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Chatterjee P, Modak R. One-dimensional Lévy quasicrystal. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2023; 35:505602. [PMID: 37708897 DOI: 10.1088/1361-648x/acf9d4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/07/2023] [Accepted: 09/14/2023] [Indexed: 09/16/2023]
Abstract
Space-fractional quantum mechanics (SFQM) is a generalization of the standard quantum mechanics when the Brownian trajectories in Feynman path integrals are replaced by Lévy flights. We introduce Lévy quasicrystal by discretizing the space-fractional Schrödinger equation using the Grünwald-Letnikov derivatives and adding on-site quasiperiodic potential. The discretized version of the usual Schrödinger equation maps to the Aubry-André (AA) Hamiltonian, which supports localization-delocalization transition even in one dimension. We find the similarities between Lévy quasicrystal and the AA model with power-law hopping, and show that the Lévy quasicrystal supports a delocalization-localization transition as one tunes the quasiperiodic potential strength and shows the coexistence of localized and delocalized states separated by mobility edge. Hence, a possible realization of SFQM in optical experiments should be a new experimental platform to test the predictions of AA models in the presence of power-law hopping.
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Affiliation(s)
- Pallabi Chatterjee
- Department of Physics, Indian Institute of Technology Tirupati, Tirupati 517619, India
| | - Ranjan Modak
- Department of Physics, Indian Institute of Technology Tirupati, Tirupati 517619, India
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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: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [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.
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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
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Zhong M, Wang L, Li P, Yan Z. Spontaneous symmetry breaking and ghost states supported by the fractional PT-symmetric saturable nonlinear Schrödinger equation. CHAOS (WOODBURY, N.Y.) 2023; 33:013106. [PMID: 36725663 DOI: 10.1063/5.0128910] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/01/2022] [Accepted: 12/05/2022] [Indexed: 06/18/2023]
Abstract
We report a novel spontaneous symmetry breaking phenomenon and ghost states existed in the framework of the fractional nonlinear Schrödinger equation with focusing saturable nonlinearity and PT-symmetric potential. The continuous asymmetric soliton branch bifurcates from the fundamental symmetric one as the power exceeds some critical value. Intriguingly, the symmetry of fundamental solitons is broken into two branches of asymmetry solitons (alias ghost states) with complex conjugate propagation constants, which is solely in fractional media. Besides, the dipole and tripole solitons (i.e., first and second excited states) are also studied numerically. Moreover, we analyze the influences of fractional Lévy index ( α) and saturable nonlinear parameters (S) on the symmetry breaking of solitons in detail. The stability of fundamental symmetric soliton, asymmetric, dipole, and tripole solitons is explored via the linear stability analysis and direct propagations. Moreover, we explore the elastic/semi-elastic collision phenomena between symmetric and asymmetric solitons. Meanwhile, we find the stable excitations from the fractional diffraction with saturation nonlinearity to integer-order diffraction with Kerr nonlinearity via the adiabatic excitations of parameters. These results will provide some theoretical basis for the study of spontaneous symmetry breaking phenomena and related physical experiments in the fractional media with PT-symmetric potentials.
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Affiliation(s)
- Ming Zhong
- KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
| | - Li Wang
- Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China
| | - Pengfei Li
- Department of Physics, Taiyuan Normal University, Taiyuan 030031, China
| | - Zhenya Yan
- KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
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Analysis of Kink Behaviour of KdV-mKdV Equation under Caputo Fractional Operator with Non-Singular Kernel. Symmetry (Basel) 2022. [DOI: 10.3390/sym14112316] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/06/2022] Open
Abstract
The KdV equation has many applications in mechanics and wave dynamics. Therefore, researchers are carrying out work to develop and analyze modified and generalized forms of the standard KdV equation. In this paper, we inspect the KdV-mKdV equation, which is a modified and generalized form of the ordinary KdV equation. We use the fractional operator in the Caputo sense to analyze the equation. We examine some theoretical results concerned with the solution’s existence, uniqueness, and stability. We employ a modified Laplace method to extract the numerical results of the considered equation. We use MATLAB-2020 to simulate the results in a few fractional orders. We report the effects of the fractional order on the wave dynamics of the proposed equation.
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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: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [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.
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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
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Diverse Forms of Breathers and Rogue Wave Solutions for the Complex Cubic Quintic Ginzburg Landau Equation with Intrapulse Raman Scattering. MATHEMATICS 2022. [DOI: 10.3390/math10111818] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/10/2022]
Abstract
This manuscript consist of diverse forms of lump: lump one stripe, lump two stripe, generalized breathers, Akhmediev breather, multiwave, M-shaped rational and rogue wave solutions for the complex cubic quintic Ginzburg Landau (CQGL) equation with intrapulse Raman scattering (IRS) via appropriate transformations approach. Furthermore, it includes homoclinic, Ma and Kuznetsov-Ma breather and their relating rogue waves and some interactional solutions, including an interactional approach with the help of the double exponential function. We have elaborated the kink cross-rational (KCR) solutions and periodic cross-rational (KCR) solutions with their graphical slots. We have also constituted some of our solutions in distinct dimensions by means of 3D and contours profiles to anticipate the wave propagation. Parameter domains are delineated in which these exact localized soliton solutions exit in the proposed model.
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Zhou W, Liu A, Huang X, Bai Y, Fu X. Propagation dynamics of Laguerre-Gaussian beams in the fractional Schrödinger equation with noise disturbance. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA. A, OPTICS, IMAGE SCIENCE, AND VISION 2022; 39:736-743. [PMID: 35471400 DOI: 10.1364/josaa.455164] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2022] [Accepted: 03/06/2022] [Indexed: 06/14/2023]
Abstract
The evolution of Laguerre-Gaussian (LG) beams in the fractional Schrödinger equation (FSE) with Gaussian noise disturbance is numerically investigated. Without noise disturbance, the peak intensity of LG beams increases with the increment of radial or azimuthal indices, and the turning point of the peak intensity between different radial indices exists. As propagation distance gets longer, the intensity of the outermost sub-lobe exceeds that of the main lobe. When Gaussian noise is added, for a given noise level, the stability of peak intensity is enhanced as the Lévy index increases, while the center of gravity shows the opposite phenomenon. Moreover, the increment of the radial index can weaken the stability of the center of gravity. We also investigate the stability of the peak intensity of Airy beams in the FSE, and generally, the stability of LG beams is better than that of Airy beams. All these properties show that LG beams modeled by the FSE have potential applications in optical manipulation and communications.
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Wang J, Jin Y, Gong X, Yang L, Chen J, Xue P. Generation of random soliton-like beams in a nonlinear fractional Schrödinger equation. OPTICS EXPRESS 2022; 30:8199-8211. [PMID: 35299566 DOI: 10.1364/oe.448972] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/18/2021] [Accepted: 02/07/2022] [Indexed: 06/14/2023]
Abstract
We investigate the generation of random soliton-like beams based on the Kuznetsov-Ma solitons in a nonlinear fractional Schrödinger equation (NLFSE). For Lévy index α = 1, the Kuznetsov-Ma solitons split into two nondiffracting beams during propagation in linear regime. According to the different input positions of the Kuznetsov-Ma solitons, the diffraction-free beams can be divided into three different types: bright-dark, dark-bright and bright-bright beams. In the nonlinear regime, the Kuznetsov-Ma solitons can be evolved into random soliton-like beams due to the collapse. The number of soliton-like beams is related to the nonlinear coefficient and the Lévy index. The bigger the nonlinear coefficient, the more beams generated. Moreover, the peak intensity of soliton-like beams presents a Gaussian distribution under the large nonlinear effect. In practice, the evolution of KM soliton can be realized by a plane wave with a Gaussian perturbation, which can be confirmed that they have the similar dynamics of propagation. In two dimensions, the plane wave with a Gaussian perturbation can be evolved into a bright-dark axisymmetric ring beam in the linear regime. Under the nonlinear modulation, the energy accumulates to the center and finally breaks apart into random beam filaments.
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Wu GZ, Dai CQ, Wang YY, Chen YX. Propagation and interaction between special fractional soliton and soliton molecules in the inhomogeneous fiber. J Adv Res 2022; 36:63-71. [PMID: 35127165 PMCID: PMC8800015 DOI: 10.1016/j.jare.2021.05.004] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/30/2020] [Revised: 05/08/2021] [Accepted: 05/12/2021] [Indexed: 12/03/2022] Open
Abstract
Analytical chirp-free and chirped fractional soliton solutions are obtained. The form conditions of soliton molecules are given. Interactions between special fractional solitons and soliton molecules are discussed.
Introduction Fractional nonlinear models have been widely used in the research of nonlinear science. A fractional nonlinear Schrödinger equation with distributed coefficients is considered to describe the propagation of pi-second pulses in inhomogeneous fiber systems. However, soliton molecules based on the fractional nonlinear Schrödinger equation are hardly reported although many fractional soliton structures have been studied. Objectives This paper discusses the propagation and interaction between special fractional soliton and soliton molecules based on analytical solutions of a fractional nonlinear Schrödinger equation. Methods Two analytical methods, including the variable-coefficient fractional mapping method and Hirota method with the modified Riemann–Liouville fractional derivative rule, are used to obtain analytical non-travelling wave solutions and multi-soliton approximate solutions. Results Analytical non-travelling wave solutions and multi-soliton approximate solutions are derived. The form conditions of soliton molecules are given, and the dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed in the periodic inhomogeneous fiber and the exponential dispersion decreasing fiber. Conclusion Analytical chirp-free and chirped non-traveling wave solutions and multi-soliton approximate solutions including soliton molecules are obtained. Based on these solutions, dynamical characteristics and interactions between special fractional solitons, multi-solitons and soliton molecules are discussed. These theoretical studies are of great help to understand the propagation of optical pulses in fibers.
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Affiliation(s)
- Gang-Zhou Wu
- College of Sciences, Zhejiang A&F University, Lin'an, Zhejiang 311300, PR China
| | - Chao-Qing Dai
- College of Sciences, Zhejiang A&F University, Lin'an, Zhejiang 311300, PR China
| | - Yue-Yue Wang
- College of Sciences, Zhejiang A&F University, Lin'an, Zhejiang 311300, PR China
| | - Yi-Xiang Chen
- School of Electronics Information, Zhejiang University of Media and Communications, Hangzhou 310018, PR China
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Liu X, Malomed BA, Zeng J. Localized Modes in Nonlinear Fractional Systems with Deep Lattices. ADVANCED THEORY AND SIMULATIONS 2022. [DOI: 10.1002/adts.202100482] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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
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Optical Solitons and Vortices in Fractional Media: A Mini-Review of Recent Results. PHOTONICS 2021. [DOI: 10.3390/photonics8090353] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
The article produces a brief review of some recent results which predict stable propagation of solitons and solitary vortices in models based on the nonlinear Schrödinger equation (NLSE) including fractional one-dimensional or two-dimensional diffraction and cubic or cubic-quintic nonlinear terms, as well as linear potentials. The fractional diffraction is represented by fractional-order spatial derivatives of the Riesz type, defined in terms of the direct and inverse Fourier transform. In this form, it can be realized by spatial-domain light propagation in optical setups with a specially devised combination of mirrors, lenses, and phase masks. The results presented in the article were chiefly obtained in a numerical form. Some analytical findings are included too, in particular, for fast moving solitons and the results produced by the variational approximation. Moreover, dissipative solitons are briefly considered, which are governed by the fractional complex Ginzburg–Landau equation.
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Meng L, Liu J, Zhang H, Yang W. Symmetrical superfission of optical solitons in a well-type nonlocal system. APPLIED OPTICS 2021; 60:5854-5859. [PMID: 34263805 DOI: 10.1364/ao.423993] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2021] [Accepted: 06/14/2021] [Indexed: 06/13/2023]
Abstract
The dynamical properties of fundamental and dipolar mode solitons, in the process of propagating in the well-type nonlocal system, are provided. During propagating in a deep well-type nonlocal system with a moderate width, a fundamental mode soliton splits into a pair of symmetrical sub-beams, and a dipolar mode soliton can divide into two pairs of symmetrical sub-beams. Furthermore, the propagation directions of these sub-beams can be effectively controlled by adjusting system parameters that include the well depth and well width. These properties could be applied to optical routing, all-optical switching, signal processing, and ultrafast optical communications.
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Li P, Malomed BA, Mihalache D. Symmetry-breaking bifurcations and ghost states in the fractional nonlinear Schrödinger equation with a PT-symmetric potential. OPTICS LETTERS 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] [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.
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