Refaie Ali Ahmed, Alam Md Nur, Parven Mst Wahida
Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Shebin El Kom 32511, Menoufia, Egypt.
Department of Mathematics, Pabna University of Science and Technology, Pabna, 6600, Bangladesh.
Sci Rep. 2024 Jan 23;14(1):2000. doi: 10.1038/s41598-024-52308-9.
The space-time fractional Fokas-Lenells (STFFL) equation serves as a fundamental mathematical model employed in telecommunications and transmission technology, elucidating the intricate dynamics of nonlinear pulse propagation in optical fibers. This study employs the Sardar sub-equation (SSE) approach within the STFFL equation framework to explore uncharted territories, uncovering a myriad of optical soliton solutions (OSSs) and conducting a thorough analysis of their bifurcations. The discovered OSSs encompass a diverse array, including bright-dark, periodic, multiple bright-dark solitons, and various other types, forming a captivating spectrum. These solutions reveal an intricate interplay among bright-dark solitons, complex periodic sequences, rhythmic breathers, coexistence of multiple bright-dark solitons, alongside intriguing phenomena like kinks, anti-kinks, and dark-bell solitons. This exploration, built upon meticulous literature review, unveils previously undiscovered wave patterns within the dynamic framework of the STFFL equation, significantly expanding the theoretical understanding and paving the way for innovative applications. Utilizing 2D, contour, and 3D diagrams, we illustrate the influence of fractional and temporal parameters on these solutions. Furthermore, comprehensive 2D, 3D, contour, and bifurcation analysis diagrams scrutinize the nonlinear effects inherent in the STFFL equation. Employing a Hamiltonian function (HF) enables detailed phase-plane dynamics analysis, complemented by simulations conducted using Python and MAPLE software. The practical implications of the discovered OSS solutions extend to real-world physical events, underlining the efficacy and applicability of the SSE scheme in solving time-space nonlinear fractional differential equations (TSNLFDEs). Hence, it is crucial to acknowledge the SSE technique as a direct, efficient, and reliable numerical tool, illuminating precise outcomes in nonlinear comparisons.
时空分数阶福卡斯 - 勒内尔斯（STFFL）方程是电信和传输技术中使用的基本数学模型，它阐明了光纤中非线性脉冲传播的复杂动力学。本研究在STFFL方程框架内采用萨达尔子方程（SSE）方法来探索未知领域，发现了大量的光学孤子解（OSSs）并对其分岔进行了深入分析。发现的OSSs种类繁多，包括亮 - 暗、周期、多个亮 - 暗孤子以及其他各种类型，形成了一个引人入胜的光谱。这些解揭示了亮 - 暗孤子、复杂周期序列、有节奏的呼吸子、多个亮 - 暗孤子的共存之间的复杂相互作用，以及诸如扭结、反扭结和暗铃孤子等有趣现象。基于细致的文献综述进行的这项探索，揭示了STFFL方程动态框架内以前未发现的波形模式，显著扩展了理论理解并为创新应用铺平了道路。利用二维、等高线和三维图，我们说明了分数阶和时间参数对这些解的影响。此外，全面的二维、三维、等高线和分岔分析图仔细研究了STFFL方程中固有的非线性效应。使用哈密顿函数（HF）能够进行详细的相平面动力学分析，并辅以使用Python和MAPLE软件进行的模拟。发现的OSS解的实际意义延伸到现实世界的物理事件，强调了SSE方案在求解时空非线性分数阶微分方程（TSNLFDEs）方面的有效性和适用性。因此，至关重要的是要认识到SSE技术是一种直接、高效且可靠的数值工具，在非线性比较中能得出精确结果。
