1
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Yin HM, Chow KW. Fermi-Pasta-Ulam-Tsingou recurrence and cascading mechanism for resonant three-wave interactions. Phys Rev E 2023; 107:064215. [PMID: 37464613 DOI: 10.1103/physreve.107.064215] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/22/2022] [Accepted: 05/26/2023] [Indexed: 07/20/2023]
Abstract
Evolution of resonant three-wave interaction is governed by quadratic nonlinearities. While propagating localized modes and inverse scattering mechanisms have been studied, transient states such as rogue waves and breathers are not fully understood. Modulation instability modes can trigger growth of disturbances and the eventual development of breathers. Here we study computationally the dynamics beyond the first formation of breathers, and demonstrate repeating patterns of breathers as a manifestation of the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT). While nonlinearity governs the actual dynamics, the range of wave numbers for modulation instability remains a useful indicator. Depending on the stability characteristics of the fundamental mode and the higher-order harmonics ("sidebands"), "regular" and "staggered" FPUT patterns can arise. A "cascading mechanism" provides analytical verification, as the fundamental and sideband modes attain the same magnitude at one particular instant, signifying the first occurrence of a breather. A triangular spectrum is also computed, similar to experimental observations of optical pulses. Such spectra can elucidate the spreading of energy among the sidebands and components of the triad resonance. The concept of "effective energy" is examined and the eigenvalues of the inverse scattering mechanism are computed. Both approaches are utilized to correlate with the occurrence of regular or staggered FPUT. These numerical and analytical studies can enhance our understanding of wave interactions in fluid mechanics and optics.
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Affiliation(s)
- H M Yin
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong
| | - K W Chow
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong
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2
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Che WJ, Liu C, Akhmediev N. Fundamental and second-order dark soliton solutions of two- and three-component Manakov equations in the defocusing regime. Phys Rev E 2023; 107:054206. [PMID: 37329094 DOI: 10.1103/physreve.107.054206] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/15/2023] [Accepted: 04/10/2023] [Indexed: 06/18/2023]
Abstract
We present exact multiparameter families of soliton solutions for two- and three-component Manakov equations in the defocusing regime. Existence diagrams for such solutions in the space of parameters are presented. Fundamental soliton solutions exist only in finite areas on the plane of parameters. Within these areas, the solutions demonstrate rich spatiotemporal dynamics. The complexity increases in the case of three-component solutions. The fundamental solutions are dark solitons with complex oscillating patterns in the individual wave components. At the boundaries of existence, the solutions are transformed into plain (nonoscillating) vector dark solitons. The superposition of two dark solitons in the solution adds more frequencies in the patterns of oscillating dynamics. These solutions admit degeneracy when the eigenvalues of fundamental solitons in the superposition coincide.
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Affiliation(s)
- Wen-Juan Che
- School of Physics, Northwest University, Xi'an 710127, China
| | - Chong Liu
- School of Physics, Northwest University, Xi'an 710127, China
- Department of Fundamental and Theoretical Physics, Research School of Physics, The Australian National University, Canberra, ACT 2600, Australia
- Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China
- Peng Huanwu Center for Fundamental Theory, Xi'an 710127, China
| | - Nail Akhmediev
- Department of Fundamental and Theoretical Physics, Research School of Physics, The Australian National University, Canberra, ACT 2600, Australia
- Arts & Sciences Division, Texas A&M University at Qatar, Doha, Qatar
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3
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Chen SC, Liu C, Yao X, Zhao LC, Akhmediev N. Extreme spectral asymmetry of Akhmediev breathers and Fermi-Pasta-Ulam recurrence in a Manakov system. Phys Rev E 2021; 104:024215. [PMID: 34525585 DOI: 10.1103/physreve.104.024215] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2020] [Accepted: 05/27/2021] [Indexed: 11/07/2022]
Abstract
The dynamics of Fermi-Pasta-Ulam (FPU) recurrence in a Manakov system is studied analytically. Exact Akhmediev breather (AB) solutions for this system are found that cannot be reduced to the ABs of a single-component nonlinear Schrödinger equation. Expansion-contraction cycle of the corresponding spectra with an infinite number of sidebands is calculated analytically using a residue theorem. A distinctive feature of these spectra is the asymmetry between positive and negative spectral modes. A practically important consequence of the spectral asymmetry is a nearly complete energy transfer from the central mode to one of the lowest-order (left or right) sidebands. Numerical simulations started with modulation instability of plane waves confirm the findings based on the exact solutions. It is also shown that the full growth-decay cycle of the AB leads to the nonlinear phase shift between the initial and final states in both components of the Manakov system. This finding shows that the final state of the FPU recurrence described by the vector ABs is not quite the same as the initial state. Our results are applicable and can be observed in a wide range of two-component physical systems such as two-component waves in optical fibers, two-directional waves in crossing seas, and two-component Bose-Einstein condensates.
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Affiliation(s)
- Shao-Chun Chen
- School of Physics, Northwest University, Xi'an 710127, China
| | - Chong Liu
- School of Physics, Northwest University, Xi'an 710127, China.,Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 2600, Australia.,Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China.,NSFC-SPTP Peng Huanwu Center for Fundamental Theory, Xi'an 710127, China
| | - Xiankun Yao
- 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.,Shaanxi Key Laboratory for Theoretical Physics Frontiers, Xi'an 710127, China.,NSFC-SPTP Peng Huanwu Center for Fundamental Theory, Xi'an 710127, China
| | - Nail Akhmediev
- Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 2600, Australia
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4
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Sun WR, Liu L, Kevrekidis PG. Rogue waves of ultra-high peak amplitude: a mechanism for reaching up to a thousand times the background level. Proc Math Phys Eng Sci 2021. [DOI: 10.1098/rspa.2020.0842] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We unveil a mechanism enabling a fundamental rogue wave, expressed by a rational function of fourth degree, to reach a peak amplitude as high as a thousand times the background level in a system of coupled nonlinear Schrödinger equations involving both incoherent and coherent coupling terms with suitable coefficients. We obtain the exact explicit vector rational solutions using a Darboux-dressing transformation. We show that
both
components of such coupled equations can reach extremely high amplitudes. The mechanism is confirmed in direct numerical simulations and its robustness is confirmed upon noisy perturbations. Additionally, we showcase the fact that extremely high peak-amplitude vector fundamental rogue waves (of about 80 times the background level) can be excited even within a
chaotic background field
.
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Affiliation(s)
- Wen-Rong Sun
- School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China
| | - Lei Liu
- Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China
| | - P. G. Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA
- Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
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5
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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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6
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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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7
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Abstract
The formation of rogue oceanic waves may be the result of different causes. Various factors (winds, currents, dispersive focussing, depth, nonlinear focussing and instability) make this subject intriguing, and yet its understanding is quite relevant to practical issues. Here, we deal only with the nonlinear character of this dynamics, which has been recognised as the main ingredient to rogue wave formation. In this perspective, the formation of rogue waves requires a non-vanishing and unstable background such as a nonlinear regular wave train with attractive self-interaction. The simplest, best known model of such dynamics is the universal nonlinear Schrödinger equation. This has proven to serve as a good approximation in various contexts and over a broad range of experimental settings. This model aims to give the slow evolution of the envelope of one monochromatic wave due to nonlinearity. Here, we naturally consider the same problem for the envelopes of two weakly resonant monochromatic waves. As for the nonlinear Schrödinger equation, which is integrable, we adopt an integrable model to describe the interaction of two waves. This is the system of two coupled nonlinear Schrödinger equations (Manakov model) with self- and cross-interactions that may be both defocussing and focussing. We first discuss the linear stability properties of the background by computing the spectrum for all values of the parameters such as coupling constants and amplitudes. In particular, we relate the instability bands to properties of the spectrum and compute the gain function (or growth rate). We also relate to the stability spectrum the value of the spectral variable, which corresponds to a rogue wave solution. In contrast with the nonlinear Schrödinger equation, different types of single rogue wave exist that correspond to different values of the spectral variable even in the same spectrum. For these critical values, which are completely classified, we give the corresponding explicit expression of the rogue wave solution that follows from the well known Darboux–Dressing transformation method. Although not all systems of two coupled nonlinear Schrödinger equations that have been derived in water wave dynamics are integrable, our investigation contributes to the understanding of new effects due to wave coupling, at least for model equations that, even if not integrable, are close enough to the model considered here. For instance, our findings lead to investigate rogue waves generated by instabilities due to self- and cross-interactions of defocusing type. An illustrative selection of two coupled rogue waves solutions is displayed.
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8
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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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9
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Mančić A, Baronio F, Hadžievski L, Wabnitz S, Maluckov A. Statistics of vector Manakov rogue waves. Phys Rev E 2018; 98:012209. [PMID: 30110823 DOI: 10.1103/physreve.98.012209] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2018] [Indexed: 06/08/2023]
Abstract
We present a statistical analysis based on the height and return-time probabilities of high-amplitude wave events in both focusing and defocusing Manakov systems. We find that analytical rational or semirational solutions, associated with extreme, rogue wave (RW) structures, are the leading high-amplitude events in this system. We define the thresholds for classifying an extreme wave event as a RW. Our results indicate that there is a strong relationship between the type of RW and the mechanism which is responsible for its creation. Initially, high-amplitude events originate from modulation instability. Upon subsequent evolution, the interaction among these events prevails as the mechanism for RW creation. We suggest a strategy for confirming the basic properties of different extreme events. This involves the definition of proper statistical measures at each stage of the RW dynamics. Our results point to the need for redefining criteria for identifying RW events.
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Affiliation(s)
- A Mančić
- Faculty of Sciences and Mathematics, University of Niš, POB 244, 18000 Niš, Serbia
| | - F Baronio
- INO CNR and Dipartimento di Ingegneria dell'Informazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy
| | - Lj Hadžievski
- P* Group, Vinča Institute of Nuclear Sciences, University of Belgrade, POB 522, 11351 Vinča, Belgrade, Serbia
| | - S Wabnitz
- INO CNR and Dipartimento di Ingegneria dell'Informazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy
- Novosibirsk State University, Novosibirsk 630090, Russia
| | - A Maluckov
- P* Group, Vinča Institute of Nuclear Sciences, University of Belgrade, POB 522, 11351 Vinča, Belgrade, Serbia
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10
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Temgoua DDE, Tchokonte MBT, Kofane TC. Combined effects of nonparaxiality, optical activity, and walk-off on rogue wave propagation in optical fibers filled with chiral materials. Phys Rev E 2018; 97:042205. [PMID: 29758712 DOI: 10.1103/physreve.97.042205] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/30/2017] [Indexed: 11/07/2022]
Abstract
The generalized nonparaxial nonlinear Schrödinger (NLS) equation in optical fibers filled with chiral materials is reduced to the higher-order integrable Hirota equation. Based on the modified Darboux transformation method, the nonparaxial chiral optical rogue waves are constructed from the scalar model with modulated coefficients. We show that the parameters of nonparaxiality, third-order dispersion, and differential gain or loss term are the main keys to control the amplitude, linear, and nonlinear effects in the model. Moreover, the influence of nonparaxiality, optical activity, and walk-off effect are also evidenced under the defocusing and focusing regimes of the vector nonparaxial NLS equations with constant and modulated coefficients. Through an algorithm scheme of wider applicability on nonparaxial beam propagation methods, the most influential effect and the simultaneous controllability of combined effects are underlined, showing their properties and their potential applications in optical fibers and in a variety of complex dynamical systems.
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Affiliation(s)
- D D Estelle Temgoua
- Laboratory of Mechanics, Materials and Structures, Post Graduate School in Sciences, Technology and Geosciences, Doctoral Research, Unit in Physics and Applications, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon.,Organization for Women in Science for the Developing World, ICTP Campus, Strada Costiera 11, 34151 Trieste, Italy.,Department of Physics and Astronomy, University of the Western Cape, Private Bag X17, Bellville, 7535 South Africa
| | - M B Tchoula Tchokonte
- Department of Physics and Astronomy, University of the Western Cape, Private Bag X17, Bellville, 7535 South Africa
| | - T C Kofane
- Laboratory of Mechanics, Materials and Structures, Post Graduate School in Sciences, Technology and Geosciences, Doctoral Research, Unit in Physics and Applications, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon.,Centre d'Excellence Africain en Technologies de l'Information et de la Communication, University of Yaounde I, P.O. Box 812, Yaounde, Cameroon
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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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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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13
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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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14
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Zhang G, Yan Z, Wen XY, Chen Y. Interactions of localized wave structures and dynamics in the defocusing coupled nonlinear Schrödinger equations. Phys Rev E 2017; 95:042201. [PMID: 28505714 DOI: 10.1103/physreve.95.042201] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/25/2016] [Indexed: 06/07/2023]
Abstract
We investigate the defocusing coupled nonlinear Schrödinger equations from a 3×3 Lax pair. The Darboux transformations with the nonzero plane-wave solutions are presented to derive the newly localized wave solutions including dark-dark and bright-dark solitons, breather-breather solutions, and different types of new vector rogue wave solutions, as well as interactions between distinct types of localized wave solutions. Moreover, we analyze these solutions by means of parameters modulation. Finally, the perturbed wave propagations of some obtained solutions are explored by means of systematic simulations, which demonstrates that nearly stable and strongly unstable solutions. Our research results could constitute a significant contribution to explore the distinct nonlinear waves (e.g., dark solitons, breather solutions, and rogue wave solutions) dynamics of the coupled system in related fields such as nonlinear optics, plasma physics, oceanography, and Bose-Einstein condensates.
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Affiliation(s)
- Guoqiang Zhang
- Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, China
- School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Zhenya Yan
- Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, China
- School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
| | - Xiao-Yong Wen
- Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, China
- Department of Mathematics, School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
| | - Yong Chen
- Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, China
- School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
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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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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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17
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Chen S, Cai XM, Grelu P, Soto-Crespo JM, Wabnitz S, Baronio F. Complementary optical rogue waves in parametric three-wave mixing. OPTICS EXPRESS 2016; 24:5886-5895. [PMID: 27136785 DOI: 10.1364/oe.24.005886] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We investigate the resonant interaction of two optical pulses of the same group velocity with a pump pulse of different velocity in a weakly dispersive quadratic medium and report on the complementary rogue wave dynamics which are unique to such a parametric three-wave mixing. Analytic rogue wave solutions up to the second order are explicitly presented and their robustness is confirmed by numerical simulations, in spite of the onset of modulation instability activated by quantum noise.
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18
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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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19
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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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20
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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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21
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Chen S, Soto-Crespo JM, Grelu P. Watch-hand-like optical rogue waves in three-wave interactions. OPTICS EXPRESS 2015; 23:349-359. [PMID: 25835681 DOI: 10.1364/oe.23.000349] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We investigate the resonant interaction of three optical pulses of different group velocity in quadratic media and report on the novel watch-hand-like super rogue wave patterns. In addition to having a giant wall-like hump, each rogue-wave hand involves a peak amplitude more than five times its background height. We attribute such peculiar structures to the nonlinear superposition of six Peregrine-type solitons. The robustness has been confirmed by numerical simulations. This stability along with the non-overlapping distribution property may facilitate the experimental diagnostics and observation of these super rogue waves.
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22
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He J, Guo L, Zhang Y, Chabchoub A. Theoretical and experimental evidence of non-symmetric doubly localized rogue waves. Proc Math Phys Eng Sci 2014; 470:20140318. [PMID: 25383023 DOI: 10.1098/rspa.2014.0318] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2014] [Accepted: 08/08/2014] [Indexed: 11/12/2022] Open
Abstract
We present determinant expressions for vector rogue wave (RW) solutions of the Manakov system, a two-component coupled nonlinear Schrödinger (NLS) equation. As a special case, we generate a family of exact and non-symmetric RW solutions of the NLS equation up to third order, localized in both space and time. The derived non-symmetric doubly localized second-order solution is generated experimentally in a water wave flume for deep-water conditions. Experimental results, confirming the characteristic non-symmetric pattern of the solution, are in very good agreement with theory as well as with numerical simulations, based on the modified NLS equation, known to model accurately the dynamics of weakly nonlinear wave packets in deep water.
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Affiliation(s)
- Jingsong He
- Department of Mathematics , Ningbo University , Ningbo, Zhejiang 315211, People's Republic of China ; DAMTP , University of Cambridge , Cambridge CB3 0WA, UK
| | - Lijuan Guo
- Department of Mathematics , Ningbo University , Ningbo, Zhejiang 315211, People's Republic of China
| | - Yongshuai Zhang
- Department of Mathematics , Ningbo University , Ningbo, Zhejiang 315211, People's Republic of China
| | - Amin Chabchoub
- Centre for Ocean Engineering Science and Technology , Swinburne University of Technology , Hawthorn, Victoria 3122, Australia
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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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Baronio F, Conforti M, Degasperis A, Lombardo S, Onorato M, Wabnitz S. Vector rogue waves and baseband modulation instability in the defocusing regime. PHYSICAL REVIEW LETTERS 2014; 113:034101. [PMID: 25083646 DOI: 10.1103/physrevlett.113.034101] [Citation(s) in RCA: 74] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/15/2014] [Indexed: 06/03/2023]
Abstract
We report and discuss analytical solutions of the vector nonlinear Schrödinger equation that describe rogue waves in the defocusing regime. This family of solutions includes bright-dark and dark-dark rogue waves. The link between modulational instability (MI) and rogue waves is displayed by showing that only a peculiar kind of MI, namely baseband MI, can sustain rogue-wave formation. The existence of vector rogue waves in the defocusing regime is expected to be a crucial progress in explaining extreme waves in a variety of physical scenarios described by multicomponent systems, from oceanography to optics and plasma physics.
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Affiliation(s)
- Fabio Baronio
- Dipartimento di Ingegneria dell'Informazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy
| | - Matteo Conforti
- PhLAM/IRCICA UMR 8523/USR 3380, CNRS-Université Lille 1, F-59655 Villeneuve d'Ascq, France
| | - Antonio Degasperis
- INFN, Dipartimento di Fisica, "Sapienza" Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy
| | - Sara Lombardo
- Department of Mathematics and Information Sciences, Northumbria University, Newcastle upon Tyne NE2 1XE, United Kingdom
| | - Miguel Onorato
- Dipartimento di Fisica, Università di Torino, Via Pietro Giuria, 10125 Torino, Italy and INFN, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy
| | - Stefan Wabnitz
- Dipartimento di Ingegneria dell'Informazione, Università di Brescia, Via Branze 38, 25123 Brescia, Italy
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