size is small, because the collapse threshold is a universal constant which is difficult to change.
Collapse
Affiliation(s)
- Zhihuan Luo
- Department of Applied Physics, South China Agricultural University, Guangzhou 510642, China
| | - Yan Liu
- Department of Applied Physics, South China Agricultural University, Guangzhou 510642, China
| | - Yongyao Li
- School of Physics and Optoelectronic Engineering, Foshan University, Foshan 528000, China
| | - Josep Batle
- CRISP Centre de Recerca Independent de sa Pobla, C. Albéniz 12, 07420 sa Pobla, Balearic Islands, Spain
| | - 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 69978, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| |
Collapse
|
23
|
Hacker N, Malomed BA. Trapping wave fields in an expulsive potential by means of linear coupling. Phys Rev E 2022; 105:034213. [PMID: 35428167 DOI: 10.1103/physreve.105.034213] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2022] [Accepted: 03/08/2022] [Indexed: 06/14/2023]
Abstract
We demonstrate the existence of confined states in one- and two-dimensional (1D and 2D) systems of two linearly coupled components, with the confining harmonic-oscillator (HO) potential acting upon one component and an expulsive anti-HO potential acting upon the other. The systems can be implemented in optical and BEC dual-core waveguides. In the 1D linear system, codimension-one solutions are found in an exact form for the ground state (GS) and dipole mode (the first excited state). Generic solutions are produced by means of the variational approximation and are found in a numerical form. Exact codimension-one solutions and generic numerical ones are also obtained for the GS and vortex states in the 2D system (the exact solutions are found for all values of the vorticity). Both the trapped and antitrapped components of the bound states may be dominant ones, in terms of the norm. The localized modes may be categorized as bound states in continuum, as they coexist with delocalized ones. The 1D states, as well as the GS in 2D, are weakly affected and remain stable if the self-attractive or repulsive nonlinearity is added to the system. The self-attraction makes the vortex states unstable against splitting, while they remain stable under the action of the self-repulsion.
Collapse
Affiliation(s)
- Nir Hacker
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter interaction, Tel Aviv University, Tel Aviv 69978, Israel
| | - 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 69978, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| |
Collapse
|
24
|
Affiliation(s)
- Xiuye Liu
- State Key Laboratory of Transient Optics and Photonics Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences Xi'an 710119 China
- University of Chinese Academy of Sciences Beijing 100049 China
| | - Boris A. Malomed
- Department of Physical Electronics School of Electrical Engineering Faculty of Engineering, and the Center for Light‐Matter Interaction Tel Aviv University Ramat Aviv Tel Aviv P.O.B. 39040 Israel
- Instituto de Alta Investigación Universidad de Tarapacá Casilla 7D Arica Chile
| | - Jianhua Zeng
- State Key Laboratory of Transient Optics and Photonics Xi'an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences Xi'an 710119 China
- University of Chinese Academy of Sciences Beijing 100049 China
| |
Collapse
|
25
|
Biasi A, Evnin O, Malomed BA. Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap. Phys Rev E 2021; 104:034210. [PMID: 34654088 DOI: 10.1103/physreve.104.034210] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2021] [Accepted: 08/27/2021] [Indexed: 06/13/2023]
Abstract
We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schrödinger (NLS) equation with an isotropic harmonic oscillator potential. The dynamics in this regime is dominated by resonant interactions between quartets of linear normal modes, accurately captured by the corresponding resonant approximation. Within this approximation, we identify Fermi-Pasta-Ulam-like recurrence phenomena, whereby the normal-mode spectrum passes in close proximity of the initial configuration, and two-mode states with time-independent mode amplitude spectra that translate into long-lived breathers of the original NLS equation. We comment on possible implications of these findings for nonlinear optics and matter-wave dynamics in Bose-Einstein condensates.
Collapse
Affiliation(s)
- Anxo Biasi
- Institute of Theoretical Physics, Jagiellonian University, Krakow 30-348, Poland
| | - Oleg Evnin
- Department of Physics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
- Theoretische Natuurkunde, Vrije Universiteit Brussel and International Solvay Institutes, Brussels 1050, Belgium
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| |
Collapse
|
26
|
Li P, Malomed BA, Mihalache D. Symmetry-breaking bifurcations and ghost states in the fractional nonlinear Schrödinger equation with a PT-symmetric potential. Opt Lett 2021; 46:3267-3270. [PMID: 34197432 DOI: 10.1364/ol.428254] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/19/2021] [Accepted: 06/09/2021] [Indexed: 06/13/2023]
Abstract
We report symmetry-breaking and restoring bifurcations of solitons in a fractional Schrödinger equation with cubic or cubic-quintic (CQ) nonlinearity and a parity-time-symmetric potential, which may be realized in optical cavities. Solitons are destabilized at the bifurcation point, and, in the case of CQ nonlinearity, the stability is restored by an inverse bifurcation. Two mutually conjugate branches of ghost states (GSs), with complex propagation constants, are created by the bifurcation, solely in the case of fractional diffraction. While GSs are not true solutions, direct simulations confirm that their shapes and results of their stability analysis provide a "blueprint" for the evolution of genuine localized modes in the system.
Collapse
|
27
|
Bhat IA, Sivaprakasam S, Malomed BA. Modulational instability and soliton generation in chiral Bose-Einstein condensates with zero-energy nonlinearity. Phys Rev E 2021; 103:032206. [PMID: 33862781 DOI: 10.1103/physreve.103.032206] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/14/2020] [Accepted: 02/17/2021] [Indexed: 11/07/2022]
Abstract
By means of analytical and numerical methods, we address the modulational instability (MI) in chiral condensates governed by the Gross-Pitaevskii equation including the current nonlinearity. The analysis shows that this nonlinearity partly suppresses the MI driven by the cubic self-focusing, although the current nonlinearity is not represented in the system's energy (although it modifies the momentum), hence it may be considered as zero-energy nonlinearity. Direct simulations demonstrate generation of trains of stochastically interacting chiral solitons by MI. In the ring-shaped setup, the MI creates a single traveling solitary wave. The sign of the current nonlinearity determines the direction of propagation of the emerging solitons.
Collapse
Affiliation(s)
- Ishfaq Ahmad Bhat
- Department of Physics, Pondicherry University, Pondicherry 605014, India
| | - S Sivaprakasam
- Department of Physics, Pondicherry University, Pondicherry 605014, India
| | - 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 69978, Israel.,Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| |
Collapse
|
28
|
Cutrona A, Hanzard PH, Rowley M, Totero-Gongora JS, Peccianti M, Malomed BA, Oppo GL, Pasquazi A. Temporal cavity solitons in a laser-based microcomb: a path to a self-starting pulsed laser without saturable absorption. Opt Express 2021; 29:6629-6646. [PMID: 33726180 DOI: 10.1364/oe.418283] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/07/2021] [Accepted: 01/30/2021] [Indexed: 06/12/2023]
Abstract
We theoretically present a design of self-starting operation of microcombs based on laser-cavity solitons in a system composed of a micro-resonator nested in and coupled to an amplifying laser cavity. We demonstrate that it is possible to engineer the modulational-instability gain of the system's zero state to allow the start-up with a well-defined number of robust solitons. The approach can be implemented by using the system parameters, such as the cavity length mismatch and the gain shape, to control the number and repetition rate of the generated solitons. Because the setting does not require saturation of the gain, the results offer an alternative to standard techniques that provide laser mode-locking.
Collapse
|
29
|
Li P, Malomed BA, Mihalache D. Metastable soliton necklaces supported by fractional diffraction and competing nonlinearities. Opt Express 2020; 28:34472-34488. [PMID: 33182916 DOI: 10.1364/oe.409908] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/09/2020] [Accepted: 10/20/2020] [Indexed: 06/11/2023]
Abstract
We demonstrate that the fractional cubic-quintic nonlinear Schrödinger equation, characterized by its Lévy index, maintains ring-shaped soliton clusters ("necklaces") carrying orbital angular momentum. They can be built, in the respective optical setting, as circular chains of fundamental solitons linked by a vortical phase field. We predict semi-analytically that the metastable necklace-shaped clusters persist, corresponding to a local minimum of an effective potential of interaction between adjacent solitons in the cluster. Systematic simulations corroborate that the clusters stay robust over extremely large propagation distances, even in the presence of strong random perturbations.
Collapse
|
30
|
Luo D, Jin Y, Nguyen JHV, Malomed BA, Marchukov OV, Yurovsky VA, Dunjko V, Olshanii M, Hulet RG. Creation and Characterization of Matter-Wave Breathers. Phys Rev Lett 2020; 125:183902. [PMID: 33196245 DOI: 10.1103/physrevlett.125.183902] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/05/2020] [Revised: 09/03/2020] [Accepted: 09/28/2020] [Indexed: 06/11/2023]
Abstract
We report the creation of quasi-1D excited matter-wave solitons, "breathers," by quenching the strength of the interactions in a Bose-Einstein condensate with attractive interactions. We characterize the resulting breathing dynamics and quantify the effects of the aspect ratio of the confining potential, the strength of the quench, and the proximity of the 1D-3D crossover for the two-soliton breather. Furthermore, we demonstrate the complex dynamics of a three-soliton breather created by a stronger interaction quench. Our experimental results, which compare well with numerical simulations, provide a pathway for utilizing matter-wave breathers to explore quantum effects in large many-body systems.
Collapse
Affiliation(s)
- D Luo
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
| | - Y Jin
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
| | - J H V Nguyen
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
| | - B A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, 6997801 Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - O V Marchukov
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, 6997801 Tel Aviv, Israel
- Institute for Applied Physics, Technical University of Darmstadt, 64289 Darmstadt, Germany
| | - V A Yurovsky
- School of Chemistry, Tel Aviv University, 6997801 Tel Aviv, Israel
| | - V Dunjko
- Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - M Olshanii
- Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - R G Hulet
- Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
| |
Collapse
|
31
|
Dos Santos MCP, Malomed BA, Cardoso WB. Double-layer Bose-Einstein condensates: A quantum phase transition in the transverse direction, and reduction to two dimensions. Phys Rev E 2020; 102:042209. [PMID: 33212641 DOI: 10.1103/physreve.102.042209] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/24/2020] [Accepted: 09/22/2020] [Indexed: 06/11/2023]
Abstract
We revisit the problem of the reduction of the three-dimensional (3D) dynamics of Bose-Einstein condensates, under the action of strong confinement in one direction (z), to a 2D mean-field equation. We address this problem for the confining potential with a singular term, viz., V_{z}(z)=2z^{2}+ζ^{2}/z^{2}, with constant ζ. A quantum phase transition is induced by the latter term, between the ground state (GS) of the harmonic oscillator and the 3D condensate split in two parallel noninteracting layers, which is a manifestation of the "superselection" effect. A realization of the respective physical setting is proposed, making use of resonant coupling to an optical field, with the resonance detuning modulated along z. The reduction of the full 3D Gross-Pitaevskii equation (GPE) to the 2D nonpolynomial Schrödinger equation (NPSE) is based on the factorized ansatz, with the z -dependent multiplier represented by an exact GS solution of the 1D Schrödinger equation with potential V_{z}(z). For both repulsive and attractive signs of the nonlinearity, the 2D NPSE produces GS and vortex states, that are virtually indistinguishable from the respective numerical solutions provided by full 3D GPE. In the case of the self-attraction, the threshold for the onset of the collapse, predicted by the 2D NPSE, is also virtually identical to its counterpart obtained from the 3D equation. In the same case, stability and instability of vortices with topological charge S=1, 2, and 3 are considered in detail. Thus, the procedure of the spatial-dimension reduction, 3D → 2D, produces very accurate results, and it may be used in other settings.
Collapse
Affiliation(s)
- Mateus C P Dos Santos
- Instituto de Física, Universidade Federal de Goiás 74.690-970, Goiânia, Goiás, Brazil
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - Wesley B Cardoso
- Instituto de Física, Universidade Federal de Goiás 74.690-970, Goiânia, Goiás, Brazil
| |
Collapse
|
32
|
Nguyen VH, The Tai LX, Bugar I, Longobucco M, Buczyński R, Malomed BA, Trippenbach M. Reversible ultrafast soliton switching in dual-core highly nonlinear optical fibers. Opt Lett 2020; 45:5221-5224. [PMID: 32932495 DOI: 10.1364/ol.404039] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/30/2020] [Accepted: 08/19/2020] [Indexed: 06/11/2023]
Abstract
We experimentally investigate a nonlinear switching mechanism in a dual-core highly nonlinear optical fiber. We focus the input stream of femtosecond pulses on one core only, to identify transitions between inter-core oscillations, self-trapping in the cross core, and self-trapping of the pulse in the straight core. A model based on the system of coupled nonlinear Schrödinger equations provides surprisingly good agreement with the experimental findings.
Collapse
|
33
|
Marchukov OV, Malomed BA, Dunjko V, Ruhl J, Olshanii M, Hulet RG, Yurovsky VA. Quantum Fluctuations of the Center of Mass and Relative Parameters of Nonlinear Schrödinger Breathers. Phys Rev Lett 2020; 125:050405. [PMID: 32794875 DOI: 10.1103/physrevlett.125.050405] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2019] [Revised: 03/06/2020] [Accepted: 07/02/2020] [Indexed: 06/11/2023]
Abstract
We study quantum fluctuations of macroscopic parameters of a nonlinear Schrödinger breather-a nonlinear superposition of two solitons, which can be created by the application of a fourfold quench of the scattering length to the fundamental soliton in a self-attractive quasi-one-dimensional Bose gas. The fluctuations are analyzed in the framework of the Bogoliubov approach in the limit of a large number of atoms N, using two models of the vacuum state: white noise and correlated noise. The latter model, closer to the ab initio setting by construction, leads to a reasonable agreement, within 20% accuracy, with fluctuations of the relative velocity of constituent solitons obtained from the exact Bethe-ansatz results [Phys. Rev. Lett. 119, 220401 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.220401] in the opposite low-N limit (for N≤23). We thus confirm, for macroscopic N, the breather dissociation time to be within the limits of current cold-atom experiments. Fluctuations of soliton masses, phases, and positions are also evaluated and may have experimental implications.
Collapse
Affiliation(s)
- Oleksandr V Marchukov
- Institute for Applied Physics, Technical University of Darmstadt, 64289 Darmstadt, Germany
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, 6997801 Tel Aviv, Israel
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, 6997801 Tel Aviv, Israel
- Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7D, Arica, Chile
| | - Vanja Dunjko
- Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - Joanna Ruhl
- Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - Maxim Olshanii
- Department of Physics, University of Massachusetts Boston, Boston, Massachusetts 02125, USA
| | - Randall G Hulet
- Department of Physics and Astronomy, Rice University, Houston,Texas 77005, USA
| | | |
Collapse
|
34
|
Huang TY, Malomed BA, Chen X. Shortcuts to adiabaticity for an interacting Bose-Einstein condensate via exact solutions of the generalized Ermakov equation. Chaos 2020; 30:053131. [PMID: 32491879 DOI: 10.1063/5.0004309] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/11/2020] [Accepted: 04/28/2020] [Indexed: 06/11/2023]
Abstract
Shortcuts to adiabatic expansion of the effectively one-dimensional Bose-Einstein condensate (BEC) loaded in the harmonic-oscillator (HO) trap are investigated by combining techniques of variational approximation and inverse engineering. Piecewise-constant (discontinuous) intermediate trap frequencies, similar to the known bang-bang forms in the optimal-control theory, are derived from an exact solution of a generalized Ermakov equation. Control schemes considered in the paper include imaginary trap frequencies at short time scales, i.e., the HO potential replaced by the quadratic repulsive one. Taking into regard the BEC's intrinsic nonlinearity, results are reported for the minimal transfer time, excitation energy (which measures deviation from the effective adiabaticity), and stability for the shortcut-to-adiabaticity protocols. These results are not only useful for the realization of fast frictionless cooling, but also help us to address fundamental problems of the quantum speed limit and thermodynamics.
Collapse
Affiliation(s)
- Tang-You Huang
- International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) and Department of Physics, Shanghai University, 200444 Shanghai, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, P.O. Box 39040, Ramat Aviv, Tel Aviv, Israel
| | - Xi Chen
- International Center of Quantum Artificial Intelligence for Science and Technology (QuArtist) and Department of Physics, Shanghai University, 200444 Shanghai, China
| |
Collapse
|
35
|
Govindarajan A, Malomed BA, Lakshmanan M. Tunable nonlinear spectra of anti-directional couplers. Opt Lett 2020; 45:1918-1921. [PMID: 32236032 DOI: 10.1364/ol.388883] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2020] [Accepted: 02/26/2020] [Indexed: 06/11/2023]
Abstract
We produce transmission and reflection spectra of the anti-directional coupler (ADC) composed of linearly coupled positive- and negative-refractive-index arms, with intrinsic Kerr nonlinearity. Both reflection and transmission feature two highly amplified peaks at two distinct wavelengths in a certain range of values of the gain, making it possible to design a wavelength-selective mode-amplification system. We also predict that a blend of gain and loss in suitable proportions can robustly enhance reflection spectra that are detrimentally affected by the attenuation, in addition to causing red and blue shifts owing to the Kerr effect. In particular, ADC with equal gain and loss coefficients is considered in necessary detail.
Collapse
|
36
|
Abstract
We demonstrate that the commonly known concept which treats solitons as nonsingular solutions produced by the interplay of nonlinear self-attraction and linear dispersion may be extended to include modes with a relatively weak singularity at the central point, which keeps their integral norm convergent. Such states are generated by self-repulsion, which should be strong enough, represented by septimal, quintic, and usual cubic terms in the framework of the one-, two-, and three-dimensional (1D, 2D, and 3D) nonlinear Schrödinger equations (NLSEs), respectively. Although such solutions seem counterintuitive, we demonstrate that they admit a straightforward interpretation as a result of screening of an additionally introduced attractive δ-functional potential by the defocusing nonlinearity. The strength ("bare charge") of the attractive potential is infinite in 1D, finite in 2D, and vanishingly small in 3D. Analytical asymptotics of the singular solitons at small and large distances are found, entire shapes of the solitons being produced in a numerical form. Complete stability of the singular modes is accurately predicted by the anti-Vakhitov-Kolokolov criterion (under the assumption that it applies to the model), as verified by means of numerical methods. In 2D, the NLSE with a quintic self-focusing term admits singular-soliton solutions with intrinsic vorticity too, but they are fully unstable. We also mention that dissipative singular solitons can be produced by the model with a complex coefficient in front of the nonlinear term.
Collapse
Affiliation(s)
- Hidetsugu Sakaguchi
- Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
| | - 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 69978, Israel
| |
Collapse
|
37
|
Mayteevarunyoo T, Malomed BA, Skryabin DV. Spatiotemporal dissipative solitons and vortices in a multi-transverse-mode fiber laser. Opt Express 2019; 27:37364-37373. [PMID: 31878518 DOI: 10.1364/oe.27.037364] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2019] [Accepted: 11/26/2019] [Indexed: 06/10/2023]
Abstract
We introduce a model for spatiotemporal modelocking in multimode fiber lasers, which is based on the (3+1)-dimensional cubic-quintic complex Ginzburg-Landau equation (cGLE) with conservative and dissipative nonlinearities and a 2-dimensional transverse trapping potential. Systematic numerical analysis reveals a variety of stable nonlinear modes, including stable fundamental solitons and breathers, as well as solitary vortices with winding number n = 1, while vortices with n = 2 are unstable, splitting into persistently rotating bound states of two unitary vortices. A characteristic feature of the system is bistability between the fundamental and vortex spatiotemporal solitons.
Collapse
|
38
|
Wu J, Malomed BA, Fu HY, Li Q. Self-interaction of ultrashort pulses in an epsilon-near-zero nonlinear material at the telecom wavelength. Opt Express 2019; 27:37298-37307. [PMID: 31878512 DOI: 10.1364/oe.27.037298] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2019] [Accepted: 11/23/2019] [Indexed: 06/10/2023]
Abstract
Dynamics of femtosecond pulses with the telecom carrier wavelength is investigated numerically in a subwavelength layer of an indium tin oxide (ITO) epsilon-near-zero (ENZ) material with high dispersion and high nonlinearity. Due to the subwavelength thickness of the ITO ENZ material, and the fact that the pulse's propagation time is shorter than its temporal width, multiple reflections give rise to self-interaction in both spectral and temporal domains, especially at wavelengths longer than at the ENZ point, at which the reflections are significantly stronger. A larger absolute value of the pulse's chirp strongly affects the self-interaction by redistributing energy between wavelengths, while the sign of the chirp affects the interaction in the temporal domain. It is also found that, when two identical pulses are launched simultaneously from both ends, a subwavelength counterpart of a standing-wave state can be established. It shows robust energy localization in the middle of the sample, in terms of both the spectral and temporal intensity distributions.
Collapse
|
39
|
Correa RAC, de Souza Dutra A, Frederico T, Malomed BA, Oliveira O, Sawado N. Creating oscillons and oscillating kinks in two scalar field theories. Chaos 2019; 29:103124. [PMID: 31675835 DOI: 10.1063/1.5120500] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2019] [Accepted: 09/25/2019] [Indexed: 06/10/2023]
Abstract
Oscillons are time-dependent, localized in space, extremely long-lived states in nonlinear scalar-field models, while kinks are topological solitons in one spatial dimension. In the present work, we show new classes of oscillons and oscillating kinks in a system of two nonlinearly coupled scalar fields in 1+1 spatiotemporal dimensions. The solutions contain a control parameter, the variation of which produces oscillons and kinks with a flat-top shape. The model finds applications in condensed matter, cosmology, and high-energy physics.
Collapse
Affiliation(s)
- R A C Correa
- Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea, 265, I-34136 Trieste, Italy
| | - A de Souza Dutra
- São Paulo State University (UNESP), Campus de Guaratingueta, 12516-410 Guaratinguetá, SP, Brazil
| | - T Frederico
- ITA-Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos, SP, Brazil
| | - 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 69978, Israel
| | - O Oliveira
- CFisUC, Department of Physics, University of Coimbra, P-3004 516 Coimbra, Portugal
| | - N Sawado
- Department of Physics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
| |
Collapse
|
40
|
Govindarajan A, Malomed BA, Lakshmanan M. Nonlinear anti-directional couplers with gain and loss. Opt Lett 2019; 44:4650-4653. [PMID: 31568408 DOI: 10.1364/ol.44.004650] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/29/2019] [Accepted: 08/27/2019] [Indexed: 06/10/2023]
Abstract
Following the concept of PT-symmetric couplers, we propose a linearly coupled system of nonlinear waveguides, made of positive- and negative-index materials, which carry, respectively, gain and loss. We report novel bi- and multistability states pertaining to transmitted and reflective intensities, which are controlled by the ratio of the gain and loss coefficients, and phase mismatch between the waveguides. These states offer transmission regimes with extremely low threshold intensities for transitions between coexisting states, and very large amplification ratios between the input and output intensities leading to an efficient way of controlling light with light.
Collapse
|
41
|
Fedorov EG, Zhukov AV, Bouffanais R, Malomed BA, Leblond H, Mihalache D, Rosanov NN, Belonenko MB, George TF. Asymptotic dynamics of three-dimensional bipolar ultrashort electromagnetic pulses in an array of semiconductor carbon nanotubes. Opt Express 2019; 27:27592-27609. [PMID: 31684524 DOI: 10.1364/oe.27.027592] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2019] [Accepted: 08/23/2019] [Indexed: 06/10/2023]
Abstract
We study the propagation of three-dimensional bipolar ultrashort electromagnetic pulses in an array of semiconductor carbon nanotubes at times much longer than the pulse duration, yet still shorter than the relaxation time in the system. The interaction of the electromagnetic field with the electronic subsystem of the medium is described by means of Maxwell's equations, taking into account the field inhomogeneity along the nanotube axis beyond the approximation of slowly varying amplitudes and phases. A model is proposed for the analysis of the dynamics of an electromagnetic pulse in the form of an effective equation for the vector potential of the field. Our numerical analysis demonstrates the possibility of a satisfactory description of the evolution of the pulse field at large times by means of a three-dimensional generalization of the sine-Gordon and double sine-Gordon equations.
Collapse
|
42
|
Zhang X, Xu X, Zheng Y, Chen Z, Liu B, Huang C, Malomed BA, Li Y. Semidiscrete Quantum Droplets and Vortices. Phys Rev Lett 2019; 123:133901. [PMID: 31697515 DOI: 10.1103/physrevlett.123.133901] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [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.
Collapse
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
| |
Collapse
|
43
|
Hu X, Guo J, Shao GD, Song YF, Yoo SW, Malomed BA, Tang DY. Observation of incoherently coupled dark-bright vector solitons in single-mode fibers. Opt Express 2019; 27:18311-18317. [PMID: 31252776 DOI: 10.1364/oe.27.018311] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2019] [Accepted: 06/03/2019] [Indexed: 06/09/2023]
Abstract
We report experimental observation of incoherently coupled dark-bright vector solitons in single-mode fibers. Properties of the vector solitons accord well with those predicted by the respective systems of incoherently coupled nonlinear Schrödinger equations. To our knowledge, this is the first experimental observation of temporal incoherently coupled dark-bright solitons in single-mode fibers.
Collapse
|
44
|
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] [What about the content of this article? (0)] [Affiliation(s)] [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.
Collapse
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
| |
Collapse
|
45
|
Abstract
We show that metastable ring-shaped clusters can be constructed from two-dimensional quantum droplets in systems described by the Gross-Pitaevskii equations augmented with Lee-Huang-Yang quantum corrections. The clusters exhibit dynamical behavior ranging from contraction to rotation with simultaneous periodic pulsations, or expansion, depending on the initial radius of the necklace pattern and phase shift between adjacent quantum droplets. We show that, using an energy-minimization analysis, one can predict equilibrium values of the cluster radius that correspond to rotation without radial pulsations. In such a regime, the clusters evolve as metastable states, withstanding abrupt variations in the underlying scattering lengths and keeping their azimuthal symmetry in the course of evolution, even in the presence of considerable perturbations.
Collapse
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 108840, Russia
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Centre for Light-Matter Interaction, Tel Aviv University, P. O. Box 39040 Tel Aviv, Israel
| | - 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
| |
Collapse
|
46
|
Sakaguchi H, Malomed BA. Interactions of solitons with positive and negative masses: Shuttle motion and coacceleration. Phys Rev E 2019; 99:022216. [PMID: 30934243 DOI: 10.1103/physreve.99.022216] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2018] [Indexed: 11/07/2022]
Abstract
We consider a possibility to realize self-accelerating motion of interacting states with effective positive and negative masses in the form of pairs of solitons in two-component BEC loaded in an optical-lattice (OL) potential. A crucial role is played by the fact that gap solitons may feature a negative dynamical mass, keeping their mobility in the OL. First, the respective system of coupled Gross-Pitaevskii equations (GPE) is reduced to a system of equations for envelopes of the lattice wave functions. Two generic dynamical regimes are revealed by simulations of the reduced system, viz., shuttle oscillations of pairs of solitons with positive and negative masses, and splitting of the pair. The coaccelerating motion of the interacting solitons, which keeps constant separation between them, occurs at the boundary between the shuttle motion and splitting. The position of the coacceleration regime in the system's parameter space can be adjusted with the help of an additional gravity potential, which induces its own acceleration, that may offset the relative acceleration of the two solitons, while gravity masses of both solitons remain positive. The numerical findings are accurately reproduced by a variational approximation. Collisions between shuttling or coaccelerating soliton pairs do not alter the character of the dynamical regime. Finally, regimes of the shuttle motion, coacceleration, and splitting are corroborated by simulations of the original GPE system, with the explicitly present OL potential.
Collapse
Affiliation(s)
- Hidetsugu Sakaguchi
- Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580, Japan
| | - 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 69978, Israel
| |
Collapse
|
47
|
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] [What about the content of this article? (0)] [Affiliation(s)] [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.
Collapse
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
| |
Collapse
|
48
|
Zeng L, Zeng J, Kartashov YV, Malomed BA. Purely Kerr nonlinear model admitting flat-top solitons. Opt Lett 2019; 44:1206-1209. [PMID: 30821749 DOI: 10.1364/ol.44.001206] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/19/2018] [Accepted: 01/28/2019] [Indexed: 06/09/2023]
Abstract
We elaborate one- and two-dimensional (1D and 2D) models of media with self-repulsive cubic nonlinearity, whose local strength is subject to spatial modulation that admits the existence of flat-top solitons of various types, including fundamental ones, 1D multipoles, and 2D vortices. Previously, solitons of this type were only produced by models with competing nonlinearities. The present setting may be implemented in optics and Bose-Einstein condensates. The 1D version gives rise to an exact analytical solution for stable flat-top solitons, and generic families may be predicted by means of the Thomas-Fermi approximation. Stability of the obtained flat-top solitons is analyzed by means of the linear-stability analysis and direct simulations. Fundamental solitons and 1D multipoles with k=1 and 2 nodes, as well as vortices with winding number m=1, are completely stable. For multipoles with k≥3 and vortices with m≥2, alternating stripes of stability and instability are identified in their parameter spaces.
Collapse
|
49
|
Li P, Mihalache D, Malomed BA. Optical solitons in media with focusing and defocusing saturable nonlinearity and a parity-time-symmetric external potential. Phil Trans R Soc A 2018; 376:20170378. [PMID: 29891499 PMCID: PMC6000148 DOI: 10.1098/rsta.2017.0378] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 04/02/2018] [Indexed: 05/20/2023]
Abstract
We report results for solitons in models of waveguides with focusing or defocusing saturable nonlinearity and a parity-time ([Formula: see text])-symmetric complex-valued external potential of the Scarf-II type. The model applies to the nonlinear wave propagation in graded-index optical waveguides with balanced gain and loss. We find both fundamental and multipole solitons for both focusing and defocusing signs of the saturable nonlinearity in such [Formula: see text]-symmetric waveguides. The dependence of the propagation constant on the soliton's power is presented for different strengths of the nonlinearity saturation, S The stability of fundamental, dipole, tripole and quadrupole solitons is investigated by means of the linear-stability analysis and direct numerical simulations of the corresponding (1+1)-dimensional nonlinear Schrödinger-type equation. The results show that the instability of the stationary solutions can be mitigated or completely suppressed, increasing the value of SThis article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.
Collapse
Affiliation(s)
- Pengfei Li
- Department of Physics, Taiyuan Normal University, Taiyuan 030031, People's Republic of China
| | - Dumitru Mihalache
- Horia Hulubei National Institute of Physics and Nuclear Engineering, Magurele, Bucharest 077125, Romania
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
- ITMO University, St. Petersburg 197101, Russia
| |
Collapse
|
50
|
Chen Z, Li Y, Malomed BA. 𝒫𝒯-symmetric and antisymmetric nonlinear states in a split potential box. Phil Trans R Soc A 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] [What about the content of this article? (0)] [Affiliation(s)] [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)'.
Collapse
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
| |
Collapse
|