1
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Wong CN, Yin HM, Chow KW. Transient modes for the coupled modified Korteweg-de Vries equations with negative cubic nonlinearity: Stability and applications of breathers. CHAOS (WOODBURY, N.Y.) 2024; 34:083132. [PMID: 39177957 DOI: 10.1063/5.0223458] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2024] [Accepted: 08/05/2024] [Indexed: 08/24/2024]
Abstract
Dynamics and properties of breathers for the modified Korteweg-de Vries equations with negative cubic nonlinearities are studied. While breathers and rogue waves are absent in a single component waveguide for the negative nonlinearity case, coupling can induce regimes of modulation instabilities. Such instabilities are correlated with the existence of rogue waves and breathers. Similar scenarios have been demonstrated previously for coupled systems of nonlinear Schrödinger and Hirota equations. Both real- and complex-valued modified Korteweg-de Vries equations will be treated, which are applicable to stratified fluids and optical waveguides, respectively. One special family of breathers for coupled, complex-valued equations is derived analytically. Robustness and stability of breathers are studied computationally. Knowledge of the growth rates of modulation instability of plane waves provides an instructive prelude on the robustness of breathers to deterministic perturbations. A theoretical formulation of the linear instability of breathers will involve differential equations with periodic coefficient, i.e., a Floquet analysis. Breathers associated with larger eigenvalues of the monodromy matrix tend to suffer greater instability and increased tendency of distortion. Predictions based on modulation instability and Floquet analysis show excellent agreements. The same trend is obtained for simulations conducted with random noise disturbances. Linear approaches like modulation instabilities and Floquet analysis, thus, generate a very illuminating picture of the nonlinear dynamics.
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Affiliation(s)
- C N Wong
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong, China
| | - H M Yin
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong, China
| | - K W Chow
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong, China
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2
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Guo L, Chen L, Mihalache D, He J. Dynamics of soliton interaction solutions of the Davey-Stewartson I equation. Phys Rev E 2022; 105:014218. [PMID: 35193316 DOI: 10.1103/physreve.105.014218] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2021] [Accepted: 01/06/2022] [Indexed: 06/14/2023]
Abstract
In this paper, we first modify the binary Darboux transformation to derive three types of soliton interaction solutions of the Davey-Stewartson I equation, namely the higher-order lumps, the localized rogue wave on a solitonic background, and the line rogue wave on a solitonic background. The uniform expressions of these solutions contain an arbitrary complex constant, which plays a key role in obtaining diverse interaction scenarios. The second-order dark-lump solution contains two hollows that undergo anomalous scattering after a head-on collision, and the minimum values of the two hollows evolve in time and reach the same asymptotic constant value 0 as t→±∞. The localized rogue wave on a solitonic background describes the occurrence of a waveform from the solitonic background, quickly evolving to a doubly localized wave, and finally retreating to the solitonic background. The line rogue wave on the solitonic background does not create an extreme wave at any instant of time, unlike the one on a constant background, which has a large amplitude at the intermediate time of evolution. For large t, the solitonic background has multiple parallel solitons possessing the same asymptotic velocities and heights. The obtained results improve our understanding of the generation mechanisms of rogue waves.
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Affiliation(s)
- Lijuan Guo
- College of Science, Nanjing Forestry University, Nanjing, Jiangsu, 210037, People's Republic of China
| | - Lei Chen
- College of Science, Nanjing Forestry University, Nanjing, Jiangsu, 210037, People's Republic of China
| | - Dumitru Mihalache
- Department of Theoretical Physics, Horia Hulubei National Institute for Physics and Nuclear Engineering, 077125 Bucharest-Magurele, Romania
| | - Jingsong He
- Institute for Advanced Study, Shenzhen University, Shenzhen, Guangdong 518060, People's Republic of China
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3
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Chapouto A. A Refined Well-Posedness Result for the Modified KdV Equation in the Fourier-Lebesgue Spaces. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS 2021; 35:2537-2578. [PMID: 37588032 PMCID: PMC10425523 DOI: 10.1007/s10884-021-10050-0] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/04/2021] [Revised: 07/08/2021] [Accepted: 07/10/2021] [Indexed: 08/18/2023]
Abstract
We study the well-posedness of the complex-valued modified Korteweg-de Vries equation (mKdV) on the circle at low regularity. In our previous work (2021), we introduced the second renormalized mKdV equation, based on the conservation of momentum, which we proposed as the correct model to study the complex-valued mKdV outside H 1 2 ( T ) . Here, we employ the method introduced by Deng et al. (Commun Math Phys 384(1):1061-1107, 2021) to prove local well-posedness of the second renormalized mKdV equation in the Fourier-Lebesgue spaces F L s , p ( T ) for s ≥ 1 2 and 1 ≤ p < ∞ . As a byproduct of this well-posedness result, we show ill-posedness of the complex-valued mKdV without the second renormalization for initial data in these Fourier-Lebesgue spaces with infinite momentum.
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Affiliation(s)
- Andreia Chapouto
- School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh, EH9 3FD UK
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4
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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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5
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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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6
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Ankiewicz A, Akhmediev N. Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy. Phys Rev E 2017; 96:012219. [PMID: 29347075 DOI: 10.1103/physreve.96.012219] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2017] [Indexed: 06/07/2023]
Abstract
We present rogue wave solutions of the integrable nonlinear Schrödinger equation hierarchy with an infinite number of higher-order terms. The latter include higher-order dispersion and higher-order nonlinear terms. In particular, we derive the fundamental rogue wave solutions for all orders of the hierarchy, with exact expressions for velocities, phase, and "stretching factors" in the solutions. We also present several examples of exact solutions of second-order rogue waves, including rogue wave triplets.
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Affiliation(s)
- A Ankiewicz
- 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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7
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Xing Q, Wang L, Mihalache D, Porsezian K, He J. Construction of rational solutions of the real modified Korteweg-de Vries equation from its periodic solutions. CHAOS (WOODBURY, N.Y.) 2017; 27:053102. [PMID: 28576109 DOI: 10.1063/1.4982721] [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
In this paper, we consider the real modified Korteweg-de Vries (mKdV) equation and construct a special kind of breather solution, which can be obtained by taking the limit λj → λ1 of the Lax pair eigenvalues used in the n-fold Darboux transformation that generates the order-n periodic solution from a constant seed solution. Further, this special kind of breather solution of order n can be used to generate the order-n rational solution by taking the limit λ1 → λ0, where λ0 is a special eigenvalue associated with the eigenfunction ϕ of the Lax pair of the mKdV equation. This eigenvalue λ0, for which ϕ(λ0)=0, corresponds to the limit of infinite period of the periodic solution. Our analytical and numerical results show the effective mechanism of generation of higher-order rational solutions of the mKdV equation from the double eigenvalue degeneration process of multi-periodic solutions.
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Affiliation(s)
- Qiuxia Xing
- Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, People's Republic of China
| | - Lihong Wang
- Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, People's Republic of China
| | - Dumitru Mihalache
- Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O. Box MG-6, Magurele 077125, Romania
| | | | - Jingsong He
- Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, People's Republic of China
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8
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Zhao HQ, Yu GF. Discrete rational and breather solution in the spatial discrete complex modified Korteweg-de Vries equation and continuous counterparts. CHAOS (WOODBURY, N.Y.) 2017; 27:043113. [PMID: 28456174 DOI: 10.1063/1.4982204] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
In this paper, a spatial discrete complex modified Korteweg-de Vries equation is investigated. The Lax pair, conservation laws, Darboux transformations, and breather and rational wave solutions to the semi-discrete system are presented. The distinguished feature of the model is that the discrete rational solution can possess new W-shape rational periodic-solitary waves that were not reported before. In addition, the first-order rogue waves reach peak amplitudes which are at least three times of the background amplitude, whereas their continuous counterparts are exactly three times the constant background. Finally, the integrability of the discrete system, including Lax pair, conservation laws, Darboux transformations, and explicit solutions, yields the counterparts of the continuous system in the continuum limit.
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Affiliation(s)
- Hai-Qiong Zhao
- Department of Applied Mathematics, Shanghai University of International Business and Economics, 1900 Wenxiang Road, Shanghai 201620, People's Republic of China
| | - Guo-Fu Yu
- School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, People's Republic of China
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9
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Chen S, Soto-Crespo JM, Baronio F, Grelu P, Mihalache D. Rogue-wave bullets in a composite (2+1)D nonlinear medium. OPTICS EXPRESS 2016; 24:15251-15260. [PMID: 27410802 DOI: 10.1364/oe.24.015251] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/06/2023]
Abstract
We show that nonlinear wave packets localized in two dimensions with characteristic rogue wave profiles can propagate in a third dimension with significant stability. This unique behavior makes these waves analogous to light bullets, with the additional feature that they propagate on a finite background. Bulletlike rogue-wave singlet and triplet are derived analytically from a composite (2+1)D nonlinear wave equation. The latter can be interpreted as the combination of two integrable (1+1)D models expressed in different dimensions, namely, the Hirota equation and the complex modified Korteweg-de Vries equation. Numerical simulations confirm that the generation of rogue-wave bullets can be observed in the presence of spontaneous modulation instability activated by quantum noise.
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10
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He J, Xu S, Porsezian K, Cheng Y, Dinda PT. Rogue wave triggered at a critical frequency of a nonlinear resonant medium. Phys Rev E 2016; 93:062201. [PMID: 27415249 DOI: 10.1103/physreve.93.062201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/12/2014] [Indexed: 06/06/2023]
Abstract
We consider a two-level atomic system interacting with an electromagnetic field controlled in amplitude and frequency by a high intensity laser. We show that the amplitude of the induced electric field admits an envelope profile corresponding to a breather soliton. We demonstrate that this soliton can propagate with any frequency shift with respect to that of the control laser, except a critical frequency, at which the system undergoes a structural discontinuity that transforms the breather in a rogue wave. A mechanism of generation of rogue waves by means of an intense laser field is thus revealed.
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Affiliation(s)
- Jingsong He
- Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, P. R. China
| | - Shuwei Xu
- College of Mathematics Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, P. R. China
- School of Mathematical Sciences, USTC, Hefei, Anhui 230026, P. R. China
| | - K Porsezian
- Department of Physics, Pondicherry University, Puducherry 605014, India
| | - Yi Cheng
- School of Mathematical Sciences, USTC, Hefei, Anhui 230026, P. R. China
| | - P Tchofo Dinda
- Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS No. 6303, 9 Avenue A. Savary, B.P. 47 870, 21078 Dijon Cédex, France
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11
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Qiu D, He J, Zhang Y, Porsezian K. The Darboux transformation of the Kundu–Eckhaus equation. Proc Math Phys Eng Sci 2015. [DOI: 10.1098/rspa.2015.0236] [Citation(s) in RCA: 68] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
We construct an analytical and explicit representation of the Darboux transformation (DT) for the Kundu–Eckhaus (KE) equation. Such solution and
n
-fold DT
T
n
are given in terms of determinants whose entries are expressed by the initial eigenfunctions and ‘seed’ solutions. Furthermore, the formulae for the higher order rogue wave (RW) solutions of the KE equation are also obtained by using the Taylor expansion with the use of degenerate eigenvalues
λ
2
k
−
1
→
λ
1
=
−
1
2
a
+
β
c
2
+
i
c
,
k
=1,2,3,…, all these parameters will be defined latter. These solutions have a parameter
β
, which denotes the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effects. We apply the contour line method to obtain analytical formulae of the length and width for the first-order RW solution of the KE equation, and then use it to study the impact of the
β
on the RW solution. We observe two interesting results on localization characters of
β
, such that if
β
is increasing from
a
/2: (i) the length of the RW solution is increasing as well, but the width is decreasing; (ii) there exist a significant rotation of the RW along the clockwise direction. We also observe the oppositely varying trend if
β
is increasing to
a
/2. We define an area of the RW solution and find that this area associated with
c
=1 is invariant when
a
and
β
are changing.
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Affiliation(s)
- Deqin Qiu
- Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, Peoples Republic of China
| | - Jingsong He
- Department of Mathematics, Ningbo University, Ningbo, Zhejiang 315211, Peoples Republic of China
| | - Yongshuai Zhang
- School of Mathematical Sciences, USTC, Hefei, Anhui 230026, Peoples Republic of China
| | - K. Porsezian
- Department of Physics, Pondicherry University, Puducherry 605014, India
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12
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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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