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Wu C, Wei B, Shi C, Feng BF. Multi-breather solutions to the Sasa–Satsuma equation. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2021.0711] [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
General breather solution to the Sasa–Satsuma equation (SSE) is systematically investigated in this paper. We firstly transform the SSE into a set of three Hirota bilinear equations under a proper plane wave boundary condition. Starting from a specially arranged tau-function of the Kadomtsev–Petviashvili hierarchy and a set of 11 bilinear equations satisfied, we implement a series steps of reduction procedure, i.e. C-type reduction, dimension reduction and complex conjugate reduction, and reduce these 11 equations to three bilinear equations for the SSE. Meanwhile, the general breather solution to the SSE is found in determinant of even order. The one- and two-breather solutions are calculated and analysed in detail.
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Affiliation(s)
- Chengfa Wu
- Institute for Advanced Study, Shenzhen University, Shenzhen 518060, People’s Republic of China
| | - Bo Wei
- Institute for Advanced Study, Shenzhen University, Shenzhen 518060, People’s Republic of China
| | - Changyan Shi
- Institute for Advanced Study, Shenzhen University, Shenzhen 518060, People’s Republic of China
| | - Bao-Feng Feng
- School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, Edinburg, TX 78541-2999, USA
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2
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Crabb M, Akhmediev N. Doubly periodic solutions of the class-I infinitely extended nonlinear Schrödinger equation. Phys Rev E 2019; 99:052217. [PMID: 31212469 DOI: 10.1103/physreve.99.052217] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2019] [Indexed: 11/07/2022]
Abstract
We present doubly periodic solutions of the infinitely extended nonlinear Schrödinger equation with an arbitrary number of higher-order terms and corresponding free real parameters. Solutions have one additional free variable parameter that allows one to vary periods along the two axes. The presence of infinitely many free parameters provides many possibilities in applying the solutions to nonlinear wave evolution. Being general, this solution admits several particular cases which are also given in this article.
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Affiliation(s)
- M Crabb
- Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 2600, Australia
| | - N Akhmediev
- Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 2600, Australia
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3
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Bandelow U, Ankiewicz A, Amiranashvili S, Akhmediev N. Sasa-Satsuma hierarchy of integrable evolution equations. CHAOS (WOODBURY, N.Y.) 2018; 28:053108. [PMID: 29857667 DOI: 10.1063/1.5030604] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
We present the infinite hierarchy of Sasa-Satsuma evolution equations. The corresponding Lax pairs are given, thus proving its integrability. The lowest order member of this hierarchy is the nonlinear Schrödinger equation, while the next one is the Sasa-Satsuma equation that includes third-order terms. Up to sixth-order terms of the hierarchy are given in explicit form, while the provided recurrence relation allows one to explicitly write all higher-order terms. The whole hierarchy can be combined into a single general equation. Each term in this equation contains a real independent coefficient that provides the possibility of adapting the equation to practical needs. A few examples of exact solutions of this general equation with an infinite number of terms are also given explicitly.
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Affiliation(s)
- U Bandelow
- Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany
| | - A Ankiewicz
- Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra, ACT 2600, Australia
| | - Sh Amiranashvili
- Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin, Germany
| | - N 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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Liu L, Tian B, Chai HP, Yuan YQ. Certain bright soliton interactions of the Sasa-Satsuma equation in a monomode optical fiber. Phys Rev E 2017; 95:032202. [PMID: 28415232 DOI: 10.1103/physreve.95.032202] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/23/2016] [Indexed: 11/07/2022]
Abstract
Under investigation in this paper is the Sasa-Satsuma equation, which describes the propagation of ultrashort pulses in a monomode fiber with the third-order dispersion, self-steepening, and stimulated Raman scattering effects. Based on the known bilinear forms, through the modified expanded formulas and symbolic computation, we construct the bright two-soliton solutions. Through classifying the interactions under different parameter conditions, we reveal six cases of interactions between the two solitons via an asymptotic analysis. With the help of the analytic and graphic analysis, we find that such interactions are different from those of the nonlinear Schrödinger equation and Hirota equation. When those solitons interact with each other, the singular-I soliton is shape-preserving, while the singular-II and nonsingular solitons may be shape preserving or shape changing. Such elastic and inelastic interaction phenomena in a scalar equation might enrich the knowledge of soliton behavior, which could be expected to be experimentally observed.
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Affiliation(s)
- Lei Liu
- State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Bo Tian
- State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Han-Peng Chai
- State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Yu-Qiang Yuan
- 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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5
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Liu L, Tian B, Xie XY, Guan YY. Vector bright soliton behaviors of the coupled higher-order nonlinear Schrödinger system in the birefringent or two-mode fiber. CHAOS (WOODBURY, N.Y.) 2017; 27:013108. [PMID: 28147489 DOI: 10.1063/1.4972114] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
Studied in this paper are the vector bright solitons of the coupled higher-order nonlinear Schrödinger system, which describes the simultaneous propagation of two ultrashort pulses in the birefringent or two-mode fiber. With the help of auxiliary functions, we obtain the bilinear forms and construct the vector bright one- and two-soliton solutions via the Hirota method and symbolic computation. Two types of vector solitons are derived. Single-hump, double-hump, and flat-top solitons are displayed. Elastic and inelastic interactions between the Type-I solitons, between the Type-II solitons, and between the two combined types of the solitons are revealed, respectively. Especially, from the interaction between a Type-I soliton and a Type-II soliton, we see that the Type-II soliton exhibits the oscillation periodically before such an interaction and becomes the double-hump soliton after the interaction, which is different from the previously reported.
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Affiliation(s)
- Lei Liu
- State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Bo Tian
- 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
| | - Yue-Yang Guan
- 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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6
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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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7
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Zhao LC, Li SC, Ling L. W-shaped solitons generated from a weak modulation in the Sasa-Satsuma equation. Phys Rev E 2016; 93:032215. [PMID: 27078352 DOI: 10.1103/physreve.93.032215] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2015] [Indexed: 06/05/2023]
Abstract
We study rational solutions of continuous wave backgrounds with the critical frequencies of the Sasa-Satsuma equation, which can be used to describe the evolution of the optical field in a nonlinear fiber with some high-order effects. We find a striking dynamical process that two W-shaped solitons are generated from a weak modulation signal on the continuous wave backgrounds. This provides a possible way to obtain stable high-intensity pulses from a low-intensity continuous wave background. The process involves both modulational instability and modulational stability regimes, in contrast to the rogue waves and W-shaped solitons reported before which involve modulational instability and stability, respectively. Furthermore, we present a phase diagram on a modulational instability spectrum plane for the fundamental nonlinear localized waves obtained already in the Sasa-Satsuma equation. The interactions between different types of nonlinear localized waves are discussed based on the phase diagram.
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Affiliation(s)
- Li-Chen Zhao
- Department of Physics, Northwest University, 710069, Xi'an, China
| | - Sheng-Chang Li
- School of Science, Xi'an Jiaotong University, 710049, Xi'an, China
| | - Liming Ling
- School of Mathematics, South China University of Technology, 510640, Guangzhou, China
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8
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Soto-Crespo JM, Devine N, Hoffmann NP, Akhmediev N. Rogue waves of the Sasa-Satsuma equation in a chaotic wave field. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032902. [PMID: 25314499 DOI: 10.1103/physreve.90.032902] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2014] [Indexed: 06/04/2023]
Abstract
We study the properties of the chaotic wave fields generated in the frame of the Sasa-Satsuma equation (SSE). Modulation instability results in a chaotic pattern of small-scale filaments with a free parameter-the propagation constant k. The average velocity of the filaments is approximately given by the group velocity calculated from the dispersion relation for the plane-wave solution. Remarkably, our results reveal the reason for the skewed profile of the exact SSE rogue-wave solutions, which was one of their distinctive unexplained features. We have also calculated the probability density functions for various values of the propagation constant k, showing that probability of appearance of rogue waves depends on k.
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Affiliation(s)
- J M Soto-Crespo
- Instituto de Óptica, C.S.I.C., Serrano 121, 28006 Madrid, Spain
| | - N Devine
- Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia
| | - N P Hoffmann
- Dynamics Group, Hamburg University of Technology, 21073 Hamburg, Germany and Department of Mechanical Engineering, Imperial College, London SW7 2AZ, United Kingdom
| | - N Akhmediev
- Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia
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9
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Zhao LC, Li SC, Ling L. Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:023210. [PMID: 25353598 DOI: 10.1103/physreve.89.023210] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/13/2013] [Indexed: 06/04/2023]
Abstract
We investigate the solution in rational form for the Sasa-Satsuma equation on a continuous background which describes a nonlinear fiber system with higher-order effects including the third-order dispersion, Kerr dispersion, and stimulated inelastic scattering. The W-shaped soliton in the system is obtained analytically. It is found that the height of hump for the soliton increases with decreasing the background frequency in certain parameter regime. The maximum height of the soliton can be three times the background's height and the corresponding profile is identical with the one for the well-known eye-shaped rogue wave with maximum peak. The numerical simulations indicate that the W-shaped soliton is stable with small perturbations. Particularly, we show that the W-shaped soliton corresponds to a stable supercontinuum pulse by performing exact spectrum analysis.
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Affiliation(s)
- Li-Chen Zhao
- Department of Physics, Northwest University, Xi'an 710069, China
| | - Sheng-Chang Li
- School of Science, Xi'an Jiaotong University, Xi'an 710049, China
| | - Liming Ling
- Department of Mathematics, South China University of Technology, Guangzhou 510640, China
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10
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Ankiewicz A, Wang Y, Wabnitz S, Akhmediev N. Extended nonlinear Schrödinger equation with higher-order odd and even terms and its rogue wave solutions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:012907. [PMID: 24580297 DOI: 10.1103/physreve.89.012907] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/04/2013] [Indexed: 06/03/2023]
Abstract
We consider an extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms with variable coefficients. The resulting equation has soliton solutions and approximate rogue wave solutions. We present these solutions up to second order. Moreover, specific constraints on the parameters of higher-order terms provide integrability of the resulting equation, providing a corresponding Lax pair. Particular cases of this equation are the Hirota and the Lakshmanan-Porsezian-Daniel equations. The resulting integrable equation admits exact rogue wave solutions. In particular cases, mentioned above, these solutions are reduced to the rogue wave solutions of the corresponding equations.
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Affiliation(s)
- Adrian Ankiewicz
- Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia
| | - Yan Wang
- Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia
| | - Stefan Wabnitz
- Dipartimento di Ingegneria dell' Informazione, Università di Brescia, via Branze 38, 25123 Brescia, Italy
| | - Nail Akhmediev
- Optical Sciences Group, Research School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia
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11
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Bandelow U, Akhmediev N. Sasa-Satsuma equation: soliton on a background and its limiting cases. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:026606. [PMID: 23005875 DOI: 10.1103/physreve.86.026606] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2012] [Indexed: 06/01/2023]
Abstract
We present a multiparameter family of a soliton on a background solution to the Sasa-Satsuma equation. The solution is controlled by a set of several free parameters that control the background amplitude as well as the soliton itself. This family of solutions admits a few nontrivial limiting cases that are considered in detail. Among these special cases is the nonlinear Schrödinger equation limit and the limit of rogue wave solutions.
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Affiliation(s)
- U Bandelow
- Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany
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12
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Yang G, Li L, Jia S. Peregrine rogue waves induced by the interaction between a continuous wave and a soliton. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:046608. [PMID: 22680599 DOI: 10.1103/physreve.85.046608] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/28/2011] [Revised: 03/26/2012] [Indexed: 06/01/2023]
Abstract
Based on the soliton solution on a continuous wave background for an integrable Hirota equation, the reduction mechanism and the characteristics of the Peregrine rogue wave in the propagation of femtosecond pulses of optical fiber are discussed. The results show that there exist two processes of the formation of the Peregrine rogue wave: one is the localized process of the continuous wave background, and the other is the reduction process of the periodization of the bright soliton. The characteristics of the Peregrine rogue wave are exhibited by strong temporal and spatial localization. Also, various initial excitations of the Peregrine rogue wave are performed and the results show that the Peregrine rogue wave can be excited by a small localized (single peak) perturbation pulse of the continuous wave background, even for the nonintegrable case. The numerical simulations show that the Peregrine rogue wave is unstable. Finally, through a realistic example, the influence of the self-frequency shift to the dynamics of the Peregrine rogue wave is discussed. The results show that in the absence of the self-frequency shift, the Peregrine rogue wave can split into several subpulses; however, when the self-frequency shift is considered, the Peregrine rogue wave no longer splits and exhibits mainly a peak changing and an increasing evolution property of the field amplitude.
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Affiliation(s)
- Guangye Yang
- Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi, China
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13
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Wang L, Li L, Li Z, Zhou G, Mihalache D. Generation, compression, and propagation of pulse trains in the nonlinear Schrödinger equation with distributed coefficients. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:036614. [PMID: 16241599 DOI: 10.1103/physreve.72.036614] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2005] [Indexed: 05/05/2023]
Abstract
The generalized nonlinear Schrödinger model with distributed dispersion, nonlinearity, and gain or loss is considered and the explicit, analytical solutions describing the dynamics of bright solitons on a continuous-wave background are obtained in quadratures. Then, the generation, compression, and propagation of pulse trains are discussed in detail. The numerical results show that solitons can be compressed by choosing the appropriate control fiber system, and pulse trains generated by modulation instability can propagate undistorsted along fibers with distributed parameters by controlling appropriately the energy of each pulse in the pulse train.
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Affiliation(s)
- Luyun Wang
- College of Physics and Electronics Engineering, and Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China
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14
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Gilson C, Hietarinta J, Nimmo J, Ohta Y. Sasa-Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:016614. [PMID: 12935277 DOI: 10.1103/physreve.68.016614] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2003] [Indexed: 05/24/2023]
Abstract
Higher-order and multicomponent generalizations of the nonlinear Schrödinger equation are important in various applications, e.g., in optics. One of these equations, the integrable Sasa-Satsuma equation, has particularly interesting soliton solutions. Unfortunately, the construction of multisoliton solutions to this equation presents difficulties due to its complicated bilinearization. We discuss briefly some previous attempts and then give the correct bilinearization based on the interpretation of the Sasa-Satsuma equation as a reduction of the three-component Kadomtsev-Petviashvili hierarchy. In the process, we also get bilinearizations and multisoliton formulas for a two-component generalization of the Sasa-Satsuma equation (the Yajima-Oikawa-Tasgal-Potasek model), and for a (2+1)-dimensional generalization.
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Affiliation(s)
- C Gilson
- Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom.
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15
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Xu Z, Li L, Li Z, Zhou G. Modulation instability and solitons on a cw background in an optical fiber with higher-order effects. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:026603. [PMID: 12636837 DOI: 10.1103/physreve.67.026603] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/05/2002] [Indexed: 05/24/2023]
Abstract
We construct the Lax pair for a higher-order nonlinear Schrödinger equation that includes terms accounting for the third-order dispersion, the self-steepening effect, and the delayed nonlinear response effect. Two exact analytic solutions that describe (i) modulation instability and (ii) soliton propagation on a continuous wave background are obtained by using the Darboux transformation. In addition, we analyze the amplification-absorption and quintic nonlinearity effects on the second solution in the adiabatic approximation.
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Affiliation(s)
- Zhiyong Xu
- Department of Electronics and Information Technology, Shanxi University, Taiyuan 030006, China.
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16
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Gerdjikov VS, Doktorov EV, Yang J. Adiabatic interaction of N ultrashort solitons: universality of the complex Toda chain model. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:056617. [PMID: 11736126 DOI: 10.1103/physreve.64.056617] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/09/2001] [Indexed: 05/23/2023]
Abstract
Using the Karpman-Solov'ev method we derive the equations for the two-soliton adiabatic interaction for solitons of the modified nonlinear Schrödinger equation (MNSE). Then we generalize these equations to the case of N interacting solitons with almost equal velocities and widths. On the basis of this result we prove that the N MNSE-soliton train interaction (N>2) can be modeled by the completely integrable complex Toda chain (CTC). This is an argument in favor of universality of the complex Toda chain that was previously shown to model the soliton train interaction for nonlinear Schrödinger solitons. The integrability of the CTC is used to describe all possible dynamical regimes of the N-soliton trains that include asymptotically free propagation of all N solitons, N-soliton bound states, various mixed regimes, etc. It allows also to describe analytically the manifolds in the 4N-dimensional space of initial soliton parameters that are responsible for each of the regimes mentioned above. We compare the results of the CTC model with the numerical solutions of the MNSE for two and three-soliton interactions and find a very good agreement.
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Affiliation(s)
- V S Gerdjikov
- Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria.
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17
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Mahalingam A, Porsezian K. Propagation of dark solitons with higher-order effects in optical fibers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 64:046608. [PMID: 11690171 DOI: 10.1103/physreve.64.046608] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/03/2000] [Revised: 03/07/2001] [Indexed: 05/23/2023]
Abstract
In this paper, we analyze dark soliton propagation in nonlinear optical fibers with higher-order effects such as third order dispersion, self-steepening, and stimulated Raman scattering. We consider the Hirota equation and the higher-order nonlinear Schrödinger equation, and identify conditions for dark soliton propagation through Painlevé analysis. We also construct an explicit Lax pair, and Hirota bilinear form is used to generate one and two dark solitons.
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Affiliation(s)
- A Mahalingam
- Department of Physics, Anna University, Chennai-600 025, India
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18
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Panoiu NC, Mel'nikov IV, Mihalache D, Etrich C, Lederer F. Soliton generation in optical fibers for a dual-frequency input. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:4868-76. [PMID: 11970352 DOI: 10.1103/physreve.60.4868] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/1999] [Indexed: 04/18/2023]
Abstract
We analyze scenarios of soliton generation in an ideal fiber for an input that consists of either two in-phase or out-of-phase solitonlike optical pulses at different frequencies. In both cases the relationship between the structure of the emerging solitons and the frequency separation of the initial solitons is studied both analytically and numerically. Depending on the value of the frequency detuning, if the two initial solitons are in phase (symmetric input), two bound solitons with equal amplitudes (breather), a single soliton, or a pair of solitons, which have equal amplitudes and exhibit opposite velocities, can be generated. When the two initial solitons are out-of-phase (antisymmetric input), only the last scenario takes place. Also, we calculated the threshold values of the frequency separation at which the structure of the emerging solitons changes. Moreover, we demonstrated that two of these critical frequencies correspond to cusplike maxima of the energy density of the radiative modes. Finally, we show that these analytical results are entirely verified by numerical simulations.
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Affiliation(s)
- N C Panoiu
- Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA
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