Ukrainian Journal of Physical Optics


2026 Volume 27, Issue 4


ISSN 1816-2002 (Online), ISSN 1609-1833 (Print)

STOCHASTIC (3+1)-DIMENSIONAL SPATIO-TEMPORAL SOLITONS IN A GENERALIZED EFFECTIVE FIBER BRAGG-GRATING ENVELOPE MODEL

Metwally, A.S.M., Alngar, M.E.M., Shohib, R.M.A., Shirsath, S.E. and Biswas, A.

Author Information
1Metwally, A.S.M. SCOPUS logo, 2Alngar, M.E.M. ORCID logo SCOPUS logo, 3Shohib, R.M.A. SCOPUS logo, 4Shirsath, S.E. ORCID logo SCOPUS logo, 5,6,7,8,9,*Biswas, A. SCOPUS logo
1Center of Excellence for Research in Science and Mathematics Education Development, DSR, King Saud University, P.O. Box 11451, Riyadh, Saudi Arabia
2Mathematics Education Program, Faculty of Education and Arts, Sohar University, Sohar 311, Oman
3Basic Science Department, Faculty of Economics and Business Administration, Innovation University, Sharqia, Egypt
4School of Materials Science and Engineering, University of New South Wales, Sydney, NSW, 2052, Australia
5Department of Mathematics & Physics, Grambling State University, Grambling, LA 71245—2715, USA
6Department of Mathematics, Faculty of Science, Karadeniz Technical University, Trabzon—61080, Türkiye
7Department of Physics and Electronics, Khazar University, Baku, AZ—1096, Azerbaijan
8Applied Science Research Center, Applied Science Private University, Amman—11937, Jordan
9Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Medunsa–0204, Pretoria, South Africa
*Corresponding author: biswas.anjan@gmail.com

ABSTRACT

This paper investigates a (3+1)-dimensional stochastic coupled nonlinear Schrödinger (NLS) system interpreted as a generalized effective envelope model for Fiber Bragg grating (FBG) media. The mathematical structure of the coupled stochastic framework was originally introduced for birefringent optical fibers, while this study extends it to the FBG setting to describe interacting optical field components under the combined influence of chromatic and spatiotemporal dispersion, self- and cross-phase modulation, intermodal coupling, four-wave mixing, and multiplicative white-noise perturbations. By applying the modified Sub-ODE method, we obtain several exact analytical solution families, including bright, dark, and singular solitons, periodic waveforms, and solutions expressed in terms of Jacobi elliptic and Weierstrass functions. The corresponding parameter and admissibility conditions are derived explicitly, providing a unified analytical description of the resulting stochastic nonlinear wave structures. Representative solutions illustrate the influence of the stochastic parameter on the phase characteristics and wave profiles within the adopted effective FBG framework. These results extend the analytical treatment of stochastic higher-dimensional coupled-wave systems and provide further insight into nonlinear, dispersive, and noise-driven dynamics in generalized grating-fiber media

Keywords: stochastic nonlinear Schrodinger equation, fiber Bragg grating, bright and dark solitons, elliptic function solutions, noise-driven dynamics

UDC: 535.18

    1. Eggleton, B. J., Slusher, R. E., de Sterke, C. M., Krug, P. A., & Sipe, J. E. (1996). Bragg grating solitons. Physical Review Letters, 76, 1627.
      doi:10.1103/PhysRevLett.76.1627
    2. Zayed, E. M., Alngar, M. E., Shohib, R. M., Biswas, A., Yıldırım, Y., Moraru, L., & Iticescu, C. (2022). Highly dispersive optical solitons in fiber Bragg gratings with quadratic-cubic nonlinearity. Electronics, 12, 125.
      doi:10.3390/electronics12010125
    3. Zayed, E. M., Alngar, M. E., Shohib, R. M., Biswas, A., Yıldırım, Y., Khan, S., & Iticescu, C. (2022). Highly dispersive optical solitons in fiber Bragg gratings with Kerr law of nonlinear refractive index. Mathematics, 10, 2968.
      doi:10.3390/math10162968
    4. 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-Rapid Publications, 19, 30.
      doi:10.1051/jeos/2023025
    5. Ghayad, M. S., Badra, N. M., Ahmed, H. M., Rabie, W. B., Mirzazadeh, M., & Hashemi, M. S. (2024). Highly dispersive optical solitons in fiber Bragg gratings with cubic-quadratic nonlinearity using improved modified extended tanh-function method. Optical and Quantum Electronics, 56, 1184.
      doi:10.1007/s11082-024-07064-5
    6. Meng, L., Li, J., & Tao, J. (2015). Blow-up for the stochastic nonlinear Schrödinger equations with quadratic potential and additive noise. Boundary Value Problems, 2015, 132.
      doi:10.1186/s13661-015-0394-5
    7. Zayed, E. M., Arnous, A. H., Seçer, A., Ozisik, M., Bayram, M., Shah, N. A., & Chung, J. D. (2023). Highly dispersive optical solitons in fiber Bragg gratings for stochastic Lakshmanan-Porsezian-Daniel equation with spatio-temporal dispersion and multiplicative white noise. Results in Physics, 55, 107177.
      doi:10.1016/j.rinp.2023.107177
    8. Zayed, E. M., Alngar, M. E., Shohib, R. M., Biswas, A., Yıldırım, Y., Moraru, L., & Georgescu, P. L. (2023). Dispersive optical solitons with differential group delay having multiplicative white noise by Itô calculus. Electronics, 12, 634.
      doi:10.3390/electronics12030634
    9. Lashkin, V. M. (2024). Dynamics of multidimensional fundamental and vortex solitons in random media. Physical Review E, 109, 064216.
      doi:10.1103/PhysRevE.109.064216
    10. Zayed, E. M., Gepreel, K. A., & El-Horbaty, M. (2023). Highly dispersive optical solitons in fiber Bragg gratings with stochastic perturbed Fokas-Lenells model. Optik, 286, 170975.
      doi:10.1016/j.ijleo.2023.170975
    11. Tang, L. (2025). Qualitative analysis, chaotic behavior and optical solitons perturbation for the coupled Fokas-Lenells system in birefringent fibers. Qualitative Theory of Dynamical Systems, 24, 59.
      doi:10.1007/s12346-024-01213-7
    12. Jahan, M., Ullah, M., Rahman, Z., & Akter, R. (2025). Novel dynamics of the Fokas-Lenells model in birefringent fibers applying different integration algorithms. International Journal of Mathematical and Computational Engineering, 3, 1-12.
      doi:10.2478/ijmce-2025-0001
    13. Zayed, E. M., El-Horbaty, M., Alngar, M. E., & El-Shater, M. (2022). Dispersive optical solitons for stochastic Fokas-Lenells equation with multiplicative white noise. Eng, 3, 523-540.
      doi:10.3390/eng3040037
    14. Lashkin, V. M. (2021). Perturbation theory for solitons of the Fokas-Lenells equation: Inverse scattering transform approach. Physical Review E, 103, 042203.
      doi:10.1103/PhysRevE.103.042203
    15. Ai, L., & Xu, J. (2019). On a Riemann-Hilbert problem for the Fokas-Lenells equation. Applied Mathematics Letters, 87, 57-63.
      doi:10.1016/j.aml.2018.07.027
    16. Cheng, Q., & Fan, E. (2023). The Fokas-Lenells equation on the line: Global well-posedness with solitons. Journal of Differential Equations, 366, 320-344.
      doi:10.1016/j.jde.2023.04.016
    17. Zhang, Y., Qiu, D., & He, J. (2023). Explicit nth-order solutions of Fokas-Lenells equation based on revised Riemann-Hilbert approach. Journal of Mathematical Physics, 64.
      doi:10.1063/5.0148086
    18. He, J., Xu, S., & Porsezian, K. (2012). Rogue waves of the Fokas-Lenells equation. Journal of the Physical Society of Japan, 81, 124007.
      doi:10.1143/JPSJ.81.124007
    19. Zhao, Y., & Fan, E. (2021). Inverse scattering transformation for the Fokas-Lenells equation with nonzero boundary conditions. Journal of Nonlinear Mathematical Physics, 28, 38-52.
      doi:10.2991/jnmp.k.200922.003
    20. Ullah, M. S., Roshid, H. O., & Ali, M. Z. (2023). New wave behaviors of the Fokas-Lenells model using three integration techniques. PLoS ONE, 18, e0291071.
      doi:10.1371/journal.pone.0291071
    21. Li, Z. L. (2010). Periodic wave solutions of a generalized KdV-mKdV equation with higher-order nonlinear terms. Zeitschrift für Naturforschung A, 65, 649-657.
      doi:10.1515/zna-2010-8-905
    22. Metwally, A. S., Alngar, M. E., Shohib, R. M., Shah, S. A., & Biswas, A. (2026). Highly dispersive optical solitons with differential group delay and multiplicative white noise having Kerr law of self-phase modulation. Ukrainian Journal of Physical Optics, 27(3), 03196-03224.
      doi:10.3116/16091833/Ukr.J.Phys.Opt.2026.03196
    23. Li, S., Qian, S., Chen, H., Song, J., & Cao, A. (2021). An extended nonlinear Schrödinger equation for water waves with linear shear flow, wind, and dissipation. AIP Advances, 11.
      doi:10.1063/5.0033921
    24. Zabolotnykh, A. A. (2023). Nonlinear Schrödinger equation for a two-dimensional plasma: Solitons, breathers, and plane wave stability. Physical Review B, 108, 115424.
      doi:10.1103/PhysRevB.108.115424
    25. Kengne, E., Liu, W. M., & Malomed, B. A. (2021). Spatiotemporal engineering of matter-wave solitons in Bose-Einstein condensates. Physics Reports, 899, 1-62.
      doi:10.1016/j.physrep.2020.11.001
    26. Hasan, W. M., Ahmed, H. M., Ahmed, A. M., Rezk, H. M., & Rabie, W. B. (2025). Exploring highly dispersive optical solitons and modulation instability in nonlinear Schrödinger equations with nonlocal self-phase modulation and polarization dispersion. Scientific Reports, 15, 27070.
      doi:10.1038/s41598-025-09710-8
    27. Kolkowski, R., Hakala, T. K., Shevchenko, A., & Huttunen, M. J. (2023). Nonlinear nonlocal metasurfaces. Applied Physics Letters, 122.
      doi:10.1063/5.0140483
    28. Sharma, M., Tal, M., McDonnell, C., & Ellenbogen, T. (2023). Electrically and all-optically switchable nonlocal nonlinear metasurfaces. Science Advances, 9, eadh2353.
      doi:10.1126/sciadv.adh2353
    29. Jiang, H., Cai, Y., & Han, Z. (2024). Enhanced high-harmonic generation with spectral tunability in nonlocal metasurfaces enabled by quasiguided modes. Physical Review B, 110, 245424.
      doi:10.1103/PhysRevB.110.245424
    30. Zhang, D. (2023). Stochastic nonlinear Schrödinger equations in the defocusing mass and energy critical cases. Annals of Applied Probability, 33, 3652-3705.
      doi:10.1214/22-AAP1903
    31. Khan, A., Muhammad, J., Younas, U., Thinakaran, R., Abdeljawad, T., & Alqudah, M. A. (2025). Investigating the stochastic higher dimensional nonlinear Schrödinger equation for telecommunication engineering. Scientific Reports, 15, 27309.
      doi:10.1038/s41598-025-12747-4
    32. Cui, J., Liu, S., & Zhou, H. (2023). Optimal control for stochastic nonlinear Schrödinger equation on graph. SIAM Journal on Control and Optimization, 61, 2021-2042.
      doi:10.1137/22M1524175
    33. Kaur, L., Arnous, A. H., Murad, M. A. S., & Biswas, A. (2026). Dispersive optical soliton perturbation with Schrödinger-Hirota equation by Lie symmetry. Ukrainian Journal of Physical Optics, 27(3), 03239.
      doi:10.3116/16091833/Ukr.J.Phys.Opt.2026.03238
    34. Arnous, A. H., Murad, M. A. S., Yıldırım, Y., & Biswas, A. (2026). Quiescent vector optical solitons with nonlinear dispersion. Ukrainian Journal of Physical Optics, 27(4), 04054-04076.
      doi:10.3116/16091833/Ukr.J.Phys.Opt.2026.04054
    35. Zayed, E. M., Ahmed, M. S., Arnous, A. H., & Yıldırım, Y. (2025). Novel soliton solutions of the (3+1)-dimensional stochastic nonlinear Schrödinger equation in birefringent fibers. Chaos, Solitons & Fractals, 194, 116152.
      doi:10.1016/j.chaos.2025.116152

    У цій статті досліджується нова (3+1)-вимірна зв’язана нелінійна система рівнянь Шредінґера (NLS), яка враховує стохастичні ефекти білого шуму у волоконних брегівських ґратках (FBG). Модель описує взаємодію двох ортогонально поляризованих комплексних функцій обвідної, які зазнають впливу хроматичної та просторово-часової дисперсії, само- та крос-фазової модуляції, інтермодового зв’язку та випадкових флуктуацій. Застосовуючи модифікований метод Sub-ODE, отримано широкий клас точних аналітичних розв’язків, зокрема яскраві та темні солітони, сингулярні розв’язки й періодичні хвильові розв’язки. Також отримано явні розв’язки, виражені через еліптичні функції Якобі та Вейєрштраса, що забезпечує єдиний опис нелінійних хвильових структур, зумовлених стохастичними ефектами. Фізично результати демонструють, як інтенсивність шуму впливає на амплітуду солітона, його локалізацію та фазову модуляцію, тим самим пов’язуючи детерміновані та стохастичні режими. Теоретичні результати мають значення для систем оптичного зв’язку, конденсатів Бозе–Ейнштейна, плазмових середовищ, а також нелінійних біологічних і хімічних систем, у яких випадкові флуктуації відіграють важливу роль.

    Ключові слова: стохастичне нелінійне рівняння Шредінгера, волоконні бреґівські ґратки, яскраві та темні солітони, розв'язки еліптичних функцій, динаміка, керована шумом

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