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Ren Y, Guo L, Cao XW, Duan L. Various solitons induced by relative phase in the nonlinear Schrödinger Maxwell-Bloch system. CHAOS (WOODBURY, N.Y.) 2024; 34:013111. [PMID: 38198679 DOI: 10.1063/5.0167062] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/10/2023] [Accepted: 12/11/2023] [Indexed: 01/12/2024]
Abstract
We study the effect of relative phase on the characteristics of rogue waves and solitons described by rational solutions in the nonlinear Schrödinger Maxwell-Bloch system. We derived the rational rogue wave and soliton solutions with adjustable relative phase and present the parameter range of different types of rogue waves and solitons. Our findings show that the relative phase can alter the distribution of rational solitons and even change the type of rational solitons, leading to a rich array of rational soliton types by adjusting the relative phase. However, the relative phase does not affect the structure of the rogue wave, because the relative phase of the rogue wave changes during evolution. In particular, we confirm that the rational solitons with varying relative phases and the rogue waves at corresponding different evolution positions share the same distribution mode. This relationship holds true for rogue waves or breathers and their stable counterparts solitons or periodic waves in different nonlinear systems. The implications of our study are significant for exploring fundamental excitation elements in nonlinear systems.
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Affiliation(s)
- Yang Ren
- School of Mechanical and Material Engineering, Xi'an University, 710065 Xi'an, China
| | - Liang Guo
- School of Mechanical and Material Engineering, Xi'an University, 710065 Xi'an, China
| | - Xin-Wei Cao
- School of Mechanical and Material Engineering, Xi'an University, 710065 Xi'an, China
| | - Liang Duan
- School of Physics and Astronomy, Shanghai Jiao Tong University, 200240 Shanghai, China
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2
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Liu H, Wang L, Zhang Y, Lu G, Liu L. Prediction of phase transition and time-varying dynamics of the (2+1)-dimensional Boussinesq equation by parameter-integrated physics-informed neural networks with phase domain decomposition. Phys Rev E 2023; 108:045303. [PMID: 37978704 DOI: 10.1103/physreve.108.045303] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2023] [Accepted: 09/13/2023] [Indexed: 11/19/2023]
Abstract
A meaningful topic that needs to be explored in the field of nonlinear waves is whether a neural network can reveal the phase transition of different types of waves and novel dynamical properties. In this paper, a physics-informed neural network (PINN) with parameters is used to explore the phase transition and time-varying dynamics of nonlinear waves of the (2+1)-dimensional Boussinesq equation describing the propagation of gravity waves on the surface of water. We embed the physical parameters into the neural network for this purpose. Via such algorithm, we find the exact boundary of the phase transition that distinguishes the periodic lump chain and transformed wave, and the inexact boundaries of the phase transition for various transformed waves are detected through PINNs with phase domain decomposition. In particular, based only on the simple soliton solution, we discover types of nonlinear waves as well as their interesting time-varying properties for the (2+1)-dimensional Boussinesq equation. We further investigate the stability by adding noise to the initial data. Finally, we perform the parameters discovery of the equation in the case of data with and without noise, respectively. Our paper introduces deep learning into the study of the phase transition of nonlinear waves and paves the way for intelligent explorations of the unknown properties of waves by means of the PINN technique with a simple solution and small data set.
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Affiliation(s)
- Haiyi Liu
- School of Mathematics and Physics, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Lei Wang
- School of Mathematics and Physics, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Yabin Zhang
- School of Control and Computer Engineering, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Gui Lu
- School of Energy Power and Mechanical Engineering, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Lei Liu
- College of Mathematics and Statistics, Chongqing University, Chongqing 401331, People's Republic of China
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3
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Liu XS, Xu HX, Qin YH, Duan L. Excitation of mirror symmetry higher-order rational soliton in modulation stability regimes on continuous wave background. CHAOS (WOODBURY, N.Y.) 2022; 32:123105. [PMID: 36587323 DOI: 10.1063/5.0106915] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2022] [Accepted: 11/03/2022] [Indexed: 06/17/2023]
Abstract
We study the relationship between the structures of the nonlinear localized waves and the distribution characteristics of the modulation stability regime in a nonlinear fiber with both third-order and fourth-order effects. On the background frequency and background amplitude plane, the modulation stability region consists of two symmetric curves on the left and right and a point on the symmetry axis. We find that the higher-order excitation characteristics are obviously different at different positions in the modulation stability region. Their excitation characteristics are closely related to the modulation instability distribution characteristics of the system. It is shown that asymmetric high-order rational solitons are excited at the left and right stable curves, and the symmetric one is excited at the stable points. Interestingly, the asymmetric higher-order rational solitons on the left and right sides are mirror-symmetrical to each other, which coincides with the symmetry of the modulation instability distribution. These results can deepen our understanding of the relationship between nonlinear excitation and modulation instability and enrich our knowledge about higher-order nonlinear excitations.
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Affiliation(s)
- Xiang-Shu Liu
- School of Physics and Electrical Engineering, Liupanshui Normal University, Liupanshui 553004, China
| | - Han-Xiang Xu
- Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan
| | - Yan-Hong Qin
- School of Mathematics, South China University of Technology, Guangzhou 510640, China
| | - Liang Duan
- School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
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4
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Qin YH, Zhang X, Ling L, Zhao LC. Phase characters of optical dark solitons with third-order dispersion and delayed nonlinear response. Phys Rev E 2022; 106:024213. [PMID: 36109962 DOI: 10.1103/physreve.106.024213] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/01/2022] [Accepted: 08/04/2022] [Indexed: 06/15/2023]
Abstract
Dark soliton is usually seen as one of the simplest topological solitons, due to phase shift across its intensity dip. We investigate phase characters of single-valley dark soliton (SVDS) and double-valley dark soliton (DVDS) in a single-mode optical fiber with third-order dispersion and delayed nonlinear response. Notably, two different phase shifts can produce an SVDS with the same velocity under some conditions, which is not admitted for a dark soliton with only the second-order dispersion and self-phase modulation, whose phase shift and velocity is a one-to-one match. This phase property of SVDS can be used to explain the generation of previously reported DVDS in Hirota equation and make DVDSs show two types of phase profiles. Moreover, the different topological vector potentials underlying the distinct phase profiles have been uncovered. We further explore the collision properties of the DVDSs by analyzing their topological phases. Strikingly, the inelastic collision can lead to the conversion between the two types of phase profiles for DVDS. The results reveal that inelastic or elastic collision can be judged by analyzing virtual topological magnetic fields.
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Affiliation(s)
- Yan-Hong Qin
- School of Mathematics, South China University of Technology, Guangzhou 510640, China
| | - Xiaoman Zhang
- School of Mathematics, South China University of Technology, Guangzhou 510640, China
| | - Liming Ling
- School of Mathematics, South China University of Technology, Guangzhou 510640, China
| | - Li-Chen Zhao
- School of Physics, Northwest University, Xi'an 710127, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
- Peng Huanwu Center for Fundamental Theory, Xi'an 710127, China
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5
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Zhang X, Wang L, Liu C, Li M, Zhao YC. High-dimensional nonlinear wave transitions and their mechanisms. CHAOS (WOODBURY, N.Y.) 2020; 30:113107. [PMID: 33261354 DOI: 10.1063/5.0019596] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/25/2020] [Accepted: 10/05/2020] [Indexed: 06/12/2023]
Abstract
In this paper, the dynamics of transformed nonlinear waves in the (2+1)-dimensional Ito equation are studied by virtue of the analysis of characteristic line and phase shift. First, the N-soliton solution is obtained via the Hirota bilinear method, from which the breath-wave solution is derived by changing values of wave numbers into complex forms. Then, the transition condition for the breath waves is obtained analytically. We show that the breath waves can be transformed into various nonlinear wave structures including the multi-peak soliton, M-shaped soliton, quasi-anti-dark soliton, three types of quasi-periodic waves, and W-shaped soliton. The correspondence of the phase diagram for such nonlinear waves on the wave number plane is presented. The gradient property of the transformed solution is discussed through the wave number ratio. We study the mechanism of wave formation by analyzing the nonlinear superposition between a solitary wave component and a periodic wave component with different phases. The locality and oscillation of transformed waves can also be explained by the superposition mechanism. Furthermore, the time-varying characteristics of high-dimensional transformed waves are investigated by analyzing the geometric properties (angle and distance) of two characteristic lines of waves, which do not exist in (1+1)-dimensional systems. Based on the high-order breath-wave solutions, the interactions between those transformed nonlinear waves are investigated, such as the completely elastic mode, semi-elastic mode, inelastic mode, and collision-free mode. We reveal that the diversity of transformed waves, time-varying property, and shape-changed collision mainly appear as a result of the difference of phase shifts of the solitary wave and periodic wave components. Such phase shifts come from the time evolution as well as the collisions. Finally, the dynamics of the double shape-changed collisions are presented.
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Affiliation(s)
- Xue Zhang
- School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
| | - Lei Wang
- School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
| | - Chong Liu
- School of Physics, Northwest University, Xi'an 710069, People's Republic of China
| | - Min Li
- School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
| | - Yin-Chuan Zhao
- School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
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6
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Özkan YS, Yaşar E, Seadawy AR. A third-order nonlinear Schrödinger equation: the exact solutions, group-invariant solutions and conservation laws. JOURNAL OF TAIBAH UNIVERSITY FOR SCIENCE 2020. [DOI: 10.1080/16583655.2020.1760513] [Citation(s) in RCA: 91] [Impact Index Per Article: 22.8] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
Affiliation(s)
- Yeşim Sağlam Özkan
- Department of Mathematics, Faculty of Arts and Science, Bursa Uludag University, Bursa, Turkey
| | - Emrullah Yaşar
- Department of Mathematics, Faculty of Arts and Science, Bursa Uludag University, Bursa, Turkey
| | - Aly R. Seadawy
- Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia
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7
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Ye Y, Zhou Y, Chen S, Baronio F, Grelu P. General rogue wave solutions of the coupled Fokas-Lenells equations and non-recursive Darboux transformation. Proc Math Phys Eng Sci 2019; 475:20180806. [PMID: 31105455 DOI: 10.1098/rspa.2018.0806] [Citation(s) in RCA: 21] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2018] [Accepted: 03/11/2019] [Indexed: 11/12/2022] Open
Abstract
We formulate a non-recursive Darboux transformation technique to obtain the general nth-order rational rogue wave solutions to the coupled Fokas-Lenells system, which is an integrable extension of the noted Manakov system, by considering both the double-root and triple-root situations of the spectral characteristic equation. Based on the explicit fundamental and second-order rogue wave solutions, we demonstrate several interesting rogue wave dynamics, among which are coexisting rogue waves and anomalous Peregrine solitons. Our solutions are generalized to include the complete background-field parameters and therefore helpful for future experimental study.
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Affiliation(s)
- Yanlin Ye
- School of Physics, Southeast University, Nanjing 211189, People's Republic of China
| | - Yi Zhou
- School of Physics, Southeast University, Nanjing 211189, People's Republic of China
| | - Shihua Chen
- School of Physics, Southeast University, Nanjing 211189, People's Republic of China
| | - Fabio Baronio
- INO CNR and Dipartimento di Ingegneria dell'Informazione, Università di Brescia, Via Branze 38, Brescia 25123, Italy
| | - Philippe Grelu
- Laboratoire ICB, U.M.R. 6303 C.N.R.S., Université Bourgogne Franche-Comté, 9 avenue A. Savary, 21078 Dijon, France
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8
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Chowdury A, Ankiewicz A, Akhmediev N, Chang W. Modulation instability in higher-order nonlinear Schrödinger equations. CHAOS (WOODBURY, N.Y.) 2018; 28:123116. [PMID: 30599532 DOI: 10.1063/1.5053941] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2018] [Accepted: 11/19/2018] [Indexed: 06/09/2023]
Abstract
We investigate the dynamics of modulation instability (MI) and the corresponding breather solutions to the extended nonlinear Schrödinger equation that describes the full scale growth-decay cycle of MI. As an example, we study modulation instability in connection with the fourth-order equation in detail. The higher-order equations have free parameters that can be used to control the growth-decay cycle of the MI; that is, the growth rate curves, the time of evolution, the maximal amplitude, and the spectral content of the Akhmediev Breather strongly depend on these coefficients.
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Affiliation(s)
- Amdad Chowdury
- School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798
| | - Adrian Ankiewicz
- Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra, Australian Capital Territory 2600, Australia
| | - Nail Akhmediev
- Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra, Australian Capital Territory 2600, Australia
| | - Wonkeun Chang
- School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798
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Liu C, Yang ZY, Yang WL. Growth rate of modulation instability driven by superregular breathers. CHAOS (WOODBURY, N.Y.) 2018; 28:083110. [PMID: 30180607 DOI: 10.1063/1.5025632] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/11/2018] [Accepted: 07/24/2018] [Indexed: 06/08/2023]
Abstract
We report an exact link between Zakharov-Gelash super-regular (SR) breathers (formed by a pair of quasi-Akhmediev breathers) with interesting different nonlinear propagation characteristics and modulation instability (MI). This shows that the absolute difference of group velocities of SR breathers coincides exactly with the linear MI growth rate. This link holds for a series of nonlinear Schrödinger equations with infinite-order terms. For the particular case of SR breathers with opposite group velocities, the growth rate of SR breathers is consistent with that of each quasi-Akhmediev breather along the propagation direction. Numerical simulations reveal the robustness of different SR breathers generated from various non-ideal single and multiple initial excitations. Our results provide insight into the MI nature described by SR breathers and could be helpful for controllable SR breather excitations in related nonlinear systems.
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Affiliation(s)
- Chong Liu
- School of Physics, Northwest University, Xi'an 710069, China
| | - Zhan-Ying Yang
- School of Physics, Northwest University, Xi'an 710069, China
| | - Wen-Li Yang
- School of Physics, Northwest University, Xi'an 710069, China
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10
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Chowdury A, Krolikowski W, Akhmediev N. Breather solutions of a fourth-order nonlinear Schrödinger equation in the degenerate, soliton, and rogue wave limits. Phys Rev E 2018; 96:042209. [PMID: 29347542 DOI: 10.1103/physreve.96.042209] [Citation(s) in RCA: 35] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2017] [Indexed: 11/07/2022]
Abstract
We present one- and two-breather solutions of the fourth-order nonlinear Schrödinger equation. With several parameters to play with, the solution may take a variety of forms. We consider most of these cases including the general form and limiting cases when the modulation frequencies are 0 or coincide. The zero-frequency limit produces a combination of breather-soliton structures on a constant background. The case of equal modulation frequencies produces a degenerate solution that requires a special technique for deriving. A zero-frequency limit of this degenerate solution produces a rational second-order rogue wave solution with a stretching factor involved. Taking, in addition, the zero limit of the stretching factor transforms the second-order rogue waves into a soliton. Adding a differential shift in the degenerate solution results in structural changes in the wave profile. Moreover, the zero-frequency limit of the degenerate solution with differential shift results in a rogue wave triplet. The zero limit of the stretching factor in this solution, in turn, transforms the triplet into a singlet plus a low-amplitude soliton on the background. A large value of the differential shift parameter converts the triplet into a pure singlet.
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Affiliation(s)
- Amdad Chowdury
- Science Program, Texas A&M University at Qatar, Doha, Qatar
| | | | - N Akhmediev
- Optical Sciences Group, Research School of Physics and Engineering, Australian National University, Canberra, ACT 2600, Australia
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11
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Sun WR, Wang L. Matter rogue waves for the three-component Gross-Pitaevskii equations in the spinor Bose-Einstein condensates. Proc Math Phys Eng Sci 2018; 474:20170276. [PMID: 29434501 DOI: 10.1098/rspa.2017.0276] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2017] [Accepted: 11/29/2017] [Indexed: 11/12/2022] Open
Abstract
To show the existence and properties of matter rogue waves in an F=1 spinor Bose-Einstein condensate (BEC), we work on the three-component Gross-Pitaevskii (GP) equations. Via the Darboux-dressing transformation, we obtain a family of rational solutions describing the extreme events, i.e. rogue waves. This family of solutions includes bright-dark-bright and bright-bright-bright rogue waves. The algebraic construction depends on Lax matrices and their Jordan form. The conditions for the existence of rogue wave solutions in an F=1 spinor BEC are discussed. For the three-component GP equations, if there is modulation instability, it is of baseband type only, confirming our analytic conditions. The energy transfers between the waves are discussed.
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Affiliation(s)
- Wen-Rong Sun
- School of Mathematics and Physics, Composite and Interface Science, University of Science and Technology Beijing, Beijing 100083, People's Republic of China.,Beijing Key Laboratory for Magneto-Photoelectrical, Composite and Interface Science, University of Science and Technology Beijing, Beijing 100083, People's Republic of China
| | - Lei Wang
- Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, People's Republic of China
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12
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Wang X, Liu C, Wang L. Rogue waves and W-shaped solitons in the multiple self-induced transparency system. CHAOS (WOODBURY, N.Y.) 2017; 27:093106. [PMID: 28964150 DOI: 10.1063/1.4986609] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We study localized nonlinear waves on a plane wave background in the multiple self-induced transparency (SIT) system, which describes an important enhancement of the amplification and control of optical waves compared to the single SIT system. A hierarchy of exact multiparametric rational solutions in a compact determinant representation is presented. We demonstrate that this family of solutions contain known rogue wave solutions and unusual W-shaped soliton solutions. State transitions between the fundamental rogue waves and W-shaped solitons as well as higher-order nonlinear superposition modes are revealed in the zero-frequency perturbation region by the suitable choice for the background wavenumber of the electric field component. Particularly, it is found that the multiple SIT system can admit both stationary and nonstationary W-shaped solitons in contrast to the stationary results in the single SIT system. Moreover, the W-shaped soliton complex which is formed by a certain number of fundamental W-shaped solitons with zero phase parameters and its decomposition mechanism in the case of the nonzero phase parameters are shown. Meanwhile, some important characteristics of the nonlinear waves including trajectories and spectrum are discussed through the numerical and analytical methods.
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Affiliation(s)
- Xin Wang
- College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China
| | - Chong Liu
- School of Physics, Northwest University, Xi'an 710069, China
| | - Lei Wang
- School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
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13
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Liu C, Ren Y, Yang ZY, Yang WL. Superregular breathers in a complex modified Korteweg-de Vries system. CHAOS (WOODBURY, N.Y.) 2017; 27:083120. [PMID: 28863480 DOI: 10.1063/1.4999916] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We study superregular (SR) breathers (i.e., the quasi-Akhmediev breather collision with a certain phase shift) in a complex modified Korteweg-de Vries equation. We demonstrate that such SR waves can exhibit intriguing nonlinear structures, including the half-transition and full-suppression modes, which have no analogues in the standard nonlinear Schrödinger equation. In contrast to the standard SR breather formed by pairs of quasi-Akhmediev breathers, the half-transition mode describes a mix of quasi-Akhmediev and quasi-periodic waves, whereas the full-suppression mode shows a non-amplifying nonlinear dynamics of localized small perturbations associated with the vanishing growth rate of modulation instability. Interestingly, we show analytically and numerically that these different SR modes can be evolved from an identical localized small perturbation. In particular, our results demonstrate an excellent compatibility relation between SR modes and the linear stability analysis.
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Affiliation(s)
- Chong Liu
- School of Physics, Northwest University, Xi'an 710069, China
| | - Yang Ren
- School of Physics, Northwest University, Xi'an 710069, China
| | - Zhan-Ying Yang
- School of Physics, Northwest University, Xi'an 710069, China
| | - Wen-Li Yang
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, China
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14
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Chowdury A, Krolikowski W. Breather-to-soliton transformation rules in the hierarchy of nonlinear Schrödinger equations. Phys Rev E 2017; 95:062226. [PMID: 28709292 DOI: 10.1103/physreve.95.062226] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2017] [Indexed: 11/07/2022]
Abstract
We study the exact first-order soliton and breather solutions of the integrable nonlinear Schrödinger equations hierarchy up to fifth order. We reveal the underlying physical mechanism which transforms a breather into a soliton. Furthermore, we show how the dynamics of the Akhmediev breathers which exist on a constant background as a result of modulation instability, is connected with solitons on a zero background. We also demonstrate that, while a first-order rogue wave can be directly transformed into a soliton, higher-order rogue wave solutions become rational two-soliton solutions with complex collisional structure on a background. Our results will have practical implications in supercontinuum generation, turbulence, and similar other complex nonlinear scenarios.
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Affiliation(s)
- Amdad Chowdury
- Science Program, Texas A&M University at Qatar, Doha, Qatar
| | - Wieslaw Krolikowski
- Science Program, Texas A&M University at Qatar, Doha, Qatar.,Laser Physics Centre, Australian National University, Canberra, Australia
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15
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Sun WR, Liu DY, Xie XY. Vector semirational rogue waves and modulation instability for the coupled higher-order nonlinear Schrödinger equations in the birefringent optical fibers. CHAOS (WOODBURY, N.Y.) 2017; 27:043114. [PMID: 28456173 DOI: 10.1063/1.4981907] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We report the existence and properties of vector breather and semirational rogue-wave solutions for the coupled higher-order nonlinear Schrödinger equations, which describe the propagation of ultrashort optical pulses in birefringent optical fibers. Analytic vector breather and semirational rogue-wave solutions are obtained with Darboux dressing transformation. We observe that the superposition of the dark and bright contributions in each of the two wave components can give rise to complicated breather and semirational rogue-wave dynamics. We show that the bright-dark type vector solitons (or breather-like vector solitons) with nonconstant speed interplay with Akhmediev breathers, Kuznetsov-Ma solitons, and rogue waves. By adjusting parameters, we note that the rogue wave and bright-dark soliton merge, generating the boomeron-type bright-dark solitons. We prove that the rogue wave can be excited in the baseband modulation instability regime. These results may provide evidence of the collision between the mixed ultrashort soliton and rogue wave.
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Affiliation(s)
- Wen-Rong Sun
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
| | - De-Yin Liu
- State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Xi-Yang Xie
- State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
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16
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Zhang JH, Wang L, Liu C. Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects. Proc Math Phys Eng Sci 2017; 473:20160681. [PMID: 28413335 DOI: 10.1098/rspa.2016.0681] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/08/2016] [Accepted: 02/03/2017] [Indexed: 11/12/2022] Open
Abstract
We study the higher-order generalized nonlinear Schrödinger (NLS) equation describing the propagation of ultrashort optical pulse in optical fibres. By using Darboux transformation, we derive the superregular breather solution that develops from a small localized perturbation. This type of solution can be used to characterize the nonlinear stage of the modulation instability (MI) of the condensate. In particular, we show some novel characteristics of the nonlinear stage of MI arising from higher-order effects: (i) coexistence of a quasi-Akhmediev breather and a multipeak soliton; (ii) two multipeak solitons propagation in opposite directions; (iii) a beating pattern followed by two multipeak solitons in the same direction. It is found that these patterns generated from a small localized perturbation do not have the analogues in the standard NLS equation. Our results enrich Zakharov's theory of superregular breathers and could provide helpful insight on the nonlinear stage of MI in presence of the higher-order effects.
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Affiliation(s)
- Jian-Hui Zhang
- Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Lei Wang
- Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Chong Liu
- School of Physics, Northwest University, Xi'an 710069, People's Republic of China.,Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, People's Republic of China
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17
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Chin SA, Ashour OA, Nikolić SN, Belić MR. Peak-height formula for higher-order breathers of the nonlinear Schrödinger equation on nonuniform backgrounds. Phys Rev E 2017; 95:012211. [PMID: 28208381 DOI: 10.1103/physreve.95.012211] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2016] [Indexed: 11/07/2022]
Abstract
Given any background (or seed) solution of the nonlinear Schrödinger equation, the Darboux transformation can be used to generate higher-order breathers with much greater peak intensities. In this work, we use the Darboux transformation to prove, in a unified manner and without knowing the analytical form of the background solution, that the peak height of a high-order breather is just a sum of peak heights of first-order breathers plus that of the background, irrespective of the specific choice of the background. Detailed results are verified for breathers on a cnoidal background. Generalizations to more extended nonlinear Schrödinger equations, such as the Hirota equation, are indicated.
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Affiliation(s)
- Siu A Chin
- Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA
| | - Omar A Ashour
- Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA.,Science Program, Texas A&M University at Qatar, P.O. Box 23874 Doha, Qatar
| | - Stanko N Nikolić
- Science Program, Texas A&M University at Qatar, P.O. Box 23874 Doha, Qatar.,Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
| | - Milivoj R Belić
- Science Program, Texas A&M University at Qatar, P.O. Box 23874 Doha, Qatar
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18
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Liu C, Yang ZY, Zhao LC, Duan L, Yang G, Yang WL. Symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime. Phys Rev E 2016; 94:042221. [PMID: 27841651 DOI: 10.1103/physreve.94.042221] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/08/2016] [Indexed: 06/06/2023]
Abstract
We study symmetric and asymmetric optical multipeak solitons on a continuous wave background in the femtosecond regime of a single-mode fiber. Key characteristics of such multipeak solitons, such as the formation mechanism, propagation stability, and shape-changing collisions, are revealed in detail. Our results show that this multipeak (symmetric or asymmetric) mode could be regarded as a single pulse formed by a nonlinear superposition of a periodic wave and a single-peak (W-shaped or antidark) soliton. In particular, a phase diagram for different types of nonlinear excitations on a continuous wave background, including the unusual multipeak soliton, the W-shaped soliton, the antidark soliton, the periodic wave, and the known breather rogue wave, is established based on the explicit link between exact solution and modulation instability analysis. Numerical simulations are performed to confirm the propagation stability of the multipeak solitons with symmetric and asymmetric structures. Further, we unveil a remarkable shape-changing feature of asymmetric multipeak solitons. It is interesting that these shape-changing interactions occur not only in the intraspecific collision (soliton mutual collision) but also in the interspecific interaction (soliton-breather interaction). Our results demonstrate that each multipeak soliton exhibits the coexistence of shape change and conservation of the localized energy of a light pulse against the continuous wave background.
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Affiliation(s)
- Chong Liu
- School of Physics, Northwest University, Xi'an 710069, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, China
| | - Zhan-Ying Yang
- School of Physics, Northwest University, Xi'an 710069, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, China
| | - Li-Chen Zhao
- School of Physics, Northwest University, Xi'an 710069, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, China
| | - Liang Duan
- School of Physics, Northwest University, Xi'an 710069, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, China
| | - Guangye Yang
- Department of Physics, Shanxi Medical University, Taiyuan, Shanxi 030001, China
| | - Wen-Li Yang
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710069, China
- Institute of Modern Physics, Northwest University, Xi'an 710069, China
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19
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Wang L, Zhang JH, Liu C, Li M, Qi FH. Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects. Phys Rev E 2016; 93:062217. [PMID: 27415265 DOI: 10.1103/physreve.93.062217] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2016] [Indexed: 06/06/2023]
Abstract
We study a variable-coefficient nonlinear Schrödinger (vc-NLS) equation with higher-order effects. We show that the breather solution can be converted into four types of nonlinear waves on constant backgrounds including the multipeak solitons, antidark soliton, periodic wave, and W-shaped soliton. In particular, the transition condition requiring the group velocity dispersion (GVD) and third-order dispersion (TOD) to scale linearly is obtained analytically. We display several kinds of elastic interactions between the transformed nonlinear waves. We discuss the dispersion management of the multipeak soliton, which indicates that the GVD coefficient controls the number of peaks of the wave while the TOD coefficient has compression effect. The gain or loss has influence on the amplitudes of the multipeak soliton. We further derive the breather multiple births and Peregrine combs by using multiple compression points of Akhmediev breathers and Peregrine rogue waves in optical fiber systems with periodic GVD modulation. In particular, we demonstrate that the Peregrine comb can be converted into a Peregrine wall by the proper choice of the amplitude of the periodic GVD modulation. The Peregrine wall can be seen as an intermediate state between rogue waves and W-shaped solitons. We finally find that the modulational stability regions with zero growth rate coincide with the transition condition using rogue wave eigenvalues. Our results could be useful for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in diverse physical systems modeled by vc-NLS equation with higher-order effects.
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Affiliation(s)
- Lei Wang
- Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Jian-Hui Zhang
- School of Energy Power and Mechanical Engneering, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Chong Liu
- School of Physics, Northwest University, Xi'an 710069, People's Republic of China
| | - Min Li
- Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Feng-Hua Qi
- School of Information, Beijing Wuzi University, Beijing 101149, People's Republic of China
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20
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Wang L, Zhang JH, Wang ZQ, Liu C, Li M, Qi FH, Guo R. Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation. Phys Rev E 2016; 93:012214. [PMID: 26871080 DOI: 10.1103/physreve.93.012214] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/26/2015] [Indexed: 06/05/2023]
Abstract
We study the nonlinear waves on constant backgrounds of the higher-order generalized nonlinear Schrödinger (HGNLS) equation describing the propagation of ultrashort optical pulse in optical fibers. We derive the breather, rogue wave, and semirational solutions of the HGNLS equation. Our results show that these three types of solutions can be converted into the nonpulsating soliton solutions. In particular, we present the explicit conditions for the transitions between breathers and solitons with different structures. Further, we investigate the characteristics of the collisions between the soliton and breathers. Especially, based on the semirational solutions of the HGNLS equation, we display the novel interactions between the rogue waves and other nonlinear waves. In addition, we reveal the explicit relation between the transition and the distribution characteristics of the modulation instability growth rate.
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Affiliation(s)
- Lei Wang
- Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, P. R. China
| | - Jian-Hui Zhang
- School of Energy Power and Mechanical Engneering, North China Electric Power University, Beijing 102206, P. R. China
| | - Zi-Qi Wang
- School of Energy Power and Mechanical Engneering, North China Electric Power University, Beijing 102206, P. R. China
| | - Chong Liu
- School of Physics, Northwest University, Xi'an 710069, P. R. China
| | - Min Li
- Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, P. R. China
| | - Feng-Hua Qi
- School of Information, Beijing Wuzi University, Beijing 101149, P. R. China
| | - Rui Guo
- School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, P. R. China
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