Effect of Fourth-Order Dispersion on Solitonic Interactions

  • K. Khelil University Badji Mokhtar–Annaba, Faculty of Sciences, Department of Physics
  • K. Saouchi University Badji Mokhtar-Annaba, Faculty of Engineering Sciences, Department of Electronics
  • D. Bahloul University Hadj Lakhdar-Batna, Faculty of Sciences, Department of Physics
Keywords: interaction of solitons, non-linearity, dispersion, optical fiber, transmission channel, Schr¨odinger equation


Solitons became important in optical communication systems thanks to their robust nature. However, the interaction of solitons is considered as a bad effect. To avoid interactions, the obvious solution is to respect the temporal separation between two adjacent solitons determined as a bit rate. Nevertheless, many better solutions exist to decrease the bit rate error. In this context, the aim of our work is to study the possibility to delete the interaction of adjacent solitons, by using a special dispersion management system, precisely by introducing both of the third- and fourth-order dispersions in the presence of a group velocity dispersion. To study the influence of the fourth- and third-order dispersions, we use the famous non-linear Schr¨odinger equation solved with the Fast Fourier Transform method. The originality of this work is to bring together the dispersion of the fourth, third, and second orders to separate two solitons close enough to create the Kerr-induced interaction and consequently to improve the propagation by decreasing the bit rate error. This study illustrates the influence of the fourth-order dispersion on one single soliton and two co-propagative solitons with different values of the temporal separation. Then the third order dispersion is introduced in the presence of the fourth-order dispersion in the propagation of one and two solitons in order to study its influence on the interaction. Finally, we show the existence of a precise dispersion management system that allows one to avoid the interaction of solitons.


T. Yu, E.A. Golovchenko, A.N. Pilipetskii, C.R. Menyuk. Dispersion-managed soliton interactions in optical fibers. Opt. Lett. 22, (11) 793 (1997). https://doi.org/10.1364/OL.22.000793

A. Hasegawa, F. Tappert. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142 (1973). https://doi.org/10.1063/1.1654836

L.F. Mollenauer, R.H. Stolen, J.P. Gordon. Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Appl. Phys. Lett. 45, 1095 (1980). https://doi.org/10.1103/PhysRevLett.45.1095

L.F. Mollenauer, R.H. Stolen. The soliton laser. Optics Letters 9(1), 13(1984). https://doi.org/10.1364/OL.9.000013

B. Guo, Z. Gan, L. Kong, J. Zhang. The Zakharov System and Its Soliton Solutions (Springer, 2016). https://doi.org/10.1007/978-981-10-2582-2

Physics and Applications of Optical Solitons in Fibres '95, Proceedings of the Symposium held in Kyoto, edited by A. Hasegawa (Springer, 1996).

A. Hasegawa, M. Matsumoto. Optical Solitons in Fibers (Springer, 2003). https://doi.org/10.1007/978-3-540-46064-0

L.F. Mollenauer, R.H. Stolen, J.P. Gordon. Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys. Rev. 45, 1095 (1980). https://doi.org/10.1103/PhysRevLett.45.1095

J.-P. Wei, J.Wang, X.-F. Jiang, B. Tang. Characteristics for the soliton based on nonlinear Schr¨odinger equation. Acta Photon. Sinica 42(6), 674 (2013). https://doi.org/10.3788/gzxb20134206.0674

N.J. Zabusky, M.D. Kruskal. Interaction of "solitons" in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240 (1965). https://doi.org/10.1103/PhysRevLett.15.240

G.P. Agrawal. Nonlinear Fiber Optics. (Academic Press, 2007). https://doi.org/10.1016/B978-012369516-1/50011-X

Y. Kodama. Optical solitons in a monomode fiber. J. Stat. Phys. 39, 597 (1985). https://doi.org/10.1007/BF01008354

Y. Kodama, A. Hasegawa. Nonlinear pulse propagation in a monomode dielectric guide. IEEE J. Quantum Electron. 23 (5), 510 (1987). https://doi.org/10.1109/JQE.1987.1073392

B. Guo, Z. Gan, L. Kong, J. Zhang, The Zakharov System and Its Soliton Solutions (Springer, 2016). https://doi.org/10.1007/978-981-10-2582-2

G. Agrawal, R. Boyd. Contemporary Nonlinear Optics. (Academic Press, 1992).

V. Aref. Nonlinear Fourier transform of truncated multi-soliton pulses. In: 12th ITG Conference on Systems, Communications and Coding (SCC), (2019).

A.G. Bakaoukas. An all-optical soliton FFT computational arrangement in the 3NLSE-domain. Nat. Comput. 17(2), 231 (2018). https://doi.org/10.1007/s11047-017-9642-1

C.Y. Yang, W.Y. Li, W.T. Yu, M.L. Liu, Y.J. Zhang, G.L. Ma, M. Lei, W.J. Liu. Amplification, reshaping, fission and annihilation of optical solitons in dispersion-decreasing fiber. Nonlin. Dyn. 92(2), 203 (2018). https://doi.org/10.1007/s11071-018-4049-9

W.J. Liu, C.Y. Yang, M.L. Liu, W.T. Yu, Y.J. Zhang, M. Lei. Effect of high-order dispersion on three-soliton interactions for the variable-coefficients Hirota equation. Phys. Rev. E 96(4), 042201(2017). https://doi.org/10.1103/PhysRevE.96.042201

X. L¨u, W-X. Ma, S-T. Chen, C.M. Khalique. A note on rational solutions to a Hirota-Satsuma-like equation. Appl. Math. Letters 58, 13 (2016). https://doi.org/10.1016/j.aml.2015.12.019

H.N. Xu, W.Y. Ruan, Y. Zhang, X.L¨u. Multi-exponential wave solutions to two extended Jimbo-Miwa equations and the resonance behavior. Appl. Math. Letters 99, 105976 (2020). https://doi.org/10.1016/j.aml.2019.07.007

Y.F. Hua, B.L. Guo, W.X. Ma, X.L¨u. Interaction behavior associated with a generalized (2 + 1)-dimensional Hirota bilinear equation for nonlinear waves. Appl. Math. Model. 74, 184 (2019). https://doi.org/10.1016/j.apm.2019.04.044

Y.H. Yin, W.X. Ma' ГC, J.G. Liu, X. L¨u. Diversity of exact solutions to a (3+1)-dimensional nonlinear evolution equation and its reduction. Comput. Math. Applic. 76, 1275 (2018). https://doi.org/10.1016/j.camwa.2018.06.020

Li-Na Gao, Xue-Ying Zhao, Yao-Yao Zi, Jun Yu, Xing L¨u. Resonant behavior of multiple wave solutions to a Hirota bilinear equation, Comp. Math. Appl. 72 (8), 1225 (2016). https://doi.org/10.1016/j.camwa.2016.06.008

Si-Jia Chen,Yu-Hang Yin, Wen-Xiu Ma, Xing L¨u. Abundant exact solutions and interaction phenomena of the (2 + 1)-dimensional YTSF equation. Anal. Math. Phys. 9, 2329 (2019). https://doi.org/10.1007/s13324-019-00338-2

X. L¨u, W.X. Ma, Y. Zhou, C.M. Khalique. Rational solutions to an extended Kadomtsev-Petviashvili-like equation with symbolic computation. Comp. Math. Appl. 71, 1560 (2016). https://doi.org/10.1016/j.camwa.2016.02.017

H.Q. Zhang, W.X. Ma. Resonant multiple wave solutions for a (3 + 1)-dimensional nonlinear evolution equation by linear superposition principle. Comp. Math. Appl. 73, 2339 (2017). https://doi.org/10.1016/j.camwa.2017.03.014

Y. Yue, L. Huang, Y. Chen. N-solitons, breathers, lumps and rogue wave solutions to a (3+1)-dimensional nonlinear evolution equation. Comp. Math. Appl. 75 (7), 2538 (2018). https://doi.org/10.1016/j.camwa.2017.12.022

E.M.E. Zayed, A.G. Al-Nowehy. The solitary wave ansatz method for finding the exact bright and dark soliton solutions of two nonlinear Schr¨odinger equations. J. Assoc. Arab Univ. Basic Appl. Sci. 24 (1), 184 (2017). https://doi.org/10.1016/j.jaubas.2016.09.003

S.H. Dong. The ansatz method for analyzing Schr¨odinger's equation with three anharmonic potentials in D dimensions. Found. Phys. Lett. 15 (4), 385 (2002).

Wahyulianti, Suparmi, Cari, Fuad Anwar. The solutions of the D-dimensional Schr¨odinger equation for the potential V(r) = ar−6+br−5+cr−4+dr−3+er−2+fr−1. J. Phys. Confer. Ser. 795 (1), 012022 (2017). https://doi.org/10.1088/1742-6596/795/1/012022

K.S. Al-Ghafri, E.V. Krishnan, Anjan Biswas. Optical solitons for the cubic-quintic nonlinear Schr¨odinger equation. AIP Conference Proceedings 2046, 020002 (2018). https://doi.org/10.1063/1.5081522

H. Triki, A.M. Wazwaz. Soliton solutions of the cubicquintic nonlinear Schr¨odinger equation with variable coefficients. Rom. J. Phys. 61 (3-4), 360 (2016). https://doi.org/10.2478/s13531-013-0119-4

M.A. Ablowitz, P.A. Clarkson. Solitons of Nonlinear Evolution Equations and Inverse Scattering (Cambridge Univ. Press, 1991). https://doi.org/10.1017/CBO9780511623998

J. Garnier, K. Kalimeris. Inverse scattering perturbation theory for the nonlinear Schr¨odinger equation with non-vanishing background. J. Phys. A: Math. Theor. 45 (3), 035202 (2012). https://doi.org/10.1088/1751-8113/45/3/035202

S. Randoux, P. Suret, G. El. Inverse scattering transform analysis of rogue waves using local periodization procedure. Sci. Rep. 6, 29238 (2016). https://doi.org/10.1038/srep29238

T. Kawata, N. Kobayashi, H. Inoue. Soliton solutions of the derivative nonlinear Schr¨odinger equation. J. Phys. Soc. Jpn. 46, 1008 (1979). https://doi.org/10.1143/JPSJ.46.1008

B. Prinari, A.K. Ortiz, C. van der Mee, M. Grabowski. Inverse scattering transform and solitons for square matrix nonlinear Sch¨odinger equation. Stud. Appl. Math. 141 (3), 308 (2018). https://doi.org/10.1111/sapm.12223

D. Qiu, J. He, Y. Zhang, K. Porsezian. The Darboux transformation of the Kundu-Eckhaus equation. Proc. R. Soc. A 471, 2180 (2015). https://doi.org/10.1098/rspa.2015.0236

M. Manas. Darboux transformations for the nonlinear Schr¨odinger equations. J. Phys. A Math. Theor. 29 (23), 7721 (1996). https://doi.org/10.1088/0305-4470/29/23/029

S. Xu, J. He, L. Wang. The Darboux transformation of the derivative nonlinear Schr¨odinger equation. J. Phys. A Math. Theor. 44 (30), 305203 (2011). https://doi.org/10.1088/1751-8113/44/30/305203

H.Q. Zhang, M.Y. Zhang, R.Hu. Darboux transformation and soliton solutions in the parity-time-symmetric nonlocal vector nonlinear Sch¨odinger equation. Appl. Math. Lett. 76, 170 (2018). https://doi.org/10.1016/j.aml.2017.09.002

P. Wang, B. Tian, WJ. Liu et al. N-soliton solutions, B¨acklund transformation and conservation laws for the integro-differential nonlinear Schr¨obinger equation from the isotropic inhomogeneous Heisenberg spin magnetic chain, Comput. Math. and Math. Phys. 54 (4), 727 (2014). https://doi.org/10.1134/S0965542514040125

X. L¨u , H.W. Zhu, X.H. Meng, Z.C. Yang, B. Tian. Soliton solutions and a B¨acklund transformation for a generalized nonlinear Schr¨odinger equation with variable coefficients from optical fiber communications. J. Math. Anal. Appl. 336 (2), 1305 (2007). https://doi.org/10.1016/j.jmaa.2007.03.017

H. Eichhorn. B¨acklund transformation and N-soliton solution for the generalized nonlinear Schr¨odinger equation. Annalen der physic 499 (4), 261 (1987). https://doi.org/10.1002/andp.19874990404

Li-Na Gao, Yao-Yao Zi, Yu-Hang Yin, Wen-Xiu Ma, Xing L¨u. B¨acklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation. Nonlin. Dyn. 89, 2233 (2017). https://doi.org/10.1007/s11071-017-3581-3

A.P. Fordy. Soliton Theory A Survey of Results (Manchester Univ. Press, 1990).

S. Giulio. Solitons and Particles (World Scientific, 1984).

I.M. Uzunov, V.D. Stoev, T.I. Tzoleva. Influence of the initial phase difference between pulses on the N-soliton interaction in trains of unequal solitons in optical fibers. Optics Comm. 97, 307 (1993). https://doi.org/10.1016/0030-4018(93)90494-P

S. Zentner, L. Sumichrast. Computer simulation of the propagation and interaction of soliton sequences in nonlinear optical fibers. J. Electr. Engin. 52, 57 (2001).

P. Balla, S. Buch, G.P. Agrawal. Effect of Raman scattering on soliton interactions in optical fibers. J. Opt. Soc. Am. B 34, 1247 (2017). https://doi.org/10.1364/JOSAB.34.001247

Y. Kodama, M. Romagnoli, S. Wabnitz, M. Midrio. Role of third-order dispersion on soliton instabilities and interactions in optical fibers. Optics Letters 19 (3), 165 (1994). https://doi.org/10.1364/OL.19.000165

A. Biswas, D. Milovic. Optical solitons with fourth order dispersion and dual-power law nonlinearity. Int. J. Nonlin. Sci. 7 (4), 443 (2009).

M. Pich'e, J.F. Cormier, X. Zhu. Bright optical soliton in the presence of fourth-order dispersion. Optics Lett. 21 (12), 845 (1996). https://doi.org/10.1364/OL.21.000845

M.E.M. Elshater, E.M.E. Zayed, A. Al-Nowehy. Solitons and other solutions to nonlinear Schr¨odinger equation with fourth-order dispersion and dual power law nonlinearity using several different techniques. Eur. Phys. J. Plus. 132 (6), 259 (2017). https://doi.org/10.1140/epjp/i2017-11527-4

J. Xingfang, S. Kai. The influence of the fourth-order dispersion coefficient for the information transmission in fiber. In: Web Information Systems and Mining, edited by Z. Gong, X. Luo, J. Chen, J. Lei, F.L. Wang (Springer, 2011), p. 148. https://doi.org/10.1007/978-3-642-23971-7_21

S. Roy, S.K. Bhadra, G.P. Agrawal. Perturbation of higher-order solitons by fourth-order dispersion in optical fibers. Optics Comm. 282 (18), 3798 (2009). https://doi.org/10.1016/j.optcom.2009.06.018

How to Cite
Khelil, K., Saouchi, K., & Bahloul, D. (2020). Effect of Fourth-Order Dispersion on Solitonic Interactions. Ukrainian Journal of Physics, 65(5), 378. https://doi.org/10.15407/ujpe65.5.378
Optics, atoms and molecules