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Manikandan K, Sakkaravarthi K, Sudharsan JB, Aravinthan D. Deformation of optical solitons in a variable-coefficient nonlinear Schrödinger equation with three distinct PT-symmetric potentials and modulated nonlinearities. CHAOS (WOODBURY, N.Y.) 2023; 33:2895997. [PMID: 37307166 DOI: 10.1063/5.0145283] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/04/2023] [Accepted: 05/22/2023] [Indexed: 06/14/2023]
Abstract
We investigate deformed/controllable characteristics of solitons in inhomogeneous parity-time (PT)-symmetric optical media. To explore this, we consider a variable-coefficient nonlinear Schrödinger equation involving modulated dispersion, nonlinearity, and tapering effect with PT-symmetric potential, which governs the dynamics of optical pulse/beam propagation in longitudinally inhomogeneous media. By incorporating three physically interesting and recently identified forms of PT-symmetric potentials, namely, rational, Jacobian periodic, and harmonic-Gaussian potentials, we construct explicit soliton solutions through similarity transformation. Importantly, we investigate the manipulation dynamics of such optical solitons due to diverse inhomogeneities in the medium by implementing step-like, periodic, and localized barrier/well-type nonlinearity modulations and revealing the underlying phenomena. Also, we corroborate the analytical results with direct numerical simulations. Our theoretical exploration will provide further impetus in engineering optical solitons and their experimental realization in nonlinear optics and other inhomogeneous physical systems.
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Affiliation(s)
- K Manikandan
- Center for Computational Modeling, Chennai Institute of Technology, Chennai 600069, Tamil Nadu, India
| | - K Sakkaravarthi
- Young Scientist Training Program, Asia-Pacific Center for Theoretical Physics (APCTP), POSTECH Campus, Pohang 37673, South Korea
| | - J B Sudharsan
- Center for Computational Modeling, Chennai Institute of Technology, Chennai 600069, Tamil Nadu, India
| | - D Aravinthan
- Center for Computational Modeling, Chennai Institute of Technology, Chennai 600069, Tamil Nadu, India
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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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Chen Y, Song J, Li X, Yan Z. Stability and modulation of optical peakons in self-focusing/defocusing Kerr nonlinear media with PT-δ-hyperbolic-function potentials. CHAOS (WOODBURY, N.Y.) 2022; 32:023122. [PMID: 35232047 DOI: 10.1063/5.0080485] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/01/2021] [Accepted: 01/28/2022] [Indexed: 06/14/2023]
Abstract
In this paper, we introduce a class of novel PT- δ-hyperbolic-function potentials composed of the Dirac δ(x) and hyperbolic functions, supporting fully real energy spectra in the non-Hermitian Hamiltonian. The threshold curves of PT symmetry breaking are numerically presented. Moreover, in the self-focusing and defocusing Kerr-nonlinear media, the PT-symmetric potentials can also support the stable peakons, keeping the total power and quasi-power conserved. The unstable PT-symmetric peakons can be transformed into other stable peakons by the excitations of potential parameters. Continuous families of additional stable numerical peakons can be produced in internal modes around the exact peakons (even unstable). Further, we find that the stable peakons can always propagate in a robust form, remaining trapped in the slowly moving potential wells, which opens the way for manipulations of optical peakons. Other significant characteristics related to exact peakons, such as the interaction and power flow, are elucidated in detail. These results will be useful in explaining the related physical phenomena and designing the related physical experiments.
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Affiliation(s)
- Yong Chen
- School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
| | - Jin Song
- Key Lab of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
| | - Xin Li
- School of Mathematics and Statistics, Changshu Institute of Technology, Changshu 215500, China
| | - Zhenya Yan
- Key Lab of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
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Chen Y, Yan Z, Mihalache D. Soliton formation and stability under the interplay between parity-time-symmetric generalized Scarf-II potentials and Kerr nonlinearity. Phys Rev E 2020; 102:012216. [PMID: 32795035 DOI: 10.1103/physreve.102.012216] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2020] [Accepted: 07/01/2020] [Indexed: 11/07/2022]
Abstract
We present an alternative type of parity-time (PT)-symmetric generalized Scarf-II potentials, which makes possible for non-Hermitian Hamiltonians in the classical linear Schrödinger system to possess fully real spectra with unique features such as the multiple PT-symmetric breaking behaviors and to support one-dimensional (1D) stable PT-symmetric solitons of power-law waveform, namely power-law solitons, in focusing Kerr-type nonlinear media. Moreover, PT-symmetric high-order solitons are also derived numerically in 1D and 2D settings. Around the exactly obtained nonlinear propagation constants, families of 1D and 2D localized nonlinear modes are also found numerically. The majority of fundamental nonlinear modes can still keep steady in general, whereas the 1D multipeak solitons and 2D vortex solitons are usually susceptible to suffering from instability. Likewise, similar results occur in the defocusing Kerr-nonlinear media. The obtained results will be useful for understanding the complex dynamics of nonlinear waves that form in PT-symmetric nonlinear media in other physical contexts.
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Affiliation(s)
- Yong Chen
- School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
| | - Zhenya Yan
- Key Lab 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
| | - Dumitru Mihalache
- Department of Theoretical Physics, Horia Hulubei National Institute for Physics and Nuclear Engineering, Bucharest-Magurele, RO-077125, Romania
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Chen Y, Yan Z, Mihalache D. Stable flat-top solitons and peakons in the PT-symmetric δ-signum potentials and nonlinear media. CHAOS (WOODBURY, N.Y.) 2019; 29:083108. [PMID: 31472484 DOI: 10.1063/1.5100294] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2019] [Accepted: 07/18/2019] [Indexed: 06/10/2023]
Abstract
We discover that the physically interesting PT-symmetric Dirac delta-function potentials can not only make sure that the non-Hermitian Hamiltonians admit fully-real linear spectra but also support stable peakons (nonlinear modes) in the Kerr nonlinear Schrödinger equation. For a specific form of the delta-function PT-symmetric potentials, the nonlinear model investigated in this paper is exactly solvable. However, for a class of PT-symmetric signum-function double-well potentials, a novel type of exact flat-top bright solitons can exist stably within a broad range of potential parameters. Intriguingly, the flat-top solitons can be characterized by the finite-order differentiable waveforms and admit the novel features differing from the usual solitons. The excitation features and the direction of transverse power flow of flat-top bright solitons are also explored in detail. These results are useful for the related experimental designs and applications in nonlinear optics and other related fields.
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Affiliation(s)
- Yong Chen
- School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
| | - Zhenya Yan
- Key Lab of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
| | - Dumitru Mihalache
- Horia Hulubei National Institute of Physics and Nuclear Engineering, P.O. Box MG-6, Magurele RO-077125, Romania
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Wang L, Malomed BA, Yan Z. Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media. Phys Rev E 2019; 99:052206. [PMID: 31212420 DOI: 10.1103/physreve.99.052206] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/19/2019] [Indexed: 11/07/2022]
Abstract
We introduce a model based on the one-dimensional nonlinear Schrödinger equation with critical (quintic) or supercritical self-focusing nonlinearity. We demonstrate that a family of solitons, which are unstable in this setting against the critical or supercritical collapse, is stabilized by pinning to an attractive defect, that may also include a parity-time (PT)-symmetric gain-loss component. The model can be realized as a planar waveguide in nonlinear optics, and in a super-Tonks-Girardeau bosonic gas. For the attractive defect with the delta-functional profile, a full family of the pinned solitons is found in an exact analytical form. In the absence of the gain-loss term, the solitons' stability is investigated in an analytical form too, by means of the Vakhitov-Kolokolov criterion; in the presence of the PT-balanced gain and loss, the stability is explored by means of numerical methods. In particular, the entire family of pinned solitons is stable in the quintic (critical) medium if the gain-loss term is absent. A stability region for the pinned solitons persists in the model with an arbitrarily high power of the self-focusing nonlinearity. A weak gain-loss component gives rise to intricate alternations of stability and instability in the system's parameter plane. Those solitons which are unstable under the action of the supercritical self-attraction are destroyed by the collapse. On the other hand, if the self-attraction-driven instability is weak and the gain-loss term is present, unstable solitons spontaneously transform into localized breathers, while the collapse does not occur. The same outcome may be caused by a combination of the critical nonlinearity with the gain and loss. Instability of the solitons is also possible when the PT-symmetric gain-loss term is added to the subcritical nonlinearity. The system with self-repulsive nonlinearity is briefly considered too, producing completely stable families of pinned localized states.
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Affiliation(s)
- Li Wang
- Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.,School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 59978, Israel
| | - Zhenya Yan
- Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.,School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
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Chen Y, Yan Z, Liu W. Impact of near-𝒫𝒯 symmetry on exciting solitons and interactions based on a complex Ginzburg-Landau model. OPTICS EXPRESS 2018; 26:33022-33034. [PMID: 30645460 DOI: 10.1364/oe.26.033022] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2018] [Accepted: 11/09/2018] [Indexed: 06/09/2023]
Abstract
We theoretically report the influence of a class of near-parity-time-(𝒫𝒯-) symmetric potentials on solitons in the complex Ginzburg-Landau (CGL) equation. Although the linear spectral problem with the potentials does not admit entirely-real spectra due to the existence of spectral filtering parameter α2 or nonlinear gain-loss coefficient β2, we do find stable exact solitons in the second quadrant of the (α2, β2) space including on the corresponding axes. Other fascinating properties associated with the solitons are also examined, such as the interactions and energy flux. Moreover, we study the excitations of nonlinear modes by considering adiabatic changes of parameters in a generalized CGL model. These results are useful for the related experimental designs and applications.
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Shen Y, Wen Z, Yan Z, Hang C. Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media. CHAOS (WOODBURY, N.Y.) 2018; 28:043104. [PMID: 31906637 DOI: 10.1063/1.5018107] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
We study the three-wave interaction that couples an electromagnetic pump wave to two frequency down-converted daughter waves in a quadratic optical crystal and PT-symmetric potentials. PT symmetric potentials are shown to modulate stably nonlinear modes in two kinds of three-wave interaction models. The first one is a spatially extended three-wave interaction system with odd gain-and-loss distribution in the channel. Modulated by the PT-symmetric single-well or multi-well Scarf-II potentials, the system is numerically shown to possess stable soliton solutions. Via adiabatical change of system parameters, numerical simulations for the excitation and evolution of nonlinear modes are also performed. The second one is a combination of PT-symmetric models which are coupled via three-wave interactions. Families of nonlinear modes are found with some particular choices of parameters. Stable and unstable nonlinear modes are shown in distinct families by means of numerical simulations. These results will be useful to further investigate nonlinear modes in three-wave interaction models.
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Affiliation(s)
- Yujia Shen
- Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
| | - Zichao Wen
- Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
| | - Zhenya Yan
- Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
| | - Chao Hang
- State Key Laboratory of Precision Spectroscopy, School of Physical and Material Sciences, East China Normal University, Shanghai 200062, China
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