Analysis of time-fractional soliton solutions for Kudryashov's law with dual nonlocal nonlinearity and refractive index in optical fibers

Salim S. Mahmood; Muhammad Amin S. Murad; Taha Radwan; Karim K. Ahmed; Abeer S. Khalifa; Salim S. Mahmood; Muhammad Amin S. Murad; Taha Radwan; Karim K. Ahmed; Abeer S. Khalifa

doi:10.3934/math.2026097

AIMS Mathematics

2026, Volume 11, Issue 1: 2384-2405. doi: 10.3934/math.2026097

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Research article

Analysis of time-fractional soliton solutions for Kudryashov's law with dual nonlocal nonlinearity and refractive index in optical fibers

1.
Department of Mathematics, Faculty of Science, Soran University, Erbil, Iraq
2.
Department of Mathematics, College of Science, University of Duhok, Duhok, Iraq
3.
Department of Management Information Systems, College of Business and Economics, Qassim University, Buraydah 51452, Saudi Arabia
4.
Department of Mathematics, Saveetha School of Engineering, SIMATS, Saveetha University, Chennai 602105, Tamil Nadu, India
5.
Department of Mathematics, Faculty of Engineering, German International University (GIU), New Administrative Capital, Cairo 11835, Egypt

Received: 13 December 2025 Revised: 29 December 2025 Accepted: 08 January 2026 Published: 26 January 2026
MSC : 35Q51, 35Q55, 37K10

Optical solitons in nonlinear fiber optics represent fundamental wave phenomena that maintain their shape during propagation, yet existing analytical methods often fail to capture the complex dynamics of models incorporating dual nonlocal nonlinearity effects. Recent studies have shown significant research gaps in solving Kudryashov's law with simultaneous refractive index variations and multiple nonlinear terms, particularly when conformable derivatives are involved. In this study, we investigate the analytical solutions of Kudryashov's equation with dual nonlocal nonlinearity and refractive index effects in optical fibers using the improved modified Sardar Sub-equation expansion method (IMSSEM). By applying systematic traveling wave transformations and rigorous mathematical analysis, we derive an extensive collection of novel optical soliton solutions, including rational, hyperbolic, and trigonometric function families. The solutions exhibit diverse wave structures encompassing bright, dark, kink, and W-shaped soliton profiles. Through comprehensive 2D and 3D graphical representations, we demonstrate the wave propagation characteristics under various parameter conditions. Furthermore, we analyze the influence of different conformable derivative orders ($ \tau $) on soliton behavior, revealing significant variations in wave amplitude, width, and propagation velocity. These findings provide crucial insights for understanding nonlinear wave dynamics in optical communication systems, particularly in wavelength-division multiplexing, ultrashort pulse propagation, and nonlinear photonic devices. The derived solutions offer practical applications in designing optical amplifiers, mode-locked lasers, and soliton-based communication protocols. This work contributes significantly to the mathematical physics community by providing a comprehensive analytical framework for solving complex nonlinear optical models and advances the field of integrable systems theory with direct applications to modern fiber optic technology.
- Kudryashov's law,
- optical solitons,
- conformable derivatives,
- nonlocal nonlinearity,
- refractive index,
- partial differential equations,
- IMSSEM,
- fiber optics,
- nonlinear wave equations,
- traveling wave solutions
Citation: Salim S. Mahmood, Muhammad Amin S. Murad, Taha Radwan, Karim K. Ahmed, Abeer S. Khalifa. Analysis of time-fractional soliton solutions for Kudryashov's law with dual nonlocal nonlinearity and refractive index in optical fibers[J]. AIMS Mathematics, 2026, 11(1): 2384-2405. doi: 10.3934/math.2026097

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Abstract

Optical solitons in nonlinear fiber optics represent fundamental wave phenomena that maintain their shape during propagation, yet existing analytical methods often fail to capture the complex dynamics of models incorporating dual nonlocal nonlinearity effects. Recent studies have shown significant research gaps in solving Kudryashov's law with simultaneous refractive index variations and multiple nonlinear terms, particularly when conformable derivatives are involved. In this study, we investigate the analytical solutions of Kudryashov's equation with dual nonlocal nonlinearity and refractive index effects in optical fibers using the improved modified Sardar Sub-equation expansion method (IMSSEM). By applying systematic traveling wave transformations and rigorous mathematical analysis, we derive an extensive collection of novel optical soliton solutions, including rational, hyperbolic, and trigonometric function families. The solutions exhibit diverse wave structures encompassing bright, dark, kink, and W-shaped soliton profiles. Through comprehensive 2D and 3D graphical representations, we demonstrate the wave propagation characteristics under various parameter conditions. Furthermore, we analyze the influence of different conformable derivative orders ($ \tau $) on soliton behavior, revealing significant variations in wave amplitude, width, and propagation velocity. These findings provide crucial insights for understanding nonlinear wave dynamics in optical communication systems, particularly in wavelength-division multiplexing, ultrashort pulse propagation, and nonlinear photonic devices. The derived solutions offer practical applications in designing optical amplifiers, mode-locked lasers, and soliton-based communication protocols. This work contributes significantly to the mathematical physics community by providing a comprehensive analytical framework for solving complex nonlinear optical models and advances the field of integrable systems theory with direct applications to modern fiber optic technology.

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