时空分数阶Sasa-Satsuma方程的行波解和分岔分析

doi:10.16088/j.issn.1001-6600.2024062102

摘要/Abstract

摘要： 本文研究时空分数阶Sasa-Satsuma方程行波解的分岔及其动力学行为。首先对时空分数阶Sasa-Satsuma方程进行分数阶复变换,将其转化为等价的常微分系统,推导出对应的平面动力系统;然后对平面动力系统参数不同取值进行讨论,获得对应相图;再根据系统分岔情况,求解时空分数阶Sasa-Satsuma方程不同轨线各类行波解的精确表达式;最后给出部分解的三维图。

关键词: 时空分数阶Sasa-Satsuma方程, 行波解, 动力系统, 分岔

Abstract: In order to study the bifurcation and dynamical behavior of traveling wave solutions of the time-space fractional Sasa-Satsuma equation, the fractional-order complex transformation of the time-space fractional Sasa-Satsuma equation is performed to transform them into an equivalent ordinary differential system,the corresponding plane dynamic system is derived,and the corresponding phase diagram is obtained by discussing the different values of the parameters of the plane dynamic system. According to the bifurcation of the system, the exact expressions of various traveling wave solutions for the time-space fractional Sasa-Satsuma equations with different trajectories are obtained.

Key words: time-space fractional, Sasa-Satsuma equation, traveling wave solutions, dynamical system, bifurcation

中图分类号: O175.29

徐健淞, 孙峪怀. 时空分数阶Sasa-Satsuma方程的行波解和分岔分析[J]. 广西师范大学学报（自然科学版）, 2025, 43(4): 120-128.

XU Jiansong, SUN Yuhuai. Bifurcation of Traveling Wave Solutions for the Time-Space Fractional Sasa-Satsuma Equation[J]. Journal of Guangxi Normal University(Natural Science Edition), 2025, 43(4): 120-128.

参考文献

[1] IQBAL I, REHMAN H U, ASHRAF H, et al. Soliton unveilings in optical fiber transmission: Examining soliton structures through the Sasa-Satsuma equation[J]. Results in Physics, 2024, 60: 107648. DOI: 10.1016/j.rinp.2024.107648.
[2] GHOSH P K. Conformal symmetry and the nonlinear Schrödinger equation[J]. Physical Review A, 2001, 65(1): 012103. DOI: 10.1103/PhysRevA.65.012103.
[3] CHABCHOUB A, KIBLER B, FINOT C, et al. The nonlinear Schrödinger equation and the propagation of weakly nonlinear waves in optical fibers and on the water surface[J]. Annals of Physics, 2015, 361: 490-500. DOI: 10.1016/j.aop.2015.07.003.
[4] WANG K J, ZHANG P L. Investigation of the periodic solution of the time-space fractional Sasa-Satsuma equation arising in the monomode optical fibers[J]. Europhysics Letters, 2022, 137(6): 62001. DOI: 10.1209/0295-5075/ac2a62.
[5] MIHALACHE D, TORNER L, MOLDOVEANU F, et al. Inverse-scattering approach to femtosecond solitons in monomode optical fibers[J]. Physical Review E, 1993,48(6): 4699-4709. DOI: 10.1103/physreve.48.4699.
[6] KODAMA Y. Optical solitons in a monomode fiber[J]. Journal of Statistical Physics, 1985, 39(5): 597-614. DOI: 10.1007/BF01008354.
[7] AKRAM G, SADAF M, ARSHED S, et al. Optical soliton solutions of fractional Sasa-Satsuma equation with beta and conformable derivatives[J]. Optical and Quantum Electronics, 2022, 54(11): 741. DOI: 10.1007/s11082-022-04153-1.
[8] LI C C, CHEN L W, LI G H. Optical solitons of space-time fractional Sasa-Satsuma equation by F-expansion method[J]. Optik, 2020, 224: 165527. DOI: 10.1016/j.ijleo.2020.165527.
[9] YÉPEZ-MARTÍNEZ H, REZAZADEH H, INC M, et al. Optical solitons of the fractional nonlinear Sasa-Satsuma equation with third-order dispersion and with Kerr nonlinearity law in modulation instability[J]. Optical and Quantum Electronics, 2022, 54(12): 804. DOI: 10.1007/s11082-022-04207-4.
[10] RAHEEL M, ZAFAR A, INC M, et al. Optical solitons to time-fractional Sasa-Satsuma higher-order non-linear Schrödinger equation via three analytical techniques[J]. Optical and Quantum Electronics, 2023, 55(4): 307. DOI: 10.1007/s11082-023-04565-7.
[11] WANG K J, LIU J H. Periodic solution of the time-space fractional Sasa-Satsuma equation in the monomode optical fibers by the energy balance theory[J]. Europhysics Letters, 2022, 138(2): 25002. DOI: 10.1209/0295-5075/ac5c78.
[12] MURAD M A S, ISMAEL H F, SULAIMAN T A, et al. Higher-order time-fractional Sasa-Satsuma equation: various optical soliton solutions in optical fiber[J]. Results in Physics, 2023, 55: 107162. DOI: 10.1016/j.rinp.2023.107162.
[13] 何进春,陈宗蕴,黄念宁.求解DNLS方程的反散射法的基本问题[J].物理学报,2009,58(9):6063-6067. DOI: 10.3321/j.issn:1000-3290.2009.09.028.
[14] 周建军,洪宝剑,卢殿臣.耦合Schrödinger-Boussinesq方程组的显式精确解[J].广西师范大学学报(自然科学版),2013,31(1):11-15. DOI: 10.16088/j.issn.1001-6600.2013.01.016.
[15] FAROOQ A, KHAN M I, NISAR K S, et al. A detailed analysis of the improved modified Korteweg-de Vries equation via the Jacobi elliptic function expansion method and the application of truncated M-fractional derivatives[J]. Results in Physics, 2024, 59: 107604. DOI: 10.1016/j.rinp.2024.107604.
[16] 曹娜,徐丽阳,尹晓军.变系数F展开法求解非线性Schrödinger方程的精确解[J].应用数学,2023,36(1):161-169. DOI: 10.13642/j.cnki.42-1184/o1.2023.01.009.
[17] BORG M, BADRA N M, AHMED H M, et al. Solitons behavior of Sasa-Satsuma equation in birefringent fibers with Kerr law nonlinearity using extended F-expansion method[J]. Ain Shams Engineering Journal, 2024, 15(1): 102290. DOI: 10.1016/j.asej.2023.102290.
[18] 王丽真,沈翔.利用首次积分法求解一致时空分数阶微分方程[J].西北大学学报(自然科学版),2022,52(2):279-287. DOI: 10.16152/j.cnki.xdxbzr.2022-02-014.
[19] LIU H Z. A modification to the first integral method and its applications[J]. Applied Mathematics and Computation, 2022,419: 126855. DOI: 10.1016/j.amc.2021.126855.
[20] KHATER M M A, LU D C, ATTIA R A M. Dispersive long wave of nonlinear fractional Wu-Zhang system via a modified auxiliary equation method[J]. AIP Advances, 2019, 9(2): 025003. DOI: 10.1063/1.5087647.
[21] REHMAN H U, HASSAN M U, SALEEM M S, et al. Soliton solutions of Zakhrov equation in ionized plasma using new extended direct algebraic method[J]. Results in Physics, 2023,46: 106325. DOI: 10.1016/j.rinp.2023.106325.
[22] 胡艳,孙峪怀.应用多项式完全判别系统方法求解时空分数阶复Ginzburg-Landau方程[J].应用数学和力学,2021,42(8):874-880. DOI: 10.21656/1000-0887.410392.
[23] LI Z, FAN W J, MIAO F. Chaotic pattern, phase portrait, sensitivity and optical soliton solutions of coupled conformable fractional Fokas-Lenells equation with spatio-temporal dispersion in birefringent fibers[J]. Results in Physics, 2023,47: 106386. DOI: 10.1016/j.rinp.2023.106386.
[24] TANG L. Bifurcation analysis and optical soliton solutions for the fractional complex Ginzburg-Landau equation in communication systems[J]. Optik, 2023, 276: 170639. DOI: 10.1016/j.ijleo.2023.170639.
[25] AYDIN M E. Effect of local fractional derivatives on Riemann curvature tensor[J]. Examples and Counterexamples, 2024, 5: 100134. DOI: 10.1016/j.exco.2023.100134.
[26] ZHANG Y N, PANG J. He’s homotopy perturbation method and fractional complex transform for analysis time fractional Fornberg-Whitham equation[J]. Sound and Vibration, 2021, 55(4): 295-303. DOI: 10.32604/sv.2021.014445.
[27] ABDELHAQ L, SYAM S M, SYAM M I. An efficient numerical method for two-dimensional fractional integro-differential equations with modified Atangana-Baleanu fractional derivative using operational matrix approach[J]. Partial Differential Equations in Applied Mathematics, 2024, 11: 100824. DOI: 10.1016/j.padiff.2024.100824.
[28] ORTIGUEIRA M D, RODRÍGUEZ-GERMÁ L, TRUJILLO J J. Complex Complex Grünwald-Letnikov, Liouville, Riemann-Liouville, and Caputo derivatives for analytic functions[J]. Communications in Nonlinear Science and Numerical Simulation, 2011, 16(11): 4174-4182. DOI: 10.1016/j.cnsns.2011.02.022.
[29] 吴强,黄建华.分数阶微积分[M].北京:清华大学出版社,2016:185.
[30] KHALIL R, AL HORANI M, YOUSEF A, et al. A new definition of fractional derivative[J]. Journal of Computational and Applied Mathematics, 2014, 264: 65-70. DOI: 10.1016/j.cam.2014.01.002.
[31] YAN F, LIU H H, LIU Z R. The bifurcation and exact travelling wave solutions for the modified Benjamin-Bona-Mahoney (mBBM) equation[J]. Communications in Nonlinear Science and Numerical Simulation, 2012, 17(7): 2824-2832. DOI: 10.1016/j.cnsns.2011.11.014.
[32] 张卫国,杨萌.耦合DSW方程的周期波解与孤波解[J].应用数学学报,2020,43(5):792-820.

Metrics

Viewed

Full text

Abstract

Cited

Shared

Discussed