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

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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 controlling 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 dispersion (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 constraint 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 interactions 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) . w2 w2 P2 M (ξ) = 11 43 , 3 2 w2 w2 w2 4r21 21 32 33 123 S. Liu et al.

Fig. 1 Interactions among three solitons affected with differ- −10, g(ξ) = 0.02 with a D(ξ) =−0.085; b D(ξ) =−0.07ξ; −ξ ent proﬁles of D(ξ). Parameters chosen as: r11 = 2.4, r12 = c D(ξ) =−0.15cos(ξ); d D(ξ) =−0.07e ; e D(ξ) = −ξ 2 −3.1, r21 =−3, r22 = 1.8, r31 = 3.2, r32 = 5, k11 = −0.07e −1, k12 =−2, k21 = 4, k22 = 8, k31 = 4, k32 =−2, P =

Fig. 2 Phase shift affected with the different values before 4, k22 = 8, k31 = 4, k32 =−2, P =−10, g(ξ) = 0.02 with 2 − . ξ 2 − . ξ 2 ξ . Parameters chosen as: r11 = 2.4, r12 =−3.1, r21 = a D(ξ) =−0.057e 0 0054 ; b D(ξ) =−0.057e 0 076 ; c − , = . , = . , = , =− , =− , = − . ξ 2 3 r22 1 8 r31 3 2 r32 5 k11 1 k12 2 k21 D(ξ) =−0.057e 0 9

123 Phase-shift controlling of three solitons

3 Discussions D(ξ) =−0.07ξ and D(ξ) =−0.15cos(ξ), the direction of the soliton propagation is obviously changed For solutions (7) obtained by the above process, we and the amplitude of solitons is reduced at the place of choose different correlative coefficients to observe the interaction especially in Fig. 1b. The attraction between interactions among three solitons. Subsequently, the solitons which leads to the appearance of peak can be degrees of interactions in different situations are ana- seen clearly in Fig. 1d. Compared with above profiles, 2 lyzed. Figure 1 indicates interactions among three soli- when D(ξ) =−0.07e−ξ , interactions among three tons affected with different profiles of D(ξ) such as the solitons are weakened distinctly because of the phase constant one, the linear one, the cosine one, the expo- shift of solitons in Fig. 1e. Results suggest that phase- nential one and the Gaussian one [41]. In Fig. 1a, we shift controlling of solitons in DDF can be achieved set D(ξ) =−0.085. When there is narrow distance when the dispersion profile is Gaussian. between solitons, although influence on the direction In order to present the effect of Gaussian profile on of propagation is little, sharp peaks are produced due phase shift of solitons, we sequentially change correla- to the interactions. As shown in Fig. 1b and c, when tive coefficient in Gaussian profile by ten times nearly

Fig. 3 Phase shift affected with the different values before 4, k22 = 8, k31 = 4, k32 =−2, P =−10, g(ξ) = 0.02 −0.076ξ 2 −0.076ξ 2 −0.076ξ 2 e . Parameters chosen as: r11 = 2.4, r12 =−3.1, r21 = with a D(ξ) =−0.21e ; b D(ξ) =−0.048e ; − . ξ 2 −3, r22 = 1.8, r31 = 3.2, r32 = 5, k11 =−1, k12 =−2, k21 = c D(ξ) =−0.0034e 0 076

Fig. 4 Interactions among three solitons affected with the 3.2, r32 = 5, k11 =−1, k12 =−2, k21 = 4, k22 = 8, k31 = different values of P. Parameters chosen as: D(ξ) = 4, k32 =−2, g(ξ) = 0.02 with a P =−14; b P =−11; c −ξ 2 =− −0.05e , r11 = 2.4, r12 =−3.1, r21 =−3, r22 = 1.8, r31 = P 8 123 S. Liu et al. as shown in Figs. 2 and 3. With decreasing the value References before ξ 2, the result of the interactions between solitons shows a process from the generation of peaks to the 1. Wazwaz, A.M., El-Tantawy, S.A.: A new integrable (3+1)- generation of new solitons and then to the independent dimensional KdV-like model with its multiple-soliton solutions. Nonlinear Dyn. 83(3), 1529–1534 (2016) propagation in Fig. 2. Compared with Fig. 2,Fig.3 illus- 2. Wazwaz, A.M.: A study on a two-wave mode Kadomtsev– trates the same process by increasing the value before Petviashvili equation: conditions for multiple soliton 2 e−0.076ξ . It is found that interactions of solitons can be solutions to exist. Math. Methods Appl. Sci. 40, 4128–4133 ξ 2 (2017) avoided when the value before is 10 times or larger 3. Wazwaz, A.M.: A two-mode modified KdV equation with −ξ 2 than before e in Figs. 2 and 3. That means phase- multiple soliton solutions. Appl. Math. Lett. 70, 1–6 (2017) shift controlling can also be realized better by changing 4. Wazwaz, A.M.: Abundant solutions of various physi- ξ 2 −ξ 2 cal features for the (2+1)-dimensional modified KdV- the value before or the value before e . Calogero–Bogoyavlenskii–Schiff equation. Nonlinear Dyn. Besides, we adjust the value of P in expression (4)as 89, 1727–1732 (2017) shown in Fig. 4. The attenuation of amplitude becomes 5. Wazwaz, A.M., El-Tantawy, S.A.: Solving the (3+1)- alleviative as P increasing, which means the loss in dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota’s method. Nonlinear Dyn. 88, optical fibers can be compensated and the propagation 3017–3021 (2017) distance of solitons can be improved and extended. 6. Wazwaz, A.M.: Multiple soliton solutions and other exact solutions for a two-mode KdV equation. Math. Methods Appl. Sci. 40, 2277–2283 (2017) 4 Conclusion 7. Nawaz, B., Ali, K., Abbas, S.O., Rizvi, S.T.R., Zhou, Q.: Optical solitons for non-Kerr law nonlinear Schrödinger equation with third and fourth order dispersions. Chin. J. Phase-shift controlling of three solitons in DDF has Phys. 60, 133–140 (2019) been mainly studied in this paper. Hirota method has 8. Rizvi, S.T.R., Ali, K., Hanif, H.: Optical solitons in dual been used to obtain the analytic three solitons solu- core fibers under various nonlinearities. Mod. Phys. Lett. B 33(17), 1950189 (2019) tions (7)forEq.(1). By comparing and analyzing differ- 9. Rizvi, S.T.R., Afzal, I., Ali, K.: optical solitons for Triki– ent values of the GVD, the phase-shift controlling has Biswas equation. Mod. Phys. Lett. B 33, 1950264 (2019) been achieved so that the interactions among three soli- 10. Liu, X.Y., Triki, H., Zhou, Q., Liu, W.J.,Biswas, A.: Analytic tons can be weakened effectively by choosing a Gaus- study on interactions between periodic solitons with con- trollable parameters. Nonlinear Dyn. 94, 1703–709 (2018) sian profile. Moreover, we have found that the enhance- 11. Zhang, Y.J., Yang, C.Y., Yu, W.T., Mirzazadeh, M., Zhou, ment of phase-shift controlling can be prompted via Q., Liu, W.J.: Interactions of vector anti-dark solitons for the decreasing the value before ξ 2 or increasing the value coupled nonlinear Schrödinger equation in inhomogeneous 2 before e−ξ . In addition, the outcome has been given fibers. Nonlinear Dyn. 94, 1351–1360 (2018) 12. Liu, J., Wang, Y.G., Qu, Z.S., Fan, X.W.: 2 μm passive that the propagation distance of solitons can be pro- Q-switched mode-locked Tm3+: YAP laser with single- longed by adjusting the value of P in expression (4). walled carbon nanotube absorber. Opt. Laser Technol. Results maybe can apply in ultra-short pulse lasers and 44(4), 960–962 (2012) fiber compensation systems. 13. Zhang, C., Liu, J., Fan, X.W., Peng, Q.Q., Guo, X.S., Jiang, D.P., Qian, X.B., Su, L.B.: Compact passive Q-switching of a diode-pumped Tm, Y: CaF2 laser near 2 μm. Opt. Laser Acknowledgements The work of Wenjun Liu was supported Technol. 103, 89–92 (2018) by the National Natural Science Foundation of China (Grant 14. Lin, M.X., Peng, Q.Q., Hou, W., Fan, X.W., Liu, J.: 1.3 Nos. 11674036 and 11875008), by the Beijing Youth Top-notch μm Q-switched solid-state laser based on few-layer ReS2 Talent Support Program (Grant No. 2017000026833ZK08) and saturable absorber. Opt. Laser Technol. 109, 90–93 (2019) by the Fund of State Key Laboratory of Information Photonics 15. Biswas, A., Yildirim, Y., Yasar, E., Zhou, Q., Moshokoa, and Optical Communications (Beijing University of Posts and S.P.,Belic, M.: Optical solitons with differential group delay Telecommunications, Grant No. IPOC2017ZZ05). and four-wave mixing using two integration procedures. Optik 167, 170–188 (2018) Compliance with ethical standards 16. Zhang, N., Xia, T.C., Fan, E.G.: A Riemann–Hilbert approach to the Chen–Lee–Liu equation on the half line. Conflict of interest The authors declare that they have no con- Acta Math. Appl. Sin. 34(3), 493–515 (2018) flict of interest. 17. Zhang, N., Xia, T.C., Jin, Q.Y.: N-Fold Darboux transformation of the discrete Ragnisco–Tu system. Adv. Differ. Equ. 2018, 302 (2018)

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18. Tao, M.S., Zhang, N., Gao, D.Z., Yang, H.W.: Symmetry 30. Sun, K., Tian, B., Liu, W.J., Jiang, Y., Qu, Q.X., Wang, P.: analysis for three-dimensional dissipation Rossby waves. Soliton dynamics and interaction in the Bose–Einstein con- Adv. Differ. Equ. 2018, 300 (2018) densates with harmonic trapping potential and time-varying 19. Gu, J.Y.,Zhang, Y., Dong, H.H.: Dynamic behaviors of inter- interatomic interaction. Nonlinear Dyn. 67(1), 165–175 action solutions of (3+1)-dimensional shallow mater wave (2012) equation. Comput. Math. Appl. 76(6), 1408–1419 (2018) 31. Askari, A., Javidan, K.: Interaction of noncommutative 20. Liu, Y., Dong, H.H., Zhang, Y.: Solutions of a discrete solitons with defects. Braz. J. Phys. 38(4), 610–614 (2008) integrable hierarchy by straightening out of its continuous 32. Konyukhov, A.I., Dorokhova, M.A., Melnikov, L.A.E., and discrete constrained flows. Anal. Math. Phys. 2018, Plastun, A.S.: Controlling the interaction between optical 1–17 (2018) solitons using periodic dispersion variations in an optical 21. Guo, M., Zhang, Y., Wang, M., Chen, Y.D., Yang, H.W.: A fibre. Quantum Electron. 45(11), 1018–1022 (2015) new ZK-ILW equation for algebraic gravity solitary waves 33. Li, Y.G., She, W.L., Wang, H.C.: Perpendicular all-optical in finite depth stratified atmosphere and the research of control of interactional optical spatical soliton pair. Acta squall lines formation mechanism. Comput. Math. Appl. Phys. Sin. 56(4), 2229–2236 (2007) 143, 3589–3603 (2018) 34. Shou, Q., Wu, M., Guo, Q.: Large phase shift of (1+1)- 22. Yang, H.W., Chen, X., Guo, M., Chen, Y.D.: A new ZK-BO dimensional nonlocal spatial solitons in lead glass. Opt. equation for three-dimensional algebraic Rossby solitary Commun. 338, 133–137 (2015) waves and its solution as well as fission property. Nonlinear 35. Roy, K., Ghosh, S.K., Chatterjee, P.: Two-soliton and Dyn. 91, 2019–2032 (2018) three-soliton interactions of electron acoustic waves in 23. Zhao, B.J., Wang, R.Y., Sun, W.J., Yang, H.W.: Combined quantum plasma. Pramana 86(4), 873–883 (2016) ZK-mZK equation for Rossby solitary waves with complete 36. Sun, Q.H., Pan, N., Lei, M., Liu, W.J.: Study on phase-shift Coriolis force and its conservation laws as well as exact control in dispersion decreasing fibers. Acta Phys. Sin. solutions. Adv. Differ. Equ. 2018, 42 (2018) 63(15), 150506 (2014) 24. Lu, C., Fu, C., Yang, H.W.: Time-fractional generalized 37. Zheng, H.J., Wu, C., Wang, Z., Yu, H., Liu, S., Li, X.: Boussinesq equation for Rossby solitary waves with dissipa- Propagation characteristics of chirped soliton in periodic tion effect in stratified fluid and conservation laws as well as distributed amplification systems with variable coefficients. exact solutions. Appl. Math. Comput. 327, 104–116 (2018) Optik 123(9), 818–822 (2012) 25. Biswas, A., Rezazadeh, H., Mirzazadeh, M., Eslami, M., 38. Zolotovskii, I.O., Sementsov, D.I.: Formation of the Ekici, M., Zhou, Q., Belic, M.: Optical soliton perturbation amplification regime of quasi-soliton pulses in waveguides with Fokas–Lenells equation using three exotic and efficient with longitudinally inhomogeneous cross sections. Opt. integration schemes. Optik 165, 288–294 (2018) Spectrosc. 102(4), 594–598 (2007) 26. Triki, H., Biswas, A., Zhou, Q., Mirzazadeh, M., Mahmood, 39. Yang, C.Y., Li, W.Y., Yu, W.T., Liu, M.L., Zhang, Y.J., Ma, M.F., Moshokoa, S.P., Belic, M.: Solitons in optical meta- G.L., Lei, M., Liu, W.J.: Amplification, reshaping, fission materials having parabolic law nonlinearity with detuning and annihilation of optical solitons in dispersion-decreasing effect and Raman scattering. Optik 164, 606–609 (2018) fiber. Nonlinear Dyn. 92(2), 203–213 (2018) 27. Eslami, M., Mirzazadeh, M.: Optical solitons with Biswas– 40. Liu, W.J., Zhang, Y.J., Pang, L.H., Yan, H., Ma, G.L., Lei, Milovic equation for power law and dual-power law M.: Study on the control technology of optical solitons in nonlinearities. Nonlinear Dyn. 83(1–2), 731–738 (2016) optical fibers. Nonlinear Dyn. 86(2), 1069–1073 (2016) 28. Ekici, M., Mirzazadeh, M., Eslami, M.: Solitons and 41. Pelusi, M.D., Liu, H.F.: Higher order soliton pulse com- other solutions to Boussinesq equation with power law pression in dispersion-decreasing optical fibers. IEEE J. nonlinearity and dual dispersion. Nonlinear Dyn. 84(2), Quantum Electron. 33(8), 1430–1439 (1997) 669–676 (2016) 29. Li, X.Z., Yan, X., Li, Y.S.: Investigation of multi-soliton, multi-cuspon solutions to the Camassa–Holm equation and Publisher’s Note Springer Nature remains neutral with regard their interaction. Chin. Ann. Math. Ser. B 33(2), 225–246 to jurisdictional claims in published maps and institutional affil- (2012) iations.

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