1
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Wen XY, Lin Z, Wang DS. High-order rogue wave and mixed interaction patterns for the three-component Gross-Pitaevskii equations in F=1 spinor Bose-Einstein condensates. Phys Rev E 2024; 109:044215. [PMID: 38755818 DOI: 10.1103/physreve.109.044215] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2023] [Accepted: 04/01/2024] [Indexed: 05/18/2024]
Abstract
Under investigation are the three-component Gross-Pitaevskii equations in F=1 spinor Bose-Einstein condensates. Various localized waves' generation mechanisms have been derived from plane wave solutions using modulation instability. The perturbed continuous waves produce a large number of rogue wave structures through the split-step Fourier numerical method. Based on the known Lax pair, we construct the generalized iterative (n,N-n)-fold Darboux transformation to generate various high-order solutions, including the bright-dark-bright structure of rogue waves, periodic waves, and their mixed interaction structures. Numerical simulations show that rogue waves with a two-peaked structure have robust noise resistance and stable dynamical behavior. The asymptotic states of high-order rogue waves as the parameter approaches infinity are also predicted using the large parameter asymptotic technique. In addition, the position of these localized wave patterns can be controlled by some special parameters. These results may help us understand the dynamic behavior of spinor condensates for the mean-field approximation.
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Affiliation(s)
- Xiao-Yong Wen
- School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
| | - Zhe Lin
- School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, East China Normal University, Shanghai 200241, China and School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
| | - Deng-Shan Wang
- Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
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2
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Wen XY, Liu XK, Chen Y, Yan Z. Rogue wave excitations and hybrid wave structures of the Heisenberg ferromagnet equation with time-dependent inhomogeneous bilinear interaction and spin-transfer torque. CHAOS (WOODBURY, N.Y.) 2024; 34:033131. [PMID: 38502966 DOI: 10.1063/5.0191956] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2023] [Accepted: 02/19/2024] [Indexed: 03/21/2024]
Abstract
In this paper, we focus on the localized rational waves of the variable-coefficient Heisenberg spin chain equation, which models the local magnetization in ferromagnet with time-dependent inhomogeneous bilinear interaction and spin-transfer torque. First, we establish the iterative generalized (m,N-m)-fold Darboux transformation of the Heisenberg spin chain equation. Then, the novel localized rational solutions (LRSs), rogue waves (RWs), periodic waves, and hybrid wave structures on the periodic, zero, and nonzero constant backgrounds with the time-dependent coefficients α(t) and β(t) are obtained explicitly. Additionally, we provide the trajectory curves of magnetization and the variation of the magnetization direction for the obtained nonlinear waves at different times. These phenomena imply that the LRSs and RWs play the crucial roles in changing the circular motion of the magnetization. Finally, we also numerically simulate the wave propagations of some localized semi-rational solutions and RWs.
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Affiliation(s)
- Xiao-Yong Wen
- School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
| | - Xue-Ke Liu
- School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
| | - Yong Chen
- School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China
| | - Zhenya Yan
- KLMM, 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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3
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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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4
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Weng W, Zhang G, Yan Z. Strong and weak interactions of rational vector rogue waves and solitons to any
n
-component nonlinear Schrödinger system with higher-order effects. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2021.0670] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
The higher-order effects play an important role in the wave propagations of ultrashort (e.g. subpicosecond or femtosecond) light pulses in optical fibres. In this paper, we investigate any
n
-component fourth-order nonlinear Schrödinger (
n
-FONLS) system with non-zero backgrounds containing the
n
-Hirota equation and the
n
-Lakshmanan–Porsezian–Daniel equation. Based on the loop group theory, we find the multi-parameter family of novel rational vector rogue waves (RVRWs) of the
n
-FONLS equation starting from the plane-wave solutions. Moreover, we exhibit the weak and strong interactions of some representative RVRW structures. In particular, we also find that the W-shaped rational vector dark and bright solitons of the
n
-FONLS equation as the second- and fourth-order dispersion coefficients satisfy some relation. Furthermore, we find the higher-order RVRWs of the
n
-FONLS equation. These obtained rational solutions will be useful in the study of RVRW phenomena of multi-component nonlinear wave models in nonlinear optics, deep ocean and Bose–Einstein condensates.
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Affiliation(s)
- Weifang Weng
- Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
- School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
| | - Guoqiang Zhang
- Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
| | - Zhenya Yan
- Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
- School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
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5
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Wang X, Wang L, Wei J, Guo B, Kang J. Rogue waves in the three-level defocusing coupled Maxwell–Bloch equations. Proc Math Phys Eng Sci 2021. [DOI: 10.1098/rspa.2021.0585] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
The coupled Maxwell–Bloch (CMB) system is a fundamental model describing the propagation of ultrashort laser pulses in a resonant medium with coherent three-level atomic transitions. In this paper, we consider an integrable generalization of the CMB equations with the defocusing case. The CMB hierarchy is derived with the aid of a
3
×
3
matrix eigenvalue problem and the Lenard recursion equation, from which the defocusing CMB model is proposed as a special reduction of the general CMB equations. The
n
-fold Darboux transformation as well as the multiparametric
n
th-order rogue wave solution of the defocusing CMB equations are put forward in terms of Schur polynomials. As an application, the explicit rogue wave solutions from first to second order are presented. Apart from the traditional dark rogue wave, bright rogue wave and four-petalled rogue wave, some novel rogue wave structures such as the dark four-peaked rogue wave and the double-ridged rogue wave are found. Moreover, the second-order rogue wave triplets which contain a fixed number of these rogue waves are shown.
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Affiliation(s)
- Xin Wang
- College of Science, Zhongyuan University of Technology, Zhengzhou 450007, People’s Republic of China
| | - Lei Wang
- School of Mathematics and Physics, North China Electric Power University, Beijing 102206, People’s Republic of China
| | - Jiao Wei
- School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan 450001, People’s Republic of China
| | - Bowen Guo
- College of Science, Zhongyuan University of Technology, Zhengzhou 450007, People’s Republic of China
| | - Jingfeng Kang
- College of Science, Zhongyuan University of Technology, Zhengzhou 450007, People’s Republic of China
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6
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Qin YH, Zhao LC, Yang ZQ, Ling L. Multivalley dark solitons in multicomponent Bose-Einstein condensates with repulsive interactions. Phys Rev E 2021; 104:014201. [PMID: 34412198 DOI: 10.1103/physreve.104.014201] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/12/2021] [Accepted: 06/09/2021] [Indexed: 06/13/2023]
Abstract
We obtain multivalley dark soliton solutions with asymmetric or symmetric profiles in multicomponent repulsive Bose-Einstein condensates by developing the Darboux transformation method. We demonstrate that the width-dependent parameters of solitons significantly affect the velocity ranges and phase jump regions of multivalley dark solitons, in sharp contrast to scalar dark solitons. For double-valley dark solitons, we find that the phase jump is in the range [0,2π], which is quite different from that of the usual single-valley dark soliton. Based on our results, we argue that the phase jump of an n-valley dark soliton could be in the range [0,nπ], supported by our analysis extending up to five-component condensates. The interaction between a double-valley dark soliton and a single-valley dark soliton is further investigated, and we reveal a striking collision process in which the double-valley dark soliton is transformed into a breather after colliding with the single-valley dark soliton. Our analyses suggest that this breather transition exists widely in the collision processes involving multivalley dark solitons. The possibilities for observing these multivalley dark solitons in related Bose-Einstein condensates experiments are discussed.
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Affiliation(s)
- Yan-Hong Qin
- School of Physics, Northwest University, Xi'an 710127, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
| | - Li-Chen Zhao
- School of Physics, Northwest University, Xi'an 710127, China
- NSFC-SPTP Peng Huanwu Center for Fundamental Theory, Xi'an 710127, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
| | - Zeng-Qiang Yang
- Department of Physics, School of Arts and Sciences, Shaanxi University of Science and Technology, Xi'an 710021, China
| | - Liming Ling
- School of Mathematics, South China University of Technology, Guangzhou 510640, China
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7
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Gao P, Zhao LC, Yang ZY, Li XH, Yang WL. High-order rogue waves excited from multi-Gaussian perturbations on a continuous wave. OPTICS LETTERS 2020; 45:2399-2402. [PMID: 32287243 DOI: 10.1364/ol.389012] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/04/2020] [Accepted: 03/20/2020] [Indexed: 06/11/2023]
Abstract
Peregrine rogue wave excitation has applications in gaining high-intensity pulses, etc., and a high-order rogue wave exhibits higher intensity. An exact solution and collision between breathers are two existing ways to excite high-order ones. Here we numerically report a new, to the best of our knowledge, possible method, which is by multi-Gaussian perturbations on a continuous wave. The order and maximal intensity of rogue waves can be adjusted by the number of perturbations. The maximal intensity approaches 63.8 times that of the power of the initial background wave, and it retains a large value under the influence of fiber loss and noise. Our results provide guidance in gaining high-intensity pulses in experiment and understanding the universality of rogue wave generation.
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8
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Chen J, Chen L, Feng BF, Maruno KI. High-order rogue waves of a long-wave-short-wave model of Newell type. Phys Rev E 2019; 100:052216. [PMID: 31869955 DOI: 10.1103/physreve.100.052216] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2019] [Indexed: 11/07/2022]
Abstract
The long-wave-short-wave (LWSW) model of Newell type is an integrable model describing the interaction between the gravity wave (long wave) and the capillary wave (short wave) for the surface wave of deep water under certain resonance conditions. In the present paper, we are concerned with rogue-wave solutions to the LWSW model of Newell type. By combining the Hirota's bilinear method and the KP hierarchy reduction, we construct its general rational solution expressed by the determinant. It is found that the fundamental rogue wave for the short wave can be classified into three different patterns: bright, intermediate, and dark states, whereas the one for the long wave is always a bright state. The higher-order rogue wave corresponds to the superposition of fundamental ones. The modulation instability analysis shows that the condition of the baseband modulation instability where an unstable continuous-wave background corresponds to perturbations with infinitesimally small frequencies, coincides with the condition for the existence of rogue-wave solutions.
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Affiliation(s)
- Junchao Chen
- Department of Mathematics, Lishui University, Lishui, 323000, People's Republic of China
| | - Liangyuan Chen
- Department of Photoelectric Engineering, Lishui University, Lishui 323000, People's Republic of China
| | - Bao-Feng Feng
- School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, Texas 78541, USA
| | - Ken-Ichi Maruno
- Department of Applied Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan
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9
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Vishnu Priya N, Senthilvelan M, Rangarajan G. On the role of four-wave mixing effect in the interactions between nonlinear modes of coupled generalized nonlinear Schrödinger equation. CHAOS (WOODBURY, N.Y.) 2019; 29:123135. [PMID: 31893664 DOI: 10.1063/1.5121245] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2019] [Accepted: 12/04/2019] [Indexed: 06/10/2023]
Abstract
In this paper, we investigate the effect of four-wave mixing in the interactions among nonlinear waves such as solitons, breathers, and rogue waves of a coupled generalized nonlinear Schrödinger equation. We explore several interesting results including superposition of breather pulses, increment in the number of breather pulses and in amplitudes of breathers, and rogue waves. By strengthening the four-wave mixing parameter, we observe different transformations that occur between different localized structures. For instance, we visualize a transformation from bright soliton to breather form, bright and dark rogue wave to four-petaled rogue wave structures, four-petaled rogue wave to other rogue wave forms, and so on. Another important observation that we report here is that the interaction of a bright soliton with a rogue wave in the presence of the four-wave mixing effect provides interaction between a dark oscillatory soliton and a rogue wave.
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Affiliation(s)
- N Vishnu Priya
- Department of Mathematics, Indian Institute of Science, Bengaluru 560 012, Karnataka, India
| | - M Senthilvelan
- Department of Nonlinear Dynamics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| | - Govindan Rangarajan
- Department of Mathematics, Indian Institute of Science, Bengaluru 560 012, Karnataka, India
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10
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Gao P, Duan L, Zhao LC, Yang ZY, Yang WL. Dynamics of perturbations at the critical points between modulation instability and stability regimes. CHAOS (WOODBURY, N.Y.) 2019; 29:083112. [PMID: 31472492 DOI: 10.1063/1.5093161] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2019] [Accepted: 07/17/2019] [Indexed: 06/10/2023]
Abstract
We study numerically the evolutions of perturbations at critical points between modulational instability and stability regimes. It is demonstrated that W-shaped solitons and rogue waves can be both excited from weak resonant perturbations at the critical points. The rogue wave excitation at the critical points indicates that rogue wave comes from modulation instability with resonant perturbations, even when the baseband modulational instability is absent. The perturbation differences for generating W-shaped solitons and rogue waves are discussed in detail. These results can be used to generate W-shaped solitons and rogue waves controllably from weak perturbations.
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Affiliation(s)
- Peng Gao
- School of Physics, Northwest University, Xi'an 710069, China
| | - Liang Duan
- School of Physics, Northwest University, Xi'an 710069, China
| | - Li-Chen Zhao
- 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, 710069 Xi'an, China
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11
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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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12
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Rao J, Porsezian K, He J. Semi-rational solutions of the third-type Davey-Stewartson equation. CHAOS (WOODBURY, N.Y.) 2017; 27:083115. [PMID: 28863505 DOI: 10.1063/1.4999083] [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
General dark solitons and mixed solutions consisting of dark solitons and breathers for the third-type Davey-Stewartson (DS-III) equation are derived by employing the bilinear method. By introducing the two differential operators, semi-rational solutions consisting of rogue waves, breathers, and solitons are generated. These semi-rational solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. Under suitable parametric conditions, we derive general rogue wave solutions expressed in terms of rational functions. It is shown that the fundamental (simplest) rogue waves are line rogue waves. It is also shown that the multi-rogue waves describe interactions of several fundamental rogue waves, which would generate interesting curvy wave patterns. The higher order rogue waves originate from a localized lump and retreat back to it. Several types of hybrid solutions composed of rogue waves, breathers, and solitons have also been illustrated. Specifically, these semi-rational solutions have a new phenomenon: lumps form on dark solitons and gradual separation from the dark solitons is observed.
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Affiliation(s)
- Jiguang Rao
- Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, People's Republic of China
| | | | - Jingsong He
- Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, People's Republic of China
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13
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Ling L, Zhao LC, Yang ZY, Guo B. Generation mechanisms of fundamental rogue wave spatial-temporal structure. Phys Rev E 2017; 96:022211. [PMID: 28950590 DOI: 10.1103/physreve.96.022211] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/06/2016] [Indexed: 06/07/2023]
Abstract
We discuss the generation mechanism of fundamental rogue wave structures in N-component coupled systems, based on analytical solutions of the nonlinear Schrödinger equation and modulational instability analysis. Our analysis discloses that the pattern of a fundamental rogue wave is determined by the evolution energy and growth rate of the resonant perturbation that is responsible for forming the rogue wave. This finding allows one to predict the rogue wave pattern without the need to solve the N-component coupled nonlinear Schrödinger equation. Furthermore, our results show that N-component coupled nonlinear Schrödinger systems may possess N different fundamental rogue wave patterns at most. These results can be extended to evaluate the type and number of fundamental rogue wave structure in other coupled nonlinear systems.
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Affiliation(s)
- Liming Ling
- School of Mathematics, South China University of Technology, Guangzhou 510640, China
| | - Li-Chen Zhao
- 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
| | - Boling Guo
- Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
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14
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Rogue Wave Modes for the Coupled Nonlinear Schrödinger System with Three Components: A Computational Study. APPLIED SCIENCES-BASEL 2017. [DOI: 10.3390/app7060559] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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15
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Duan L, Zhao LC, Xu WH, Liu C, Yang ZY, Yang WL. Soliton excitations on a continuous-wave background in the modulational instability regime with fourth-order effects. Phys Rev E 2017; 95:042212. [PMID: 28505799 DOI: 10.1103/physreve.95.042212] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/27/2016] [Indexed: 06/07/2023]
Abstract
We study the correspondence between modulational instability and types of fundamental nonlinear excitation in a nonlinear fiber with both third-order and fourth-order effects. Some soliton excitations are obtained in the modulational instability regime which have not been found in nonlinear fibers with second-order effects and third-order effects. Explicit analysis suggests that the existence of solitons is related to the modulation stability circle in the modulation instability regime, and they just exist in the modulational instability regime outside of the modulational stability circle. It should be emphasized that the solitons exist only with two special profiles on a continuous-wave background at a certain frequency. The evolution stability of the solitons is tested numerically by adding some noise to initial states, which indicates that they are robust against perturbations even in the modulation instability regime. Further analysis indicates that solitons in the modulational instability regime are caused by fourth-order effects.
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Affiliation(s)
- Liang Duan
- School of Physics, Northwest University, 710069 Xi'an, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, 710069 Xi'an, China
| | - Li-Chen Zhao
- School of Physics, Northwest University, 710069 Xi'an, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, 710069 Xi'an, China
| | - Wen-Hao Xu
- School of Physics, Northwest University, 710069 Xi'an, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, 710069 Xi'an, China
| | - Chong Liu
- School of Physics, Northwest University, 710069 Xi'an, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, 710069 Xi'an, China
| | - Zhan-Ying Yang
- School of Physics, Northwest University, 710069 Xi'an, China
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, 710069 Xi'an, China
| | - Wen-Li Yang
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, 710069 Xi'an, China
- Institute of Modern Physics, Northwest University, 710069 Xian, China
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16
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Chan HN, Malomed BA, Chow KW, Ding E. Rogue waves for a system of coupled derivative nonlinear Schrödinger equations. Phys Rev E 2016; 93:012217. [PMID: 26871083 DOI: 10.1103/physreve.93.012217] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2015] [Indexed: 06/05/2023]
Abstract
Rogue waves (RWs) are unexpectedly strong excitations emerging from an otherwise tranquil background. The nonlinear Schrödinger equation (NLSE), a ubiquitous model with wide applications to fluid mechanics, optics, plasmas, etc., exhibits RWs only in the regime of modulation instability (MI) of the background. For a system of multiple waveguides, the governing coupled NLSEs can produce regimes of MI and RWs, even if each component has dispersion and cubic nonlinearity of opposite signs. A similar effect is demonstrated here for a system of coupled derivative NLSEs (DNLSEs) where the special feature is the nonlinear self-steepening of narrow pulses. More precisely, these additional regimes of MI and RWs for coupled DNLSEs depend on the mismatch in group velocities between the components, and the parameters for cubic nonlinearity and self-steepening. RWs considered in this paper differ from those of the NLSEs in terms of the amplification ratio and criteria of existence. Applications to optics and plasma physics are discussed.
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Affiliation(s)
- H N Chan
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong
| | - B A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel
| | - K W Chow
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong
| | - E Ding
- Department of Mathematics and Physics, Azusa Pacific University, Azusa, California 91702, USA
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17
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Zhong WP, Belić M, Malomed BA. Rogue waves in a two-component Manakov system with variable coefficients and an external potential. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:053201. [PMID: 26651808 DOI: 10.1103/physreve.92.053201] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2014] [Indexed: 06/05/2023]
Abstract
We construct rogue waves (RWs) in a coupled two-mode system with the self-focusing nonlinearity of the Manakov type (equal SPM and XPM coefficients), spatially modulated coefficients, and a specially designed external potential. The system may be realized in nonlinear optics and Bose-Einstein condensates. By means of a similarity transformation, we establish a connection between solutions of the coupled Manakov system with spatially variable coefficients and the basic Manakov model with constant coefficients. Exact solutions in the form of two-component Peregrine and dromion waves are obtained. The RW dynamics is analyzed for different choices of parameters in the underlying parameter space. Different classes of RW solutions are categorized by means of a naturally introduced control parameter which takes integer values.
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Affiliation(s)
- Wei-Ping Zhong
- Department of Electronic and Information Engineering, Shunde Polytechnic, Guangdong Province, Shunde 528300, China
- Texas A&M University at Qatar, P.O. Box 23874 Doha, Qatar
| | - Milivoj Belić
- Texas A&M University at Qatar, P.O. Box 23874 Doha, Qatar
| | - Boris A Malomed
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
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Wang L, Zhu YJ, Qi FH, Li M, Guo R. Modulational instability, higher-order localized wave structures, and nonlinear wave interactions for a nonautonomous Lenells-Fokas equation in inhomogeneous fibers. CHAOS (WOODBURY, N.Y.) 2015; 25:063111. [PMID: 26117105 DOI: 10.1063/1.4922025] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
In this paper, the nonautonomous Lenells-Fokas (LF) model is investigated. The modulational instability analysis of the solutions with variable coefficients in the presence of a small perturbation is studied. Higher-order soliton, breather, earthwormon, and rogue wave solutions of the nonautonomous LF model are derived via the n-fold variable-coefficient Darboux transformation. The solitons and earthwormons display the elastic collisions. It is found that the nonautonomous LF model admits the higher-order periodic rogue waves, composite rogue waves (rogue wave pair), and oscillating rogue waves, whose dynamics can be controlled by the inhomogeneous nonlinear parameters. Based on the second-order rogue wave, a diamond structure consisting of four first-order rogue waves is observed. In addition, the semirational solutions (the mixed rational-exponential solutions) of the nonautonomous LF model are obtained, which can be used to describe the interactions between the rogue waves and breathers. Our results could be helpful for the design of experiments in the optical fiber communications.
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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
| | - Yu-Jie Zhu
- Institute of Electrical and Electronic Engineering, 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
| | - Min Li
- Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Rui Guo
- School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, People's Republic of China
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19
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Chen S, Soto-Crespo JM, Grelu P. Dark three-sister rogue waves in normally dispersive optical fibers with random birefringence. OPTICS EXPRESS 2014; 22:27632-27642. [PMID: 25401907 DOI: 10.1364/oe.22.027632] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We investigate dark rogue wave dynamics in normally dispersive birefringent optical fibers, based on the exact rational solutions of the coupled nonlinear Schrödinger equations. Analytical solutions are derived up to the second order via a nonrecursive Darboux transformation method. Vector dark "three-sister" rogue waves as well as their existence conditions are demonstrated. The robustness against small perturbations is numerically confirmed in spite of the onset of modulational instability, offering the possibility to observe such extreme events in normal optical fibers with random birefringence, or in other Manakov-type vector nonlinear media.
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Mareeswaran RB, Charalampidis EG, Kanna T, Kevrekidis PG, Frantzeskakis DJ. Vector rogue waves and dark-bright boomeronic solitons in autonomous and nonautonomous settings. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:042912. [PMID: 25375572 DOI: 10.1103/physreve.90.042912] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2014] [Indexed: 06/04/2023]
Abstract
In this work we consider the dynamics of vector rogue waves and dark-bright solitons in two-component nonlinear Schrödinger equations with various physically motivated time-dependent nonlinearity coefficients, as well as spatiotemporally dependent potentials. A similarity transformation is utilized to convert the system into the integrable Manakov system and subsequently the vector rogue and dark-bright boomeronlike soliton solutions of the latter are converted back into ones of the original nonautonomous model. Using direct numerical simulations we find that, in most cases, the rogue wave formation is rapidly followed by a modulational instability that leads to the emergence of an expanding soliton train. Scenarios different than this generic phenomenology are also reported.
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Affiliation(s)
- R Babu Mareeswaran
- Post Graduate and Research Department of Physics, Bishop Heber College, Tiruchirapalli-620 017, Tamil Nadu, India
| | - E G Charalampidis
- School of Civil Engineering, Faculty of Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece and Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - T Kanna
- Post Graduate and Research Department of Physics, Bishop Heber College, Tiruchirapalli-620 017, Tamil Nadu, India
| | - P G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - D J Frantzeskakis
- Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
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21
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Zhao LC, Xin GG, Yang ZY. Rogue-wave pattern transition induced by relative frequency. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:022918. [PMID: 25215810 DOI: 10.1103/physreve.90.022918] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2014] [Indexed: 06/03/2023]
Abstract
We revisit a rogue wave in a two-mode nonlinear fiber whose dynamics is described by two-component coupled nonlinear Schrödinger equations. The relative frequency between two modes can induce different rogue wave patterns transition. In particular, we find a four-petaled flower structure rogue wave can exist in the two-mode coupled system, which possesses an asymmetric spectrum distribution. Furthermore, spectrum analysis is performed on these different type rogue waves, and the spectrum relations between them are discussed. We demonstrate qualitatively that different modulation instability gain distribution can induce different rogue wave excitation patterns. These results would deepen our understanding of rogue wave dynamics in complex systems.
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Affiliation(s)
- Li-Chen Zhao
- Department of Physics, Northwest University, Xi'an 710069, China
| | - Guo-Guo Xin
- Department of Physics, Northwest University, Xi'an 710069, China
| | - Zhan-Ying Yang
- Department of Physics, Northwest University, Xi'an 710069, China
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