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Phase-Shift Controlling of Three Solitons in Dispersion-Decreasing fibers

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Phase-Shift Controlling of Three Solitons in Dispersion-Decreasing fibers Nonlinear Dyn https://doi.org/10.1007/s11071-019-05200-5 ORIGINAL PAPER Phase-shift controlling of three solitons in dispersion-decreasing fibers Suzhi Liu · Qin Zhou · Anjan Biswas · Wenjun Liu Received: 3 July 2019 / Accepted: 16 August 2019 © Springer Nature B.V. 2019 Abstract Phase-shift controlling can attenuate the In addition, by adjusting the constraint value, propaga- interactions between solitons and gives practical tion distance of solitons can be further extended. This advantage in optical communication systems. For the may be useful in the optical logic devices and ultra- variable-coefficient nonlinear Schrödinger equation, short pulse lasers. which can be imitated the transmission of solitons in the dispersion-decreasing fiber, analytic three solitons Keywords Soliton · Analytic solution · Dispersion- solutions are derived via the Hirota method. Based on decreasing fiber · Hirota method the obtained solutions, influences of the second-order dispersion parameters and other related parameters in different function types on the soliton transmission 1 Introduction are discussed. Results declare that phase-shift control- ling of solitons in dispersion-decreasing fiber can be Optical solitons, which can be stably transmitted by achieved when the dispersion function is Gaussian one. virtue of the offset about the group velocity disper- sion (GVD) and self-phase modulation (SPM), have S. Liu · W. Liu (B) induced substantial interests of researchers in math- State Key Laboratory of Information Photonics and Optical ematics, physics and optics [1–24]. In particular, the Communications, and School of Science, nonlinearity and integrability in the solitons equation Beijing University of Posts and Telecommunications, P. O. Box 122, Beijing 100876, China are widely concerned [25–28]. Under ideal conditions, e-mail: [email protected] solitons can achieve lossless transmission [29]. How- Q. Zhou ever, when two- or multi-solitons are transmitted simul- School of Electronics and Information Engineering, Wuhan taneously in the optical fiber, these solitons can attract Donghu University, Wuhan 430212, each other [29–31]. Due to the existence of solitary People’s Republic of China interactions, which causes transmission rate of the opti- A. Biswas cal communication system to be seriously degraded, Department of Physics, Chemistry and Mathematics, studying how to minimize the interactions becomes one Alabama A&M University, Normal, AL 35762, USA of the meaningful directions [32,33]. A. Biswas Phase-shift controlling, which contributes to an Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia approach for the question mentioned above, has been investigated widely in theory and experiment [34,35]. A. Biswas Department of Mathematics and Statistics, Tshwane University Ref. [34] has researched large phase shift of solitons in of Technology, Pretoria 0008, South Africa lead glass under strong nonlocal space. Furthermore, 123 S. Liu et al. the phase shifts and trajectories during the overtak- where h(ξ, τ) is assumed as a complex differentiable ing interaction of multi-solitons have been analyzed function while f (ξ, τ) is a real one. After calculation, in Ref. [35]. The propagation of three solitons in the the following bilinear equations of Eq. (1) are obtained: dispersion-decreasing fiber (DDF) can be explained 2 by the nonlinear Schrödinger (NLS) equation as fol- (2iDξ + D(ξ)Dτ )h · f = 0, lows [36,37]: 2 2 g(ξ)dx ∗ D(ξ)Dτ f · f + 2ρ(ξ)e h · h = 0, (3) (ξ) D 2 φξ − i φττ + iρ(ξ)|φ| φ = g(ξ)φ. (1) 2 where * denotes complex conjugate. There is a con- straint relationship among D(ξ), ρ(ξ) and g(ξ): Here, φ is complex function with ξ and τ representing scaled distance and time. D(ξ) is the GVD parame- D(ξ) ter, ρ(ξ) is the Kerr nonlinear parameter, and g(ξ) is ρ(ξ) = P . (4) 2 g(ξ)dξ a parameter related to fiber loss or amplification. With e regard to Eq. (1), the formation and amplification of As Hirota bilinear operators, Dξ and Dτ have the fol- solitons have been studied [38]. Interactions between lowing form: two solitons have been attenuated by phase-shift con- ∂ ∂ m ∂ ∂ n trolling [36]. Besides, researches have done about how Dm Dn(a · b) = − − ξ τ ∂ξ ∂ξ ∂τ ∂τ to amplify, reshape, fission and annihilate solitons [39] and investigated the propagation properties of optical a(ξ, τ)b(ξ ,τ )|ξ =ξ,τ=τ , (5) solitons in the DDF [40]. where both of m and n are non-negative integers, a is the Compared with two solitons, the results of three soli- function about ξ and τ, and b is the function about ξ tons on propagation and interactions are more abun- and τ . To solve bilinear forms (3), h(ξ, τ) and f (ξ, τ) dant. In order to better understand the phase-shift char- can be expanded as: acteristics of three solitons in DDF, results of interac- tions are shown and analyzed under the GVD param- h(ξ, τ) = h (ξ, τ) + 3h (ξ, τ) + 5h (ξ, τ), eters with different functions in this paper. By setting 1 3 5 (ξ, τ) = + 2 (ξ, τ) + 4 (ξ, τ) + 6 (ξ, τ). appropriate dispersion values, phase-shift controlling f 1 f2 f4 f6 of solitons in DDF can be better achieved, which is (6) helpful to improve the transmission quality of the sig- nal in optical communication system. In addition, the Without affecting the generality, we can define the loss in the medium can be effectively compensated value of as 1, and expand the bilinear forms (3)by by changing the constraint value produced by D(ξ), using expression (6). Three solitons solutions of Eq. (1) ρ(ξ) and g(ξ), so that the propagation distance can be are constructed as follows: extended. The structure of the paper is as follows. Section 2 h (ξ, τ) + h (ξ, τ) + h (ξ, τ) (ξ) ξ φ(ξ,τ) = 1 3 5 e g d , is arranged to get analytic three solitons solutions of 1 + f2(ξ, τ) + f4(ξ, τ) + f6(ξ, τ) Eq. (1) through the Hirota method. Section 3 is reserved (7) for analyzing interactions influences under different value of related parameters and finding suitable value to where weaken the interactions among three solitons. Finally, θ1 θ2 θ3 Sect. 4 is used to compose a conclusion for this paper. h1(ξ, τ) = e + e + e , ∗ ∗ θ +θ1 θ +θ1 f2(ξ, τ) = A1(ξ)e 1 + A2(ξ)e 2 ∗ ∗ θ +θ1 θ +θ2 2 Bilinear forms and three solitons solutions + A3(ξ)e 3 + A4(ξ)e 1 ∗ ∗ θ +θ2 θ +θ2 + A5(ξ)e 2 + A6(ξ)e 3 ∗ ∗ θ +θ3 θ +θ3 Using the Hirota method to introduce the dependent + A7(ξ)e 1 + A8(ξ)e 2 ∗ variable transformation: θ +θ3 + A9(ξ)e 3 , θ∗+θ +θ θ∗+θ +θ h (ξ, τ) = B (ξ)e 1 1 2 + B (ξ)e 2 1 2 h(ξ, τ) (ξ) ξ 3 1 2 φ(ξ,τ) = g d , θ∗+θ +θ θ∗+θ +θ e (2) + (ξ) 3 1 2 + (ξ) 1 1 3 f (ξ, τ) B3 e B4 e 123 Phase-shift controlling of three solitons θ∗+θ +θ θ∗+θ +θ 2 2 2 2 2 + (ξ) 2 1 3 + (ξ) 3 1 3 w w P J P B5 e B6 e (ξ) = 41 32 , (ξ) = 21 , θ∗+θ +θ θ∗+θ +θ M4 M5 + (ξ) 1 2 3 + (ξ) 2 2 3 2 w2 w2 w2 2 2 2 B7 e B8 e 4r11 21 22 23 16r11r31 J22 θ∗+θ +θ + (ξ) 3 2 3 , B9 e w2 w2 2 θ∗+θ∗+θ +θ θ∗+θ∗+θ +θ 12 43 P (ξ, τ) = (ξ) 1 2 1 2 + (ξ) 1 3 1 2 (ξ) = , f4 M1 e M2 e M6 2 2 2 2 ∗ ∗ ∗ ∗ 4r w w w θ +θ +θ1+θ2 θ +θ +θ1+θ3 31 31 32 23 + M3(ξ)e 2 3 + M4(ξ)e 1 2 θ∗+θ∗+θ +θ θ∗+θ∗+θ +θ 2 2 2 2 2 2 + (ξ) 1 3 1 3 + (ξ) 2 3 1 3 w w P w w P M5 e M6 e (ξ) = 41 13 , (ξ) = 13 23 , θ∗+θ∗+θ +θ θ∗+θ∗+θ +θ M7 M8 + (ξ) 1 2 2 3 + (ξ) 1 3 2 3 2 w2 w2 w2 2 w2 w2 w2 M7 e M8 e 4r21 21 22 23 4r31 21 33 22 ∗ ∗ θ +θ +θ2+θ3 + M9(ξ)e 2 3 , 2 2 θ∗+θ∗+θ +θ +θ θ∗+θ∗+θ +θ +θ J31 P (ξ, τ) = (ξ) 1 2 1 2 3 + (ξ) 1 3 1 2 3 (ξ) = , h5 N1 e N2 e M9 2 2 2 ∗ ∗ 16r r J θ +θ +θ1+θ2+θ3 21 31 32 + N3(ξ)e 2 3 , ∗ ∗ ∗ θ +θ +θ +θ1+θ2+θ3 2 w2 w2 2 f6(ξ, τ) = n(ξ)e 1 2 3 , J11 12 13 P N1(ξ) = , 16r 2 r 2 w2 w2 w2 w2 with 11 21 31 21 22 23 θ = a (ξ)+ia (ξ)+(r + ir )τ + k + ik , J 2 w2 w2 P2 i i1 i2 i1 i2 i1 i2 N (ξ) = 21 11 13 , 2 2 2 w2 w2 w2 w2 16r11r31 21 32 33 22 ai1(ξ) =− ri1ri2 D(ξ)dξ, 2 w2 w2 2 1 J31 11 12 P (ξ) = ( 2 − 2 ) (ξ) ξ( = , , ), N3(ξ) = , ai2 ri1 ri2 D d i 1 2 3 2 2 w2 w2 w2 w2 2 16r21r31 31 32 33 23 P P P A (ξ) =− , A (ξ) =− , A (ξ) =− , J 2 J 2 J 2 P3 1 2 2 w2 3 w2 n(ξ) =− 11 21 31 , 4r11 31 32 2 2 2 w2 w2 w2 w2 w2 w2 64r11r21r31 31 21 32 33 22 23 P P P A (ξ) =− , A (ξ) =− , A (ξ) =− , w = r + ir − r − ir , 4 w2 5 2 6 w2 11 11 12 21 22 21 4r21 33 w12 = r11 + ir12 − r31 − ir32, P P P A (ξ) =− , A (ξ) =− , A (ξ) =− , w = + − − , 7 w2 8 w2 9 2 13 r21 ir22 r31 ir32 22 23 4r31 w21 = r11 − ir12 + r21 + ir22, w2 P w2 P B (ξ) =− 11 , B (ξ) =− 11 , w = − + + , 1 2 w2 2 2 w2 22 r11 ir12 r31 ir32 4r11 21 4r21 31 w23 = r21 − ir22 + r31 + ir32, w2 P B (ξ) =− 11 , w = + + − , 3 w2 w2 31 r11 ir12 r21 ir22 32 33 w32 = r11 + ir12 + r31 − ir32, w2 P w2 P B (ξ) =− 12 , B (ξ) =− 12 , w = + + − , 4 2 w2 5 w2 w2 33 r21 ir22 r31 ir32 4r11 22 31 23 w41 = r11 − ir12 − r21 + ir22, w2 P B (ξ) =− 12 , w = − − + , 6 2 w2 42 r11 ir12 r31 ir32 4r31 32 w43 = r21 − ir22 − r31 + ir32, w2 P w2 P B (ξ) =− 13 , B (ξ) =− 13 , = ( − )2 + ( − )2, 7 w2 w2 8 2 w2 J11 r11 r21 r12 r22 21 22 4r21 23 2 2 J12 = (r11 + r21) + (r12 − r22) , w2 P B (ξ) =− 13 , = ( − )2 + ( − )2, 9 2 w2 J21 r11 r31 r12 r32 4r31 33 J = (r + r )2 + (r − r )2, 2 2 w2 w2 2 22 11 31 12 32 J11 P 11 42 P M1(ξ) = , M2(ξ) = , J = (r − r )2 + (r − r )2, 16r 2 r 2 J 2 4r 2 w2 w2 w2 31 21 31 22 32 11 21 12 11 21 32 33 2 2 J32 = (r21 + r31) + (r22 − r32) .
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