Ukrainian Journal of Physical Optics

2024 Volume 25, Issue 2

ISSN 1609-1833 (Print)


Khalil S. Al-Ghafri1, Edamana V. Krishnan2, Anjan Biswas3,4,5,6, Yakup Yildirim7,8 and Ali Saleh Alshomrani4

1University of Technology and Applied Sciences, P.O. Box 14, Ibri 516, Oman
2Department of Mathematics, Sultan Qaboos University, P.O. Box 36, Al-Khod, Muscat 123, Oman
3Department of Mathematics and Physics, Grambling State University, Grambling, LA 71245-2715, USA
4Mathematical Modeling and Applied Computation (MMAC) Research Group, Center of Modern Mathematical Sciences and their Applications (CMMSA), Department of Mathematics, King Abdulaziz University, Jeddah-21589, Saudi Arabia
5Department of Applied Sciences, Cross-Border Faculty of Humanities, Economics and Engineering, Dunarea de Jos University of Galati, 111 Domneasca Street, 800201 Galati, Romania
6Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa-0204, Pretoria, South Africa
7Department of Computer Engineering, Biruni University, Istanbul, 34010, Turkey
8Department of Mathematics, Near East University, 99138, Nicosia, Cyprus


This study aims to investigate cubic-quartic optical solitons with Kudryashov’s law of self-phase modulation. Thus, the combination of third-order dispersion (3OD) and fourth-order dispersion (4OD) is assumed in the model to ensure the smooth existence of solitons. The study is implemented with the aid of two effective integration schemes known as the improved projective Riccati equations method and the soliton ansatz technique. The soliton solutions are derived based on two physical cases targeting the relation between 3OD and 4OD. In case 3OD is equivalent to fourfold frequency times 4OD, only dark and singular soliton profiles are extracted. However, if the former relation is not achieved, various structures of soliton pulses are generated, including kink-dark, singular, W-shaped, bright, dark, kink, and anti-kink solitons. The physical interpretations of retrieved optical solitons are represented by illustrating the wave behaviors with suitable values of model parameters. The results show that the combination of 3OD and 4OD has a significant effect on the dynamics of soliton propagation.

Keywords: optical solitons, cubic-quartic dispersion, Kudryashov's law, improved projective Riccati equations method, soliton ansatz

UDC: 535.32

    1. Kikuchi, K. (2015). Fundamentals of coherent optical fiber communications. Journal of lightwave technology, 34(1), 157-179.
    2. Zhong, K., Zhou, X., Huo, J., Yu, C., Lu, C., & Lau, A. P. T. (2018). Digital signal processing for short-reach optical communications: A review of current technologies and future trends. Journal of Lightwave Technology, 36(2), 377-400.
    3. Winzer, P. J., Neilson, D. T., & Chraplyvy, A. R. (2018). Fiber-optic transmission and networking: the previous 20 and the next 20 years. Optics express, 26(18), 24190-24239.
    4. Corcoran, B., Tan, M., Xu, X., Boes, A., Wu, J., Nguyen, T. G., Chu, S. T., Little B. E., Morandotti R., Mitchell A. & Moss, D. J. (2020). Ultra-dense optical data transmission over standard fibre with a single chip source. Nature Communications, 11(1), 2568.
    5. Doran, N., & Blow, K. (1983). Solitons in optical communications. IEEE journal of quantum electronics, 19(12), 1883-1888.
    6. Hasegawa, A. (2004). Application of Optical Solitons for Information Transfer in Fibers–A Tutorial Review. Journal of Optics, 33(3), 145-156.
    7. Amiri, I. S., & Ahmad, H. (2015). Optical soliton communication using ultra-Short pulses. Springer Singapore.
    8. Marin-Palomo, P., Kemal, J. N., Karpov, M., Kordts, A., Pfeifle, J., Pfeiffer, M. H., Trocha, P., Wolf, S., Brasch, V., Anderson, M. H., Rosenberger, R., Vijayan, K., Freude, W., Kippenberg, T. J. & Koos, C. (2017). Microresonator-based solitons for massively parallel coherent optical communications. Nature, 546(7657), 274-279.
    9. Liu, J., Lucas, E., Raja, A. S., He, J., Riemensberger, J., Wang, R. N., Karpov, M,. Guo, H., Bouchand, R. & Kippenberg, T. J. (2020). Photonic microwave generation in the X-and K-band using integrated soliton microcombs. Nature Photonics, 14(8), 486-491.
    10. Wang, S., Ma, G., Zhang, X., & Zhu, D. (2022). Dynamic behavior of optical soliton interactions in optical communication systems. Chinese Physics Letters, 39(11), 114202.
    11. Yu, W., Zhou, Q., Mirzazadeh, M., Liu, W., & Biswas, A. (2019). Phase shift, amplification, oscillation and attenuation of solitons in nonlinear optics. Journal of advanced research, 15, 69-76.
    12. Aulia, T. D. F., Astharini, D., Lubis, A., & Syahriar, A. (2019, July). Performance Analysis of Fiber with Solitons Parameters and Fiber Non-Solitons Parameters using OptiSystem. In 2019 6th International Conference on Instrumentation, Control, and Automation (ICA) (pp. 142-146). IEEE.
    13. Yang, G., Wu, F. O., Aviles, H. E. L., & Christodoulides, D. N. (2020). Optical amplification and transmission of attenuated multi-soliton based on spectral characteristics of Akhmediev breather. Optics Communications, 473, 125899.
    14. Raghuraman, P. J., Shree, S. B., & Mani Rajan, M. S. (2021). Soliton control with inhomogeneous dispersion under the influence of tunable external harmonic potential. Waves in Random and Complex Media, 31(3), 474-485.
    15. Konrad, B., Petermann, K., Berger, J., Ludwig, R., Weinert, C. M., Weber, H. G., & Schmauss, B. (2002). Impact of fiber chromatic dispersion in high-speed TDM transmission systems. Journal of lightwave technology, 20(12), 2129-2135.
    16. Yang, A., Liu, X., & Chen, X. (2017). A FrFT based method for measuring chromatic dispersion and SPM in optical fibers. Optical Fiber Technology, 34, 59-64.
    17. Terra, O. (2019). Chromatic dispersion measurement in optical fibers using optoelectronic oscillations. Optics & Laser Technology, 115, 292-297.
    18. Allured, R., & Ashcom, J. B. (2021). Broadband chromatic dispersion in fiber-coupled optical interferometry. Applied Optics, 60(22), 6371-6384.
    19. Dowluru, R. K., & Bhima, P. R. (2011). Influences of third-order dispersion on linear birefringent optical soliton transmission systems. Journal of Optics, 40, 132-142.
    20. Rottenberg, F., Nguyen, T. H., Gorza, S. P., Horlin, F., & Louveaux, J. (2017). Advanced chromatic dispersion compensation in optical fiber FBMC-OQAM systems. IEEE Photonics Journal, 9(6), 1-10.
    21. Nguyen, T. H., Rottenberg, F., Gorza, S. P., Louveaux, J., & Horlin, F. (2017). Efficient chromatic dispersion compensation and carrier phase tracking for optical fiber FBMC/OQAM systems. Journal of Lightwave Technology, 35(14), 2909-2916.
    22. Amiri, I. S., Rashed, A. N. Z., Kader, H. M. A., Al-Awamry, A. A., Abd El-Aziz, I. A., Yupapin, P., & Palai, G. (2020). Optical communication transmission systems improvement based on chromatic and polarization mode dispersion compensation simulation management. Optik, 207, 163853.
    23. Al-Kalbani, K. K., Al-Ghafri, K. S., Krishnan, E. V., & Biswas, A. (2021). Pure-cubic optical solitons by Jacobi’s elliptic function approach. Optik, 243, 167404.
    24. de Sterke, C. M., Runge, A. F., Hudson, D. D., & Blanco-Redondo, A. (2021). Pure-quartic solitons and their generalizations–Theory and experiments. APL Photonics, 6(9), 091101.
    25. Onder, I., Secer, A., Ozisik, M., & Bayram, M. (2022). Obtaining optical soliton solutions of the cubic–quartic Fokas–Lenells equation via three different analytical methods. Optical and Quantum Electronics, 54(12), 786.
    26. Malik, S., Kumar, S., Biswas, A., Yıldırım, Y., Moraru, L., Moldovanu, S., Iticescu, C. & Alshehri, H. M. (2022). Cubic-quartic optical solitons in fiber bragg gratings with dispersive reflectivity having parabolic law of nonlinear refractive index by lie symmetry. Symmetry, 14(11), 2370.
    27. Soltani, M., Triki, H., Azzouzi, F., Sun, Y., Biswas, A., Yıldırım, Y., Alshehri, H. M. & Zhou, Q. (2023). Pure–quartic optical solitons and modulational instability analysis with cubic–quintic nonlinearity. Chaos, Solitons & Fractals, 169, 113212.
    28. Tang, L. (2023). Bifurcations and optical solitons for the coupled nonlinear Schrödinger equation in optical fiber Bragg gratings. Journal of Optics, 52(3), 1388-1398.
    29. Chen, W., Shen, M., Kong, Q., & Wang, Q. (2015). The interaction of dark solitons with competing nonlocal cubic nonlinearities. Journal of Optics, 44, 271-280.
    30. Biswas, A., Ekici, M., Sonmezoglu, A., & Belic, M. R. (2019). Optical solitons in fiber Bragg gratings with dispersive reflectivity for quadratic–cubic nonlinearity by extended trial function method. Optik, 185, 50-56.
    31. Yıldırım, Y., Biswas, A., Guggilla, P., Khan, S., Alshehri, H. M., & Belic, M. R. (2021). Optical solitons in fibre Bragg gratings with third-and fourth-order dispersive reflectivities. Ukrainian Journal of Physical Optic, 22(4), 239-254.
    32. Zhou, Q., Zhong, Y., Triki, H., Sun, Y., Xu, S., Liu, W., & Biswas, A. (2022). Chirped bright and kink solitons in nonlinear optical fibers with weak nonlocality and cubic-quantic-septic nonlinearity. Chinese Physics Letters, 39(4), 044202.
    33. Zhou, Q., Triki, H., Xu, J., Zeng, Z., Liu, W., & Biswas, A. (2022). Perturbation of chirped localized waves in a dual-power law nonlinear medium. Chaos, Solitons & Fractals, 160, 112198.
    34. Triki, H., Sun, Y., Zhou, Q., Biswas, A., Yıldırım, Y., & Alshehri, H. M. (2022). Dark solitary pulses and moving fronts in an optical medium with the higher-order dispersive and nonlinear effects. Chaos, Solitons & Fractals, 164, 112622.
    35. Kopçasız, B., & Yaşar, E. (2023). The investigation of unique optical soliton solutions for dual-mode nonlinear Schrödinger’s equation with new mechanisms. Journal of Optics, 52(3), 1513-1527.
    36. Al-Ghafri, K. S., Sankar, M., Krishnan, E. V., Khan, S., & Biswas, A. (2023). Chirped gap solitons in fiber Bragg gratings with polynomial law of nonlinear refractive index. Journal of the European Optical Society, 19(1), 30.
    37. Han, T., Li, Z., Li, C., & Zhao, L. (2023). Bifurcations, stationary optical solitons and exact solutions for complex Ginzburg–Landau equation with nonlinear chromatic dispersion in non-Kerr law media. Journal of Optics, 52(2), 831-844.
    38. Biswas, A., Sonmezoglu, A., Ekici, M., Alshomrani, A. S., & Belic, M. R. (2019). Optical solitons with Kudryashov’s equation by F-expansion. Optik, 199, 163338.
    39. Kumar, S., Malik, S., Biswas, A., Zhou, Q., Moraru, L., Alzahrani, A. K., & Belic, M. R. (2020). Optical solitons with Kudryashov’s equation by Lie symmetry analysis. Physics of Wave Phenomena, 28, 299-304.
    40. Biswas, A., Vega-Guzmán, J., Ekici, M., Zhou, Q., Triki, H., Alshomrani, A. S., & Belic, M. R. (2020). Optical solitons and conservation laws of Kudryashov's equation using undetermined coefficients. Optik, 202, 163417.
    41. Hu, X., & Yin, Z. (2022). A study of the pulse propagation with a generalized Kudryashov equation. Chaos, Solitons & Fractals, 161, 112379.
    42. Arnous, A. H., Biswas, A., Ekici, M., Alzahrani, A. K., & Belic, M. R. (2021). Optical solitons and conservation laws of Kudryashov's equation with improved modified extended tanh-function. Optik, 225, 165406.
    43. Khuri, S. A., & Wazwaz, A. M. (2023). Optical solitons and traveling wave solutions to Kudryashov’s equation. Optik, 279, 170741.
    44. Kumar, S., & Niwas, M. (2023). Optical soliton solutions and dynamical behaviours of Kudryashov’s equation employing efficient integrating approach. Pramana, 97(3), 98.
    45. Zayed, E. M., & Alngar, M. E. (2021). Optical soliton solutions for the generalized Kudryashov equation of propagation pulse in optical fiber with power nonlinearities by three integration algorithms. Mathematical Methods in the Applied Sciences, 44(1), 315-324.
    46. Zayed, E. M., Alngar, M. E., Biswas, A., Ekici, M. E. H. M. E. T., Alzahrani, A. K., & Belic, M. R. (2020). Chirped and chirp-free optical solitons in fiber Bragg gratings with Kudryashov’s model in presence of dispersive reflectivity. Journal of Communications Technology and Electronics, 65(11), 1267-1287.
    47. Al-Ghafri, K. S., Sankar, M., Krishnan, E. V., Biswas, A., & Asiri, A. (2023). Chirped gap solitons with Kudryashov’s law of self-phase modulation having dispersive reflectivity. Journal of the European Optical Society-Rapid Publications, 19(2), 40.
    48. Zayed, E. M., Shohib, R. M., Biswas, A., Ekici, M., Moraru, L., Alzahrani, A. K., & Belic, M. R. (2020). Optical solitons with differential group delay for Kudryashov’s model by the auxiliary equation mapping method. Chinese Journal of Physics, 67, 631-645.
    49. Biswas, A., Sonmezoglu, A., Ekici, M., Kara, A. H., Alzahrani, A. K., & Belic, M. R. (2021). Cubic–quartic optical solitons and conservation laws with Kudryashov’s law of refractive index by extended trial function. Computational Mathematics and Mathematical Physics, 61(12), 1995-2003.
    50. Kumar, S., & Malik, S. (2021). Cubic-quartic optical solitons with Kudryashov’s law of refractive index by Lie symmetry analysis. Optik, 242, 167308.
    51. Genc, G., Ekici, M., Biswas, A., & Belic, M. R. (2020). Cubic-quartic optical solitons with Kudryashov’s law of refractive index by F-expansions schemes. Results in Physics, 18, 103273.
    52. Al-Ghafri, K. S., Krishnan, E. V., & Biswas, A. (2022). Cubic–quartic optical soliton perturbation and modulation instability analysis in polarization-controlled fibers for Fokas–Lenells equation. Journal of the European Optical Society-Rapid Publications, 18(2), 9.

    Метою цього дослідження є вивчення кубічно-квартичних оптичних солітонів з використанням закону Кудряшова щодо самомодуляції фази. Для забезпечення неперервного існування солітонів в моделі передбачено комбінацію дисперсії третього (3OD) і четвертого (4OD) порядків. Дослідження проводиться за допомогою двох ефективних методів інтегрування, відомих як метод покращених проективних рівнянь Ріккаті та техніки анзацу солітона. Рішення солітонів, отримані на основі двох фізичних випадків, спрямованих на встановлення співідношення між 3OD і 4OD. У випадку, коли 3OD дорівнює чотирикратному значенню хвильового вектора 4OD, отримуються лише темні та сингулярні профілі солітонів. Однак, якщо це співвідношення не виконується,тоді генеруються різні структури солітонних імпульсів, включаючи кінк-темні, сингулярні, W-подібні, яскраві, темні, кінк та антикінк солітони. Фізичні інтерпретації отриманих оптичних солітонів представлені шляхом ілюстрації хвильової поведінки при певних значеннях параметрів моделі. Результати показують, що поєднання 3OD і 4OD має значний вплив на динаміку поширення солітонів.

    Ключові слова: оптичні солітони, кубічно-квартова дисперсія, закон Кудряшова, вдосконалений метод проективних рівнянь Ріккаті, солітонний анзац

© Ukrainian Journal of Physical Optics ©