Exact Solutions to Space-Time Fractional Fokas-Lenells Equations With Parameters
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摘要:
利用多项式完全判别系统法求得非线性光学中带参数时空分数阶Fokas-Lenells方程在一般情况下的精确解，包括有理函数解、周期解、孤波解、Jacobi椭圆函数解和双曲函数解等，绘制了精确解的相关图像，并由此分析了参数对解的结构的影响。
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关键词:
- Fokas-Lenells方程 /
- 多项式判别系统 /
- 精确解
Abstract:The exact solutions to the space-time fractional Fokas-Lenells equations with parameters in nonlinear optics were obtained by means of the complete discrimination system for polynomial method, including rational function solutions, periodic solutions, solitary wave solutions, Jacobi elliptic function solutions and hyperbolic function solutions. The relevant graphs of the exact solutions were drawn, and the influence of parameters on the structure of the solution was analyzed.
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Key words:
- Fokas-Lenells equation /
- polynomial discriminant system /
- exact solution
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图 1
${\left| {{\varOmega _1}(x,t)} \right|^2}$ 取不同分数阶导数值时的三维图，图1(b)对应的等高线图，以及$ t = 3 $ 时${\left| {{\varOmega _1}(x,t)} \right|^2}$ 关于$ x $ 的截面图：(a)$\alpha=1/2, \beta=1/3,{\left| {{\varOmega _1}(x,t)} \right|^2}$ 的三维图； (b)$\alpha=1/3, \beta=1/3,{\left| {{\varOmega _1}(x,t)} \right|^2} $ 的三维图； (c) 图1(b)的等高线图； (d) 当$ t = 3 $ 时${\left| {{\varOmega _1}(x,t)} \right|^2}$ 关于$ x $ 的截面图注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同。
Figure 1. The 3D graph of
${\left| {{\varOmega _1}(x,t)} \right|^2}$ with different values of the fractional derivative, the contour plot of fig.1(b) and the sectional view of${\left| {{\varOmega _1}(x,t)} \right|^2}$ against$ x $ with$ t = 3 $ : (a) the graphic model of$\left| {{\varOmega _1}(x,t)} \right|^2, \alpha=1/2, \beta=1/3$ ; (b) the graphic model of$\left| {{\varOmega _1}(x,t)} \right|^2, \alpha=1/3, \beta=1/3$ ; (c) the contour plot of fig.1(b); (d) the sectional view of${\left| {{\varOmega _1}(x,t)} \right|^2}$ against$ x $ when t=3
图 2
${\left| {{\varOmega _2}(x,t)} \right|^2}$ 取不同分数阶导数值时的三维图，图2(b)对应的等高线图，以及$ t = 3 $ 时${\left| {{\varOmega _2}(x,t)} \right|^2}$ 关于$ x $ 的截面图：(a)$\alpha=1/2, \beta=1/3,{\left| {{\varOmega _2}(x,t)} \right|^2}$ 的三维图； (b)$\alpha=1/3, \beta=1/3,{\left| {{\varOmega _2}(x,t)} \right|^2} $ 的三维图； (c) 图2(b)的等高线图； (d) 当$ t = 3 $ 时${\left| {{\varOmega _2}(x,t)} \right|^2}$ 关于$ x $ 的截面图Figure 2. The 3D graph of
${\left| {{\varOmega _2}(x,t)} \right|^2}$ with different values of the fractional derivative, the contour plot of fig.2(b) and the sectional view of${\left| {{\varOmega _2}(x,t)} \right|^2}$ against$ x $ with$ t = 3 $ : (a) the graphic model of${\left| {{\varOmega _2}(x,t)} \right|^2}, \alpha=1/2, \beta=1/3$ ; (b) the graphic model of${\left| {{\varOmega _2}(x,t)} \right|^2} , \alpha=1/3, \beta=1/3$ ; (c) the contour plot of fig.2(b); (d) the sectional view of${\left| {{\varOmega _2}(x,t)} \right|^2}$ against$ x $ when t=3
图 3
${\left| {{\varOmega _3}(x,t)} \right|^2}$ 取不同分数阶导数值时的三维图，图3(b)对应的等高线图，以及$ t = 3 $ 时${\left| {{\varOmega _3}(x,t)} \right|^2}$ 关于$ x $ 的截面图：(a)$\alpha=1/2, \beta=1/3, {\left| {{\varOmega _3}(x,t)} \right|^2}$ 的三维图； (b)$\alpha=1/3, \beta=1/3,{\left| {{\varOmega _3}(x,t)} \right|^2} $ 的三维图； (c) 图3(b)的等高线图； (d) 当$ t = 3 $ 时${\left| {{\varOmega _3}(x,t)} \right|^2}$ 关于$ x $ 的截面图Figure 3. The 3D graph of
${\left| {{\varOmega _3}(x,t)} \right|^2}$ with different values of the fractional derivative, the contour map of fig. 3(b) and the sectional view of${\left| {{\varOmega _3}(x,t)} \right|^2}$ against$ x $ with$ t = 3 $ : (a) the graphic model of${\left| {{\varOmega _3}(x,t)} \right|^2},\alpha=1/2, \beta=1/3$ ; (b) the graphic model of${\left| {{\varOmega _3}(x,t)} \right|^2},\alpha=1/3, \beta=1/3$ ; (c) the contour plot of fig. 3(b); (d) the sectional view of${\left| {{\varOmega _3}(x,t)} \right|^2}$ against$ x $ when t=3
图 4
${\left| {{\varOmega _4}(x,t)} \right|^2}$ 取不同分数阶导数值时的三维图，图4(b)对应的等高线图，以及$ t = 3 $ 时${\left| {{\varOmega _4}(x,t)} \right|^2}$ 关于$ x $ 的截面图：(a)$\alpha=1/2, \beta=1/3,{\left| {{\varOmega _4}(x,t)} \right|^2}$ 的三维图； (b)$\alpha=1/3, \beta=1/3,{\left| {{\varOmega _4}(x,t)} \right|^2} $ 的三维图； (c) 图4(b)的等高线图； (d) 当$ t = 3 $ 时${\left| {{\varOmega _4}(x,t)} \right|^2}$ 关于$ x $ 的截面图Figure 4. The 3D graph of
${\left| {{{{\varOmega }}_4}(x,t)} \right|^2}$ with different values of the fractional derivative, the contour plot of fig.4(b) and the sectional view of${\left| {{\varOmega _4}(x,t)} \right|^2}$ against$ x $ with$ t = 3 $ : (a) the graphic model of${\left| {{\varOmega _4}(x,t)} \right|^2}, \alpha=1/2, \beta=1/3$ ; (b) the graphic model of${\left| {{\varOmega _4}(x,t)} \right|^2} , \alpha=1/3, \beta=1/3$ ; (c) the contour plot of fig.4(b); (d) the sectional view of${\left| {{\varOmega _4}(x,t)} \right|^2}$ against$ x $ when t=3
图 5
${\left| {{\varOmega _5}(x,t)} \right|^2}$ 取不同分数阶导数值时的三维图，图5(b)对应的等高线图，以及$ t = 3 $ 时${\left| {{\varOmega _5}(x,t)} \right|^2}$ 关于$ x $ 的截面图：(a)$\alpha=1/2, \beta=1/3,{\left| {{\varOmega _5}(x,t)} \right|^2}$ 的三维图； (b)$\alpha=1/3, \beta=1/3,{\left| {{\varOmega _5}(x,t)} \right|^2} $ 的三维图； (c) 图5(b)的等高线图； (d) 当$ t = 3 $ 时${\left| {{\varOmega _5}(x,t)} \right|^2}$ 关于$ x $ 的截面图Figure 5. The 3D graph of
${\left| {{\varOmega _5}(x,t)} \right|^2}$ with different values of the fractional derivative, the contour plot of fig. 5(b) and the sectional view of${\left| {{{{\varOmega }}_5}(x,t)} \right|^2}$ against$ x $ with$ t = 3 $ : (a) the graphic model of${\left| {{\varOmega _5}(x,t)} \right|^2} ,\alpha=1/2, \beta=1/3$ ; (b) the graphic model of${\left| {{\varOmega _5}(x,t)} \right|^2},\alpha=1/3, \beta=1/3$ ; (c) the contour plot of fig. 5(b); (d) the sectional view of${\left| {{{{\varOmega }}_5}(x,t)} \right|^2}$ against$ x $ when t=3
图 6
${\left| {{\varOmega _6}(x,t)} \right|^2}$ 取不同分数阶导数值时的三维图，图6(b)对应的等高线图，以及$ t = 3 $ 时${\left| {{\varOmega _6}(x,t)} \right|^2}$ 关于$ x $ 的截面图：(a)$\alpha=1/2, \beta=1/3,{\left| {{\varOmega _6}(x,t)} \right|^2}$ 的三维图； (b)$\alpha=1/3, \beta=1/3,{\left| {{\varOmega _6}(x,t)} \right|^2} $ 的三维图； (c) 图6(b)的等高线图；(d) 当$ t = 3 $ 时${\left| {{\varOmega _6}(x,t)} \right|^2}$ 关于$ x $ 的截面图Figure 6. The 3D graph of
${\left| {{\varOmega _6}(x,t)} \right|^2}$ with different values of the fractional derivative, the contour plot of fig. 6(b) and the sectional view of${\left| {{\varOmega _6}(x,t)} \right|^2}$ against$ x $ with$ t = 3 $ : (a) the graphic model of${\left| {{\varOmega _6}(x,t)} \right|^2},\alpha=1/2, \beta=1/3$ ; (b) the graphic model of${\left| {{\varOmega _6}(x,t)} \right|^2} ,\alpha=1/3, \beta=1/3$ ; (c) the contour plot of fig. 6(b); (d) the sectional view of${\left| {{\varOmega _6}(x,t)} \right|^2}$ against$ x $ when t=3
图 7
${\left| {{\varOmega _7}(x,t)} \right|^2}$ 取不同分数阶导数值时的三维图，图7(b)对应的等高线图，以及$ t = 3 $ 时${\left| {{\varOmega _7}(x,t)} \right|^2}$ 关于$ x $ 的截面图：(a)$\alpha=1/2, \beta=1/3, {\left| {{\varOmega _7}(x,t)} \right|^2}$ 的三维图； (b)$\alpha=1/3, \beta=1/3,{\left| {{\varOmega _7}(x,t)} \right|^2} $ 的三维图； (c) 图7(b)的等高线图； (d) 当$ t = 3 $ 时${\left| {{\varOmega _7}(x,t)} \right|^2}$ 关于$ x $ 的截面图Figure 7. The 3D graph of
${\left| {{\varOmega _7}(x,t)} \right|^2}$ with different values of the fractional derivative, the contour plot of fig.7(b) and the sectional view of${\left| {{\varOmega _7}(x,t)} \right|^2}$ against$ x $ with$ t = 3 $ : (a) the graphic model of${\left| {{\varOmega _7}(x,t)} \right|^2}, \alpha=1/2, \beta=1/3$ ; (b) the graphic model of${\left| {{\varOmega _7}(x,t)} \right|^2}, \alpha=1/3, \beta=1/3$ ; (c) the contour plot of fig.7(b); (d) the sectional view of${\left| {{\varOmega _7}(x,t)} \right|^2}$ against$ x $ when t=3
图 8
${\left| {{\varOmega _8}(x,t)} \right|^2}$ 取不同分数阶导数值时的三维图，图8(b)对应的等高线图，以及$ t = 3 $ 时${\left| {{\varOmega _8}(x,t)} \right|^2}$ 关于$ x $ 的截面图：(a)$\alpha=1/2, \beta=1/3,{\left| {{{{\varOmega }}_8}(x,t)} \right|^2}$ 的三维图； (b)$\alpha=1/3, \beta=1/3,{\left| {{{{\varOmega }}_8}(x,t)} \right|^2} $ 的三维图； (c) 图8(b)的等高线图； (d) 当$ t = 3 $ 时$\alpha=1/2, \beta=1/3, {\left| {{\varOmega _8}(x,t)} \right|^2}$ 关于$ x $ 的截面图Figure 8. The 3D graph of
${\left| {{\varOmega _8}(x,t)} \right|^2}$ with different values of the fractional derivative, the contour plot of fig. 8(b) and the sectional view of${\left| {{\varOmega _8}(x,t)} \right|^2}$ against$ x $ with$ t = 3 $ : (a) the graphic model of${\left| {{\varOmega _8}(x,t)} \right|^2},\alpha=1/2, \beta=1/3$ ; (b) the graphic model of${\left| {{{{\varOmega }}_8}(x,t)} \right|^2} ,\alpha=1/3, \beta=1/3$ ; (c) the contour plot of fig. 8(b); (d) the sectional view of${\left| {{\varOmega _8}(x,t)} \right|^2}$ against$ x $ when t=3
图 9
${\left| {{\varOmega _9}(x,t)} \right|^2}$ 取不同分数阶导数值时的三维图，图9(b)对应的等高线图，以及$ t = 3 $ 时${\left| {{\varOmega _9}(x,t)} \right|^2}$ 关于$ x $ 的截面图：(a)$\alpha=1/2, \beta=1/3,{\left| {{\varOmega _9}(x,t)} \right|^2}$ 的三维图； (b)$\alpha=1/3, \beta=1/3,{\left| {{\varOmega _9}(x,t)} \right|^2} $ 的三维图； (c) 图9(b)的等高线图； (d) 当$ t = 3 $ 时${\left| {{\varOmega _9}(x,t)} \right|^2}$ 关于$ x $ 的截面图Figure 9. The 3D graph of
${\left| {{\varOmega _9}(x,t)} \right|^2}$ with different values of the fractional derivative, the contour plot of fig.9(b) and the sectional view of${\left| {{\varOmega _9}(x,t)} \right|^2}$ against$ x $ with$ t = 3 $ : (a) the graphic model of${\left| {{\varOmega _9}(x,t)} \right|^2}, \alpha=1/2, \beta=1/3$ ; (b) the graphic model of${\left| {{\varOmega _9}(x,t)} \right|^2}, \alpha=1/3, \beta=1/3$ ; (c) the contour plot of fig.9(b); (d) the sectional view of${\left| {{\varOmega _9}(x,t)} \right|^2}$ against$ x $ when t=3 -
[1] WANG B H, WANG Y Y, DAI C Q, et al. Dynamical characteristic of analytical fractional solitons for the space-time fractional Fokas-Lenells equation[J]. Alexandria Engineering Journal, 2020, 59(6): 4699-4707. doi: 10.1016/j.aej.2020.08.027 [2] KILBAS A, SRIVASTAVA H M, TRUJILLO J J. Theory and Applications of Fractional Differential Equations[M]. Amsterdam: Elsevier Science, 2006. [3] MILLER K S, ROSS B. An Introduction to the Fractional Calculus and Fractional Differential Equations[M]. Wiley-Interscience, 1993. [4] ATANGANA A. Derivative With a New Parameter Theory, Methods and Applications[M]. Amsterdam: Elsevier Science, 2015. [5] PANDIR Y, EKIN A. Dynamics of combined soliton solutions of unstable nonlinear Schrödinger equation with new version of the trial equation method[J]. Chinese Journal of Physics, 2020, 67: 534-543. doi: 10.1016/j.cjph.2020.08.013 [6] LIU XIAOYAN, ZHOU Q, BISWAS A, et al. The similarities and differences of different plane solitons controlled by (3 + 1)-dimensional coupled variable coefficient system[J]. Journal of Advanced Research, 2020, 24: 167-173. doi: 10.1016/j.jare.2020.04.003 [7] SAJID N, AKRAM G. Optical solitons with full nonlinearity for the conformable space-time fractional Fokas-Lenells equation[J]. Optik, 2019, 196: 163131. doi: 10.1016/j.ijleo.2019.163131 [8] HASHEMI M S, BALEANU D. Lie Symmetry Analysis of Fractional Differential Equations[M]. New York: Chapman and Hall/CRC , 2020. [9] 胡彦鑫, 郭增鑫, 辛祥鹏. 一类Burgers-KdV方程的李群分析、李代数、对称约化及精确解[J]. 聊城大学学报(自然科学版), 2021, 34(2): 8-13HU Yanxin, GUO Zengxin, XIN Xiangpeng. Lie group analysis, Lie algebra, symmetric reduction and exact solutions of a class of nonlinear evolution equations[J]. Journal of Liaocheng University (Natural Science) , 2021, 34(2): 8-13.(in Chinese) [10] KALBANI K, AL-GHAFRI K S, KRISHNAN E V, et al. Pure-cubic optical solitons by Jacobi’s elliptic function approach[J]. Optik, 2021, 243: 167404. doi: 10.1016/j.ijleo.2021.167404 [11] BISWAS A, EKICI M, SONMEZOGLU A, et al. Highly dispersive optical solitons with non-local nonlinearity by extended Jacobi’s elliptic function expansion[J]. Optik, 2019, 184: 277-286. doi: 10.1016/j.ijleo.2019.03.061 [12] MANAFIAN J, AGHDAEI M F, JEDDI R S, et al. Application of the generalized G'/G-expansion method for nonlinear PDEs to obtaining soliton wave solution[J]. Optik, 2017, 135: 395-406. doi: 10.1016/j.ijleo.2017.01.078 [13] KALLEL W, ALMUSAWA H, MIRHOSSEINI-ALIZAMINI S M, et al. Optical soliton solutions for the coupled conformable Fokas-Lenells equation with spatio-temporal dispersion[J]. Results in Physics, 2021, 26: 104388. doi: 10.1016/j.rinp.2021.104388 [14] 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): 874-880HU Yan, SUN Yuhuai. Solutions to space-time fractional complex Ginzburg-Landau equations with the complete discrimination system for polynomial method[J]. Applied Mathematics and Mechanics, 2021, 42(8): 874-880.(in Chinese) [15] CAO Q, DAI C. Symmetric and anti-symmetric solitons of the fractional second-and third-order nonlinear Schrödinger equation[J]. Chinese Physics Letters, 2021, 38(9): 090501. doi: 10.1088/0256-307X/38/9/090501 [16] DAI C Q, WANG Y Y, ZHANG J F. Managements of scalar and vector rogue waves in a partially nonlocal nonlinear medium with linear and harmonic potentials[J]. Nonlinear Dynamics, 2020, 102: 179-391. [17] FEI J, CAO W. Explicit soliton-cnoidal wave interaction solutions for the (2 + 1)-dimensional negative-order breaking soliton equation[J]. Waves in Random and Complex Media, 2020, 30(1): 54-64. doi: 10.1080/17455030.2018.1479548 [18] HAN H, LI H, DAI C. Wick-type stochastic multi-soliton and soliton molecule solutions in the framework of nonlinear Schrödinger equation[J]. Applied Mathematics Letters, 2021, 120: 107302. doi: 10.1016/j.aml.2021.107302 [19] ARSHED S, RAZA N. Optical solitons perturbation of Fokas-Lenells equation with full nonlinearity and dual dispersion[J]. Chinese Journal of Physics, 2020, 63: 314-324. doi: 10.1016/j.cjph.2019.12.004 [20] YOUNIS M, SEADAWY A R, BABER M Z, et al. Analytical optical soliton solutions of the Schrödinger-Poisson dynamical system[J]. Results in Physics, 2021, 27: 104369. doi: 10.1016/j.rinp.2021.104369 [21] FANG Y, WU G Z, WANG Y Y, et al. Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order NLSE using the PINN[J]. Nonlinear Dynamics, 2021, 105: 603-616. doi: 10.1007/s11071-021-06550-9 [22] RAGHURAMAN P J, BAGHYA SHREE S, MANI RAJAN M S. Soliton control with inhomogeneous dispersion under the influence of tunable external harmonic potential[J]. Waves in Random and Complex Media, 2021, 31(3): 474-485. doi: 10.1080/17455030.2019.1598602 [23] DAI C Q, WANG Y Y. Coupled spatial periodic waves and solitons in the photovoltaic photorefractive crystals[J]. Nonlinear Dynamics, 2020, 102: 1733-1741. doi: 10.1007/s11071-020-05985-w [24] AIN Q T, HE J H, N ANJUM, et al. The fractional complex transform: a novel approach to the time-fractional Schrödinger equation[J]. Fractals, 2021, 28(7): 2150002. [25] SHEHATA M, REZAZADEH H, ZAHRAN E, et al. New optical soliton solutions of the perturbed Fokas-Lenells equation[J]. Communications in Theoretical Physics, 2019, 71(11): 13-18. [26] TRIKI H, WAZWAZ A M. Combined optical solitary waves of the Fokas-Lenells equation[J]. Waves in Random and Complex Media, 2017, 27(4): 587-593. doi: 10.1080/17455030.2017.1285449 [27] ALI KHALID K, OSMAN M S, ABDEL-ATY M. New optical solitary wave solutions of Fokas-Lenells equation in optical fiber via Sine-Gordon expansion method[J]. Alexandria Engineering Journal, 2020, 59(3): 1191-1196. doi: 10.1016/j.aej.2020.01.037 [28] BULUT H, ABDULKADIR SULAIMAN T, MEHMET BASKONUS H, et al. Optical solitons and other solutions to the conformable space-time fractional Fokas-Lenells equation[J]. Optik, 2018, 172: 20-27. doi: 10.1016/j.ijleo.2018.06.108 [29] TRIKI H, ZHOU Q, LIU W, et al. Localized pulses in optical fibers governed by perturbed Fokas-Lenells equation[J]. Physics Letters A, 2022, 421: 127782. doi: 10.1016/j.physleta.2021.127782 [30] BISWAS A, YILDIRM Y, YASAR E, et al. Optical soliton solutions to Fokas-Lenells equation using some different methods[J]. Optik, 2018, 173: 21-31. doi: 10.1016/j.ijleo.2018.07.098 [31] BISWAS A, EKICI M, SONMEZOGLU A, et al. Optical solitons with differential group delay for coupled Fokas-Lenells equation by extended trial function scheme[J]. Optik, 2018, 165: 102-110. doi: 10.1016/j.ijleo.2018.03.102 [32] ZHANG Y, YANG J W, CHOW K W, et al. Solitons, breathers and rogue waves for the coupled Fokas-Lenells system via Darboux transformation[J]. Nonlinear Analysis: Real World Applications, 2017, 33: 237-252. doi: 10.1016/j.nonrwa.2016.06.006 [33] ZHANG Q, ZHANG Y, YE R. Exact solutions of nonlocal Fokas-Lenells equation[J]. Applied Mathematics Letters, 2019, 98: 336-343. doi: 10.1016/j.aml.2019.05.015 [34] YAKUP Y, BISWAS A, DAKOVA A, et al. Cubic-quartic optical soliton perturbation with Fokas-Lenells equation by sine-Gordon equation approach[J]. Results in Physics, 2021, 26: 104409. doi: 10.1016/j.rinp.2021.104409 [35] DIEU DONNE G, TIOFACK C G L, SEADAWY A, et al. Propagation of W-shaped, M-shaped and other exotic optical solitons in the perturbed Fokas-Lenells equation[J]. The European Physical Journal Plus, 2020, 135: 371. doi: 10.1140/epjp/s13360-020-00382-z [36] NADIA M, GHAZALA A. Exact solitary wave solutions of the (1 + 1)-dimensional Fokas-Lenells equation[J]. Optik, 2020, 208: 164459. doi: 10.1016/j.ijleo.2020.164459 [37] 杨翠红, 朱思铭, 梁肇军. 多项式代数方程根的完全分类及其应用[J]. 中山大学学报(自然科学版), 2003, 42(1): 5-8 doi: 10.3321/j.issn:0529-6579.2003.01.002YANG Cuihong, ZHU Siming, LIANG Zhaojun. Complete discrimination of the roots of polynomials and its applications[J]. Acta Scientiarum Naturalium Universitatis Sunyatseni (Natural Science) , 2003, 42(1): 5-8.(in Chinese) doi: 10.3321/j.issn:0529-6579.2003.01.002 [38] 夏壁灿, 杨路. 多项式判别矩阵的若干性质及其应用[J]. 应用数学学报, 2003, 4: 652-663 doi: 10.3321/j.issn:0254-3079.2003.04.009XIA Bican, YANG Lu. Some properties of the discrimination matrix of polynomials with applications[J]. Acta Mathematicae Applicatae Sinica, 2003, 4: 652-663.(in Chinese) doi: 10.3321/j.issn:0254-3079.2003.04.009 [39] 杨路, 张景中, 侯晓荣. 非线性代数方程组与定理机器证明[M]. 上海: 上海科技教育出版社, 1996.YANG Lu, ZHANG Jingzhong, HOU Xiaorong. Machine Proof of Systems of Nonlinear Algebraic Equations and Theorems[M]. Shanghai: Shanghai Science and Technology Education Press, 1996. (in Chinese) [40] LIU C. Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations[J]. Computer Physics Communications, 2009, 181(2): 317-324. [41] 辛华. 带有微扰项的Fokas-Lenells方程的包络行波模式[J]. 数学的实践与认识, 2021, 51(11): 324-328XIN Hua. The patterns of envelope traveling waves of Fokas-Lenells equation with perturbation term[J]. Mathematics in Practice and Theory, 2021, 51(11): 324-328.(in Chinese) [42] KHALIL R, HORANI A, 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 [43] LIU C. Counterexamples on Jumarie’s two basic fractional calculus formulae[J]. Communications in Nonlinear Science and Numerical Simulation, 2015, 22(1/3): 92-94. doi: 10.1016/j.cnsns.2014.07.022 [44] LIU C. Counterexamples on Jumarie’s three basic fractional calculus formulae for non-differentiable continuous functions[J]. Chaos, Solitons and Fractals, 2018, 109: 219-222. doi: 10.1016/j.chaos.2018.02.036 [45] TARASOV V E. No nonlocality, no fractional derivative[J]. Communications in Nonlinear Science and Numerical Simulation, 2018, 62: 157-163. doi: 10.1016/j.cnsns.2018.02.019 [46] TARASOV V E. No violation of the Leibniz rule, no fractional derivative[J]. Communications in Nonlinear Science and Numerical Simulation, 2013, 18(11): 2945-2948. doi: 10.1016/j.cnsns.2013.04.001 [47] WANG B H, WANG Y Y. Fractional white noise functional soliton solutions of a wick-type stochastic fractional NLSE[J]. Applied Mathematics Letters, 2020, 110: 106583. doi: 10.1016/j.aml.2020.106583 [48] YU L J, WU G Z, WANG Y Y, et al. Traveling wave solutions constructed by Mittag-Leffler function of a (2 + 1)-dimensional space-time fractional NLS equation[J]. Results in Physics, 2020, 17: 103156. doi: 10.1016/j.rinp.2020.103156 [49] WANG B H, LU P H, DAI C Q, et al. Vector optical soliton and periodic solutions of a coupled fractional nonlinear Schrödinger equation[J]. Results in Physics, 2020, 17: 103036. doi: 10.1016/j.rinp.2020.103036 [50] DAI C Q, WU G, LI H J, et al. Wick-type stochastic fractional solitons supported by quadratic-cubic nonlinearity[J]. Fractals, 2021, 29(7): 2150192. doi: 10.1142/S0218348X21501929 -
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