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Dark two-soliton solutions for nonlinear Schrödinger equations in inhomogeneous optical fibers
¨ Dark two-soliton solutions for nonlinear Schrodinger equations in inhomogeneous optical fibers
Journal Pre-proof
¨ Dark two-soliton solutions for nonlinear Schrodinger equations in inhomogeneous optical fibers Xiaoyan Liu, Zitong Luan, Qin Zhou, Wenjun Liu, Anjan Biswas PII: DOI: Reference:
S0577-9073(19)30894-9 https://doi.org/10.1016/j.cjph.2019.08.006 CJPH 917
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
23 February 2019 19 May 2019 28 August 2019
Please cite this article as: Xiaoyan Liu, Zitong Luan, Qin Zhou, Wenjun Liu, Anjan Biswas, Dark two¨ soliton solutions for nonlinear Schrodinger equations in inhomogeneous optical fibers, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.08.006
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Dark two-soliton solutions for nonlinear Schr¨odinger equations in inhomogeneous optical fibers Xiaoyan Liua , Zitong Luanb , Qin Zhouc , Wenjun Liua , Anjan Biswasd,e,f a
State Key Laboratory of Information Photonics and Optical Communications, School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China b School of Economics and Management, Beijing University of Posts and Telecommunications, Beijing 100876, China c School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan 430212, China d Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL-35762, USA e Department of Mathematics, King Abdulaziz University, Jeddah–21589, Saudi Arabia f Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa
Abstract In this paper, the variable coefficient nonlinear Schr¨odinger equation is investigated analytically. With the bilinear method, the bilinear forms and analytic soliton solutions are obtained. Based on the obtained analytic solutions, the effect of free parameters on the control of soliton transmission is studied. Influences of second-order and third-order dispersion coefficients on dark solitons are discussed. Results in this paper would be of great significance in the generation of dark solitons. Keywords: Solitons, Bilinear method, Partial differential equation, Nonlinear Schr¨odinger equation 1. Introduction Solitons have been widely studied in theory and experiment in recent years [1-19]. Since the development of the technique of synthesizing ultrashort pulses of arbitrary shape and phase in 1988, dark solitons, as one of Email addresses: Corresponding author: [email protected] (Qin Zhou), Corresponding author: [email protected] (Wenjun Liu) Preprint submitted to Chinese Journal of Physics
September 23, 2019
solitons, have been experimentally confirmed [20, 21]. In the case of the same fiber loss, the attenuation of the amplitude of the dark soliton and the increase of the pulse duration are smaller than that of the bright one. And for various disturbance effects, such as the interaction of solitons, dark solitons have better stability and stronger resistance than bright ones, they have better self-recovery ability and higher code rate [22-25]. However, the loss, dispersion and nonlinear effects are the main factors affecting soliton transmissions, and they can cause pulse broadening and crepe generation of solitons, resulting in solitons deformation [26, 27]. In order to reduce the generation of negative effects and achieve the purpose of controlling transmission effects, it would be critical to study the effects of the loss, dispersion and other effects on the soliton transmission [28, 29]. Thus, it is necessary to discuss the effects of different order dispersion terms on the soliton transmission. The nonlinear Schr¨odinger (NLS) equation can be used to describe the transmission of optical solitons in optical fibers [26]. In inhomogeneous optical fibers, the transmission characteristics of optical solitons can be presented in a NLS equation as below [30]: iqx − (qtt − 2|q|2 q) + iε(qttt − 6|q|2 qt ) = 0,
(1)
where q(x, t) is the normalized complex amplitude of the optical soliton envelope. ε is related to the third-order effect. x represents the transmission distance of optical solitons, and t is the normalization time of the system. In Ref. [30], the binary Darboux transformation has been used to obtain dark soliton solutions of Eq. (1) under constant coefficients. Its initial value problem can be solved by the inverse scattering transformation [31]. Eq. (1) can be transformed into a complex modified KdV equation by the variable substitution [32]. Besides, the breather solutions, rogue wave solutions, bright N-soliton solutions of Eq. (1) have also been solved [33, 34]. Due to the development of nonlinear optics and the application of dispersion management system, the variable coefficient NLS equation can be introduced as [1]: iqx − α2 (x)(qtt − 2|q|2 q) + iα3 (x)(qttt − 6|q|2 qt ) = 0,
(2)
where α2 (x) and α3 (x) are real functions that related to the second-order dispersion and third-order dispersion, respectively. As far as we know, dark solitons with varying second-order and third-order dispersion coefficients have 2
not yet been solved. And no one has compared their different effects between α2 (x) and α3 (x) in Eq. (2) on the transmission of dark solitons. In this paper, we use the bilinear method to get the dark two-soliton solutions of Eq. (2) in Section 2. By analyzing the transmission characteristics of dark solitons, the dispersion and other effects of optical fibers on the dark soliton propagation are discussed. And the influences of different order dispersion coefficients α2 (x) and α3 (x) on the transmission of dark solitons are analyzed in Section 3. Finally, we will give the conclusions in Section 4. 2. Analytic dark two-soliton solutions By introducing a dependent variable [35]: q(x, t) =
g(x, t) , f (x, t)
(3)
the bilinear forms of Eq. (2) can be obtained as i h iDx − α2 (x)Dt2 + iα3 (x)Dt3 + λα2 (x) − 3iλα3 (x)Dt g · f = 0, (4) Dt2 f · f + 2gg ∗ − λf 2 = 0.
(5)
Here, f is a real fuction, and g is a complex one. ∗ represents complex conjugate. Dx and Dt are bilinear operators, which are defined as [34, 36]: ∂ ∂ n ∂ ∂ m 0 0 Dxn Dtm g(x, t) · f (x, t) = − 0 − 0 g(x, t)f (x , t )|x0 =x,t0 =t . (6) ∂x ∂x ∂t ∂t
To obtain the dark two-soliton solutions, we write g(x, t) and f (x, t) as a power series expansion with a formal parameter ε: g(x, t) = g0 (x, t)[1 + εg1 (x, t) + ε2 g2 (x, t) + ε3 g3 (x, t)] · · · , f (x, t) = 1 + εf1 (x, t) + ε2 f2 (x, t) + ε3 f3 (x, t) · · · .
(7) (8)
And then make the following assumptions: g0 (x, t) = Aeia(x) , g1 (x, t) = eη1 +2iθ1 + eη2 +2iθ2 , g2 (x, t) = Beη1 +η2 +2iθ1 +2iθ2 , f1 (x, t) = eη1 + eη2 , f2 (x, t) = Ceη1 +η2 , ηj = mj (x) + ωj t (j = 1, 2), (9) where A, B and C are all complex constants, θj and ωj are real constants, mj (x) are real functions of x. Truncating Eqs. (7) and (8) as g = g0 (x, t)[1 + 3
εg1 (x, t) + ε2 g2 (x, t)], f = 1 + εf1 (x, t) + ε2 f2 (x, t), and substituting the above assumptions into the bilinear forms Eqs. (4) and (5), combining the terms with the same power of ε, and making their coefficients equal to 0, we can derive Z 2 λ = 2|A| , a(x) = 2α2 (x)|A|2 dx, ωj2 = 4|A|2 sin2 θj , Z 2 mj (x) = 4|A| sinηj − α2 (x)cosθj + α3 (x)|A|(2 + cos2θj ) dx, B=C=
−1 + cos(θ1 − θ2 ) . −1 + cos(θ1 + θ2 )
With ε = 1, the dark two-soliton solutions for Eq. (1) can be expressed as g0 (x, t)[1 + g1 (x, t) + g2 (x, t)] 1 + f1 (x, t) + f2 (x, t) ia(x) Ae [1 + eη1 +2iθ1 + eη2 +2iθ2 + Beη1 +η2 +2iθ1 +2iθ2 ] = . 1 + eη1 + eη2 + Ceη1 +η2
q(x, t) =
3. Discussion Firstly, we discuss the effects of parameters A and θj (j = 1, 2) on the transmission of dark solitons. In Fig. 1, when the constant A changes from 0.5 + i to 1.3 + 1.5i, we find that the intensities of dark solitons become higher, and the pulse duration becomes narrower. In the next, if we keep other parameters unchanged, the corresponding soliton intensity of θ1 = 0.5 becomes smaller than that of θ1 = 1, and the other one remains completely unchanged in Fig. 1(d).
4
Figure 1: Dark two-soliton transmission under different dispersion coefficients. (a) A = 0.5+i, θ1 = 1, θ2 = 2.4, α2 (x) = 1, α3 (x) = 0.2. (b) A = 1.3+1.5i, θ1 = 1, θ2 = 2.4, α2 (x) = 1, α3 (x) = 0.2. (c) A = 0.5 + i, θ1 = 0.5, θ2 = 2.4, α2 (x) = 1, α3 (x) = 0.2. (d) The three soliton states under different parameters when x = 1.
In Figs. 2(a) and 2(b), the second-order dispersion coefficient α2 (x) and the third-order dispersion coefficient α3 (x) are equal to 0 and sinx in turn. In both cases, the transmission path of two dark solitons are sinusoidal. However, when α3 (x) equals to 0, the transmission vibration directions of two dark solitons are opposite in Fig. 2(a). When α2 (x) is equal to 0, the vibration directions are consistent in Fig. 2(b).
5
Figure 2: Dark two-soliton transmission under different dispersion coefficients A = 0.5 + i, θ1 = 1, and θ2 = 2.4. (a) α2 (x) = sinx, α3 (x) = 0. (b) α2 (x) = 0, α3 (x) = sinx. (c) α2 (x) = 0.5, α3 (x) = sinx. (d) α2 (x) = sinx, α3 (x) = 0.5.
In addition, it can be clearly seen that the lateral amplitude of the soliton transmission with the third-order dispersion term is larger than that with the second-order dispersion term. What’s more, compared with Fig. 2(b), when the value of α2 (x) changes from 0 to 0.5, the angle between two dark solitons increases significantly in Fig. 2(c). Obviously, the transmission direction of two solitons changes. Moreover, the number of peaks appearing in the unit period has increased. Then, when the value of α3 (x) is 0.5, the vibration period in Fig. 2(d) is longer than that in Fig. 2(a). Moreover, the transmission direction of dark solitons is no longer along the x-axis direction, and an angle is formed with the x-axis. 6
If α2 (x) and α3 (x) take different types of functions, the transmission path of dark solitons will produce rich changes. For example, when α2 (x) = ex + e−x and α3 (x) = x, the transmission path of dark solitons in Fig. 3 turns into cross-shaped. By appropriately adjusting the value of some parameters, the shape of the soliton transmission can be adjusted purposefully. As shown in Fig. 3(b), when the value of α3 (x) = 0.1x, the transmission path is also cross-shaped, but in the portion where t > 0, the angle between two dark solitons becomes smaller.
Figure 3: Dark two-soliton transmission under different dispersion coefficients A = 1 + i, θ1 = 1, θ2 = 2.4. (a) α2 (x) = ex + e−x , α3 (x) = x. (b) α2 (x) = ex + e−x , α3 (x) = 0.1x.
4. Conclusions In this paper, analytic dark two-soliton solutions for Eq. (1) have been obtained with the help of bilinear method. Dark solitons have been less affected by their interactions. Effects of some parameters on the dark soliton transmission have been discussed, and the second and third order dispersion coefficients have been analyzed. Among them, the parameter A is an important parameter that affects the waveform of dark soliton. As the value of A changes, the intensity of the dark soliton is adjusted. In addition, θj can be used to control the amplitude of the corresponding solitons. The larger θj is, the larger the amplitude of the dark soliton is. As for α2 (x) and α3 (x), both of them can be used to control the transmission path of dark solitons. In comparison, the third-order dispersion coefficient has a more significant effect on the amplitude of the lateral transmission than the second-order dispersion 7
coefficient. The second-order dispersion coefficient has an obvious effect on the change of angle between two dark solitons. Results are beneficial to the study of interactions between dark solitons. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 11674036 and 11875008), by the Beijing Youth Topnotch Talent Support Program (Grant No. 2017000026833ZK08), and by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications, Grant No. IPOC2017ZZ05). This work of Qin Zhou was supported by the National Natural Science Foundation of China (Grant Nos. 11975172 and 11705130), this author was also sponsored by the Chutian Scholar Program of Hubei Government in China. Conflict of interest The authors declare that they have no conflict of interest. References [1] G. P. Agrawal, Nonlinear fiber optics, 4th ed. Academic Press, San Diego 2007. [2] N. Zhang, T. C. Xia, and E. G. Fan, A Riemann-Hilbert approach to the Chen-Lee-Liu equation on the half line, Acta Math. Appl. Sin. 34(3) (2018) 493-515. [3] A. Biswas, A. H. Kara, M. Z. Ullah, Q. Zhou, H. Triki, and M. Belic, Conservation laws for cubic-quartic optical solitons in Kerr and power law media, Optik 145 (2017) 650-654. [4] N. Zhang, T. C. Xia, and Q. Y. Jin, N-Fold Darboux transformation of the discrete Ragnisco-Tu system, Adv. Differ. Equ. 2018 (2018) 302. [5] A. Biswas, M. Z. Ullah, Q. Zhou, S. P. Moshokoa, H. Triki, and M. Belic, Resonant optical solitons with quadratic-cubic nonlinearity by semi-inverse variational principle, Optik 145 (2017) 18-21. 8
[6] M. S. Tao, N. Zhang, D. Z. Gao, and H. W. Yang, Symmetry analysis for three-dimensional dissipation Rossby waves, Adv. Differ. Equ. 2018 (2018) 300. [7] A. Biswas, Q. Zhou, S. P. Moshokoa, H. Triki, M. Belic, and R. T. Alqahtani, Resonant 1-soliton solution in anti-cubic nonlinear medium with perturbations, Optik 145 (2017) 14-17. [8] J. Y. Gu, Y. Zhang, and H. H. Dong, Dynamic behaviors of interaction solutions of (3+1)-dimensional Shallow Water wave equation, Comput. Math. Appl. 76(6) (2018) 1408-1419. [9] A. Biswas, H. Triki, Q. Zhou, S. P. Moshokoa, M. Z. Ullah, and M. Belic, Cubic-quartic optical solitons in Kerr and power law media, Optik 144 (2017) 357-362. [10] Y. Liu, H. H. Dong, and Y. Zhang, Solutions of a discrete integrable hierarchy by straightening out of its continuous and discrete constrained flows, Ana. Math. Phys. 2019 (2019) 465-481. [11] A. Biswas, Q. Zhou, M. Z. Ullah, H. Triki, S. P. Moshokoa, and M. Belic, Optical soliton perturbation with ant-cubic nonlinearity by semi-inverse variational principle, Optik 143 (2017) 131-134. [12] M. Guo, Y. Zhang, M. Wang, Y. D. Chen, and H. W. Yang, A new ZK-ILW equation for algebraic gravity solitary waves in finite depth stratified atmosphere and the research of squall lines formation mechanism, Comput. Math. Appl. 75 (2018) 3589-3603. [13] A. Biswas, Q. Zhou, M. Z. Ullah, M. Asma, S. P. Moshokoa, and M. Belic, Perturbation theory and optical soliton cooling with anti-cubic nonlinearity, Optik 142 (2017) 73-76. [14] C. Lu, C. Fu, and H. W. Yang, Time-fractional generalized Boussinesq equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions, Appl. Math. Comput. 327 (2018) 104-116. [15] B. J. Zhao, R. Y. Wang, W. J. Sun, and H. W. Yang, Combined ZKmZK equation for Rossby solitary waves with complete Coriolis force 9
and its conservation laws as well as exact solutions, Adv. Differ. Equ. 2018 (2018) 42. [16] H. W. Yang, X. Chen, M. Guo, and Y. D. Chen, A new ZK-BO equation for three-dimensional algebraic Rossby solitary waves and its solution as well as fission property, Nonlinear Dyn. 91 (2018) 2019-2032. [17] M. Li, T. Xu, and D. Meng, Rational solitons in the parity-timesymmetric nonlocal nonlinear Schr¨odinger model, J. Phys. Soc. Jpn. 85 (2016) 124001. [18] M. Li and T. Xu, Dark and antidark soliton interactions in the nonlocal nonlinear Schr¨odinger equation with the self-induced parity-timesymmetric potential, Phys. Rev. E 91(3) (2015) 033202. [19] T. Xu, S. Lan, M. Li, L. L. Li, and G. W. Zhang, Mixed soliton solutions of the defocusing nonlocal nonlinear Schr¨odinger equation, Physica D 390 (2019) 47-61. [20] D. Kr¨okel, N. J. Halas, and G. Giuliani, Dark-pulse propagation in optical fibers, Phys. Rev. Lett. 60 (1988) 29-32. [21] A. M. Weiner, J. P. Heritage, and R. J. Hawkins, Experimental observation of the fundamental dark soliton in optical fibers, Phys. Rev. Lett. 61 (1988) 2445. [22] W. Zhao and E. Bourkoff, Interactions between dark solitons, Opt. Lett. 14 (1989) 1371-1373. [23] W. Zhao and E. Bourkoff, Generation, propagation, and amplification of dark solitons, J. Opt. Soc. Am. B 9 (1992) 1134-1144. [24] J. P. Hamaide, P. Emplit, and M. Haelterman, Dark-soliton jitter in amplified optical transmission systems, Opt. Lett. 16 (1991) 1578-1580. [25] I. M. Uzunov and V. S. Gerdjikov, Self-frequency shift of dark solitons in optical fibers, Phys. Rev. A 47 (1993) 1582-1585. [26] W. J. Liu, Y. J. Zhang, H. Triki, M. Mirzazadeh, M. Ekici, Q. Zhou, A. Biswas, and M. Belic, Interaction properties of solitonics in inhomogeneous optical fibers, Nonlinear Dyn. 95(1) (2019) 557-563. 10
[27] C. Y. Yang, W. J. Liu, Q. Zhou, D. Mihalache, and B. A. Malomed, Onesoliton shaping and two-soliton interaction in the fifth-order variablecoefficient nonlinear Schr¨odinger equation, Nonlinear Dyn. 95(1) (2019) 369-380. [28] X. Y. Liu, H. Triki, Q. Zhou, M. Mirzazadeh, W. J. Liu, A. Biswas, and M. Belic, Generation and control of multiple solitons under the influence of parameters, Nonlinear Dyn. 95(1) (2019) 143-150. [29] W. T. Yu, Q. Zhou, M. Mirzazadeh, W. J. Liu, and A. Biswas, Phase shift, amplification, oscillation and attenuation of solitons in nonlinear optics, J. Adv. Res. 15 (2019) 69-76. [30] H. Q. Zhang and S. S. Yuan, Dark soliton solutions of the defocusing Hirota equation by the binary Darboux transformation, Nonlinear Dyn. 89(1) (2017) 531-538. [31] L. Li, Z. H. Li, Z. Y. Xu, G. S. Zhou, and K. H. Spatschek, Gray optical dips in the subpicosecond regime, Phys. Rev. E 66 (2002) 046616. [32] A. Mahalingam and K. Porsezian, Propagation of dark solitons with higher-order effects in optical fibers, Phys. Rev. E 64 (2001) 046608. [33] A. Ankiewicz, J. M. Soto-Crespo, and N. Akhmediev, Rogue waves and rational solutions of the Hirota equation, Phys. Rev. E 81 (2010) 046602. [34] R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys. 14(7) (1973) 805-809. [35] C. Y. Yang, A. M. Wazwaz, Q. Zhou, and W. J. Liu, Transformation of soliton states for a (2+1) dimensional fourth-order nonlinear Schr¨odinger equation in the Heisenberg ferromagnetic spin chain, Laser Phys. 29(3) (2019) 035401. [36] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27(18) (1971) 1192.
11
Journal Pre-proof
¨ Dark two-soliton solutions for nonlinear Schrodinger equations in inhomogeneous optical fibers Xiaoyan Liu, Zitong Luan, Qin Zhou, Wenjun Liu, Anjan Biswas PII: DOI: Reference:
S0577-9073(19)30894-9 https://doi.org/10.1016/j.cjph.2019.08.006 CJPH 917
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
23 February 2019 19 May 2019 28 August 2019
Please cite this article as: Xiaoyan Liu, Zitong Luan, Qin Zhou, Wenjun Liu, Anjan Biswas, Dark two¨ soliton solutions for nonlinear Schrodinger equations in inhomogeneous optical fibers, Chinese Journal of Physics (2019), doi: https://doi.org/10.1016/j.cjph.2019.08.006
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.
Dark two-soliton solutions for nonlinear Schr¨odinger equations in inhomogeneous optical fibers Xiaoyan Liua , Zitong Luanb , Qin Zhouc , Wenjun Liua , Anjan Biswasd,e,f a
State Key Laboratory of Information Photonics and Optical Communications, School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China b School of Economics and Management, Beijing University of Posts and Telecommunications, Beijing 100876, China c School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan 430212, China d Department of Physics, Chemistry and Mathematics, Alabama A&M University, Normal, AL-35762, USA e Department of Mathematics, King Abdulaziz University, Jeddah–21589, Saudi Arabia f Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa
Abstract In this paper, the variable coefficient nonlinear Schr¨odinger equation is investigated analytically. With the bilinear method, the bilinear forms and analytic soliton solutions are obtained. Based on the obtained analytic solutions, the effect of free parameters on the control of soliton transmission is studied. Influences of second-order and third-order dispersion coefficients on dark solitons are discussed. Results in this paper would be of great significance in the generation of dark solitons. Keywords: Solitons, Bilinear method, Partial differential equation, Nonlinear Schr¨odinger equation 1. Introduction Solitons have been widely studied in theory and experiment in recent years [1-19]. Since the development of the technique of synthesizing ultrashort pulses of arbitrary shape and phase in 1988, dark solitons, as one of Email addresses: Corresponding author: [email protected] (Qin Zhou), Corresponding author: [email protected] (Wenjun Liu) Preprint submitted to Chinese Journal of Physics
September 23, 2019
solitons, have been experimentally confirmed [20, 21]. In the case of the same fiber loss, the attenuation of the amplitude of the dark soliton and the increase of the pulse duration are smaller than that of the bright one. And for various disturbance effects, such as the interaction of solitons, dark solitons have better stability and stronger resistance than bright ones, they have better self-recovery ability and higher code rate [22-25]. However, the loss, dispersion and nonlinear effects are the main factors affecting soliton transmissions, and they can cause pulse broadening and crepe generation of solitons, resulting in solitons deformation [26, 27]. In order to reduce the generation of negative effects and achieve the purpose of controlling transmission effects, it would be critical to study the effects of the loss, dispersion and other effects on the soliton transmission [28, 29]. Thus, it is necessary to discuss the effects of different order dispersion terms on the soliton transmission. The nonlinear Schr¨odinger (NLS) equation can be used to describe the transmission of optical solitons in optical fibers [26]. In inhomogeneous optical fibers, the transmission characteristics of optical solitons can be presented in a NLS equation as below [30]: iqx − (qtt − 2|q|2 q) + iε(qttt − 6|q|2 qt ) = 0,
(1)
where q(x, t) is the normalized complex amplitude of the optical soliton envelope. ε is related to the third-order effect. x represents the transmission distance of optical solitons, and t is the normalization time of the system. In Ref. [30], the binary Darboux transformation has been used to obtain dark soliton solutions of Eq. (1) under constant coefficients. Its initial value problem can be solved by the inverse scattering transformation [31]. Eq. (1) can be transformed into a complex modified KdV equation by the variable substitution [32]. Besides, the breather solutions, rogue wave solutions, bright N-soliton solutions of Eq. (1) have also been solved [33, 34]. Due to the development of nonlinear optics and the application of dispersion management system, the variable coefficient NLS equation can be introduced as [1]: iqx − α2 (x)(qtt − 2|q|2 q) + iα3 (x)(qttt − 6|q|2 qt ) = 0,
(2)
where α2 (x) and α3 (x) are real functions that related to the second-order dispersion and third-order dispersion, respectively. As far as we know, dark solitons with varying second-order and third-order dispersion coefficients have 2
not yet been solved. And no one has compared their different effects between α2 (x) and α3 (x) in Eq. (2) on the transmission of dark solitons. In this paper, we use the bilinear method to get the dark two-soliton solutions of Eq. (2) in Section 2. By analyzing the transmission characteristics of dark solitons, the dispersion and other effects of optical fibers on the dark soliton propagation are discussed. And the influences of different order dispersion coefficients α2 (x) and α3 (x) on the transmission of dark solitons are analyzed in Section 3. Finally, we will give the conclusions in Section 4. 2. Analytic dark two-soliton solutions By introducing a dependent variable [35]: q(x, t) =
g(x, t) , f (x, t)
(3)
the bilinear forms of Eq. (2) can be obtained as i h iDx − α2 (x)Dt2 + iα3 (x)Dt3 + λα2 (x) − 3iλα3 (x)Dt g · f = 0, (4) Dt2 f · f + 2gg ∗ − λf 2 = 0.
(5)
Here, f is a real fuction, and g is a complex one. ∗ represents complex conjugate. Dx and Dt are bilinear operators, which are defined as [34, 36]: ∂ ∂ n ∂ ∂ m 0 0 Dxn Dtm g(x, t) · f (x, t) = − 0 − 0 g(x, t)f (x , t )|x0 =x,t0 =t . (6) ∂x ∂x ∂t ∂t
To obtain the dark two-soliton solutions, we write g(x, t) and f (x, t) as a power series expansion with a formal parameter ε: g(x, t) = g0 (x, t)[1 + εg1 (x, t) + ε2 g2 (x, t) + ε3 g3 (x, t)] · · · , f (x, t) = 1 + εf1 (x, t) + ε2 f2 (x, t) + ε3 f3 (x, t) · · · .
(7) (8)
And then make the following assumptions: g0 (x, t) = Aeia(x) , g1 (x, t) = eη1 +2iθ1 + eη2 +2iθ2 , g2 (x, t) = Beη1 +η2 +2iθ1 +2iθ2 , f1 (x, t) = eη1 + eη2 , f2 (x, t) = Ceη1 +η2 , ηj = mj (x) + ωj t (j = 1, 2), (9) where A, B and C are all complex constants, θj and ωj are real constants, mj (x) are real functions of x. Truncating Eqs. (7) and (8) as g = g0 (x, t)[1 + 3
εg1 (x, t) + ε2 g2 (x, t)], f = 1 + εf1 (x, t) + ε2 f2 (x, t), and substituting the above assumptions into the bilinear forms Eqs. (4) and (5), combining the terms with the same power of ε, and making their coefficients equal to 0, we can derive Z 2 λ = 2|A| , a(x) = 2α2 (x)|A|2 dx, ωj2 = 4|A|2 sin2 θj , Z 2 mj (x) = 4|A| sinηj − α2 (x)cosθj + α3 (x)|A|(2 + cos2θj ) dx, B=C=
−1 + cos(θ1 − θ2 ) . −1 + cos(θ1 + θ2 )
With ε = 1, the dark two-soliton solutions for Eq. (1) can be expressed as g0 (x, t)[1 + g1 (x, t) + g2 (x, t)] 1 + f1 (x, t) + f2 (x, t) ia(x) Ae [1 + eη1 +2iθ1 + eη2 +2iθ2 + Beη1 +η2 +2iθ1 +2iθ2 ] = . 1 + eη1 + eη2 + Ceη1 +η2
q(x, t) =
3. Discussion Firstly, we discuss the effects of parameters A and θj (j = 1, 2) on the transmission of dark solitons. In Fig. 1, when the constant A changes from 0.5 + i to 1.3 + 1.5i, we find that the intensities of dark solitons become higher, and the pulse duration becomes narrower. In the next, if we keep other parameters unchanged, the corresponding soliton intensity of θ1 = 0.5 becomes smaller than that of θ1 = 1, and the other one remains completely unchanged in Fig. 1(d).
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Figure 1: Dark two-soliton transmission under different dispersion coefficients. (a) A = 0.5+i, θ1 = 1, θ2 = 2.4, α2 (x) = 1, α3 (x) = 0.2. (b) A = 1.3+1.5i, θ1 = 1, θ2 = 2.4, α2 (x) = 1, α3 (x) = 0.2. (c) A = 0.5 + i, θ1 = 0.5, θ2 = 2.4, α2 (x) = 1, α3 (x) = 0.2. (d) The three soliton states under different parameters when x = 1.
In Figs. 2(a) and 2(b), the second-order dispersion coefficient α2 (x) and the third-order dispersion coefficient α3 (x) are equal to 0 and sinx in turn. In both cases, the transmission path of two dark solitons are sinusoidal. However, when α3 (x) equals to 0, the transmission vibration directions of two dark solitons are opposite in Fig. 2(a). When α2 (x) is equal to 0, the vibration directions are consistent in Fig. 2(b).
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Figure 2: Dark two-soliton transmission under different dispersion coefficients A = 0.5 + i, θ1 = 1, and θ2 = 2.4. (a) α2 (x) = sinx, α3 (x) = 0. (b) α2 (x) = 0, α3 (x) = sinx. (c) α2 (x) = 0.5, α3 (x) = sinx. (d) α2 (x) = sinx, α3 (x) = 0.5.
In addition, it can be clearly seen that the lateral amplitude of the soliton transmission with the third-order dispersion term is larger than that with the second-order dispersion term. What’s more, compared with Fig. 2(b), when the value of α2 (x) changes from 0 to 0.5, the angle between two dark solitons increases significantly in Fig. 2(c). Obviously, the transmission direction of two solitons changes. Moreover, the number of peaks appearing in the unit period has increased. Then, when the value of α3 (x) is 0.5, the vibration period in Fig. 2(d) is longer than that in Fig. 2(a). Moreover, the transmission direction of dark solitons is no longer along the x-axis direction, and an angle is formed with the x-axis. 6
If α2 (x) and α3 (x) take different types of functions, the transmission path of dark solitons will produce rich changes. For example, when α2 (x) = ex + e−x and α3 (x) = x, the transmission path of dark solitons in Fig. 3 turns into cross-shaped. By appropriately adjusting the value of some parameters, the shape of the soliton transmission can be adjusted purposefully. As shown in Fig. 3(b), when the value of α3 (x) = 0.1x, the transmission path is also cross-shaped, but in the portion where t > 0, the angle between two dark solitons becomes smaller.
Figure 3: Dark two-soliton transmission under different dispersion coefficients A = 1 + i, θ1 = 1, θ2 = 2.4. (a) α2 (x) = ex + e−x , α3 (x) = x. (b) α2 (x) = ex + e−x , α3 (x) = 0.1x.
4. Conclusions In this paper, analytic dark two-soliton solutions for Eq. (1) have been obtained with the help of bilinear method. Dark solitons have been less affected by their interactions. Effects of some parameters on the dark soliton transmission have been discussed, and the second and third order dispersion coefficients have been analyzed. Among them, the parameter A is an important parameter that affects the waveform of dark soliton. As the value of A changes, the intensity of the dark soliton is adjusted. In addition, θj can be used to control the amplitude of the corresponding solitons. The larger θj is, the larger the amplitude of the dark soliton is. As for α2 (x) and α3 (x), both of them can be used to control the transmission path of dark solitons. In comparison, the third-order dispersion coefficient has a more significant effect on the amplitude of the lateral transmission than the second-order dispersion 7
coefficient. The second-order dispersion coefficient has an obvious effect on the change of angle between two dark solitons. Results are beneficial to the study of interactions between dark solitons. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 11674036 and 11875008), by the Beijing Youth Topnotch Talent Support Program (Grant No. 2017000026833ZK08), and by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications, Grant No. IPOC2017ZZ05). This work of Qin Zhou was supported by the National Natural Science Foundation of China (Grant Nos. 11975172 and 11705130), this author was also sponsored by the Chutian Scholar Program of Hubei Government in China. Conflict of interest The authors declare that they have no conflict of interest. References [1] G. P. Agrawal, Nonlinear fiber optics, 4th ed. Academic Press, San Diego 2007. [2] N. Zhang, T. C. Xia, and E. G. Fan, A Riemann-Hilbert approach to the Chen-Lee-Liu equation on the half line, Acta Math. Appl. Sin. 34(3) (2018) 493-515. [3] A. Biswas, A. H. Kara, M. Z. Ullah, Q. Zhou, H. Triki, and M. Belic, Conservation laws for cubic-quartic optical solitons in Kerr and power law media, Optik 145 (2017) 650-654. [4] N. Zhang, T. C. Xia, and Q. Y. Jin, N-Fold Darboux transformation of the discrete Ragnisco-Tu system, Adv. Differ. Equ. 2018 (2018) 302. [5] A. Biswas, M. Z. Ullah, Q. Zhou, S. P. Moshokoa, H. Triki, and M. Belic, Resonant optical solitons with quadratic-cubic nonlinearity by semi-inverse variational principle, Optik 145 (2017) 18-21. 8
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