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Saha N, Roy B, Khare A. Dipole and quadrupole nonparaxial solitary waves. CHAOS (WOODBURY, N.Y.) 2022; 32:093106. [PMID: 36182394 DOI: 10.1063/5.0096099] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2022] [Accepted: 08/09/2022] [Indexed: 06/16/2023]
Abstract
The cubic nonlinear Helmholtz equation with third and fourth order dispersion and non-Kerr nonlinearity, such as the self steepening and the self frequency shift, is considered. This model describes nonparaxial ultrashort pulse propagation in an optical medium in the presence of spatial dispersion originating from the failure of slowly varying envelope approximation. We show that this system admits periodic (elliptic) solitary waves with a dipole structure within a period and also a transition from a dipole to quadrupole structure within a period depending on the value of the modulus parameter of a Jacobi elliptic function. The parametric conditions to be satisfied for the existence of these solutions are given. The effect of the nonparaxial parameter on physical quantities, such as amplitude, pulse width, and speed of the solitary waves, is examined. It is found that by adjusting the nonparaxial parameter, the speed of solitary waves can be decelerated. The stability and robustness of the solitary waves are discussed numerically.
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Affiliation(s)
- Naresh Saha
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Barnana Roy
- Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata 700108, India
| | - Avinash Khare
- Department of Physics, Savitribai Phule Pune University, Pune 411007, India
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Triki H, Porsezian K, Choudhuri A. Solitons in the nonlinear Schrödinger equation with two power-law nonlinear terms modulated in time and space. Phys Rev E 2017; 95:062208. [PMID: 28709188 DOI: 10.1103/physreve.95.062208] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/15/2016] [Indexed: 11/07/2022]
Abstract
A nonlinear Schrödinger equation that includes two terms with power-law nonlinearity and external potential modulated both on time and on the spatial coordinates is considered. The model appears in various branches of contemporary physics, especially in the case of lower values of the nonlinearity power. A significant generalization of the similarity transformations approach to construct explicit localized solutions for the model with arbitrary power-law nonlinearities is introduced. We obtain the exact analytical bright and kink soliton solutions of the governing equation for different nonlinearities and potentials that are of particular interest in applications to Bose-Einstein condensates and nonlinear optics. Necessary conditions on the physical parameters for propagating envelope formation are presented. The obtained results can be straightforwardly applied to a large variety of nonlinear Schrödinger models and hence would be of value to understand nonlinear phenomena in a diversity of nonlinear media.
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Affiliation(s)
- Houria Triki
- Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P. O. Box 12, 23000 Annaba, Algeria
| | - K Porsezian
- Department of Physics, School of Physical, Chemical and Applied Sciences, Pondicherry University, Pondicherry 605014, India
| | - Amitava Choudhuri
- Department of Physics, The University of Burdwan, Golapbag 713104, West Bengal, India
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Wang L, Zhang JH, Liu C, Li M, Qi FH. Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schrödinger equation with higher-order effects. Phys Rev E 2016; 93:062217. [PMID: 27415265 DOI: 10.1103/physreve.93.062217] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2016] [Indexed: 06/06/2023]
Abstract
We study a variable-coefficient nonlinear Schrödinger (vc-NLS) equation with higher-order effects. We show that the breather solution can be converted into four types of nonlinear waves on constant backgrounds including the multipeak solitons, antidark soliton, periodic wave, and W-shaped soliton. In particular, the transition condition requiring the group velocity dispersion (GVD) and third-order dispersion (TOD) to scale linearly is obtained analytically. We display several kinds of elastic interactions between the transformed nonlinear waves. We discuss the dispersion management of the multipeak soliton, which indicates that the GVD coefficient controls the number of peaks of the wave while the TOD coefficient has compression effect. The gain or loss has influence on the amplitudes of the multipeak soliton. We further derive the breather multiple births and Peregrine combs by using multiple compression points of Akhmediev breathers and Peregrine rogue waves in optical fiber systems with periodic GVD modulation. In particular, we demonstrate that the Peregrine comb can be converted into a Peregrine wall by the proper choice of the amplitude of the periodic GVD modulation. The Peregrine wall can be seen as an intermediate state between rogue waves and W-shaped solitons. We finally find that the modulational stability regions with zero growth rate coincide with the transition condition using rogue wave eigenvalues. Our results could be useful for the experimental control and manipulation of the formation of generalized Peregrine rogue waves in diverse physical systems modeled by vc-NLS equation with higher-order effects.
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Affiliation(s)
- Lei Wang
- Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Jian-Hui Zhang
- School of Energy Power and Mechanical Engneering, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Chong Liu
- School of Physics, Northwest University, Xi'an 710069, People's Republic of China
| | - Min Li
- Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, People's Republic of China
| | - Feng-Hua Qi
- School of Information, Beijing Wuzi University, Beijing 101149, People's Republic of China
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Wong P, Liu WJ, Huang LG, Li YQ, Pan N, Lei M. Higher-order-effects management of soliton interactions in the Hirota equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:033201. [PMID: 25871232 DOI: 10.1103/physreve.91.033201] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/25/2014] [Indexed: 06/04/2023]
Abstract
The study of soliton interactions is of significance for improving pulse qualities in nonlinear optics. In this paper, interaction between two solitons, which is governed by the Hirota equation, is considered. Via use of the Hirota method, an analytic soliton solution is obtained. Then a two-period vibration phenomenon is observed. Moreover, turning points of the coefficients of higher-order terms, which are related with sudden delaying or leading, are found and analyzed. With different coefficient constraints, soliton interactions are discussed by different frequency separation with the split-step Fourier method, and characteristics of soliton interactions are exhibited. Through turning points, we get a pair of solitons which tend to be bound solitons but not exactly. Furthermore, we control a pair of solitons to emit at different emission angles. The stability of the two-period vibration is analyzed. Results in this paper may be helpful for the applications of optical self-routing, waveguiding, and faster switching.
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Affiliation(s)
- Pring Wong
- State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China
- and School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing 100876, China
| | - Wen-Jun Liu
- State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China
- and School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing 100876, China
| | - Long-Gang Huang
- State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China
- and School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing 100876, China
| | - Yan-Qing Li
- State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China
- and School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing 100876, China
| | - Nan Pan
- State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China
- and School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing 100876, China
| | - Ming Lei
- State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China
- and School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing 100876, China
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Dai CQ, Zhou GQ, Zhang JF. Controllable optical rogue waves in the femtosecond regime. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:016603. [PMID: 22400691 DOI: 10.1103/physreve.85.016603] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/21/2011] [Revised: 10/13/2011] [Indexed: 05/31/2023]
Abstract
We derive analytical rogue wave solutions of variable-coefficient higher-order nonlinear Schrödinger equations describing the femtosecond pulse propagation via a transformation connected with the constant-coefficient Hirota equation. Then we discuss the propagation behaviors of controllable rogue waves, including recurrence, annihilation, and sustainment in a periodic distributed fiber system and an exponential dispersion decreasing fiber. Finally, we investigate nonlinear tunneling effects for rogue waves.
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Affiliation(s)
- Chao-Qing Dai
- School of Sciences, Zhejiang Agriculture and Forestry University, Lin'an, Zhejiang CN-311300, China
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Liu WJ, Tian B, Zhang HQ. Types of solutions of the variable-coefficient nonlinear Schrödinger equation with symbolic computation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:066613. [PMID: 19256974 DOI: 10.1103/physreve.78.066613] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2008] [Indexed: 05/27/2023]
Abstract
By using Hirota's bilinear method and symbolic computation, solutions for a variable-coefficient nonlinear Schrödinger equation are obtained theoretically. It is found that the type of the solutions changes with the different choices of the group-velocity dispersion coefficient beta_{2}(z) . According to those solutions, the relevant properties and features of physical and optical interest are illustrated. In addition, an effective technique for controlling the shape of the pulses is presented. The results of this paper will be valuable to the study of the future development of ultrahigh rate and long-distance optical communication systems.
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Affiliation(s)
- Wen-Jun Liu
- School of Science, Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing 100876, China
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