2020 Volume 10 Issue 6
Article Contents

Xiangpeng Xin, Lihua Zhang, Yarong Xia, Hanze Liu. NONLOCAL SYMMETRIES AND EXACT SOLUTIONS OF A VARIABLE COEFFICIENT AKNS SYSTEM[J]. Journal of Applied Analysis & Computation, 2020, 10(6): 2669-2681. doi: 10.11948/20200022
Citation: Xiangpeng Xin, Lihua Zhang, Yarong Xia, Hanze Liu. NONLOCAL SYMMETRIES AND EXACT SOLUTIONS OF A VARIABLE COEFFICIENT AKNS SYSTEM[J]. Journal of Applied Analysis & Computation, 2020, 10(6): 2669-2681. doi: 10.11948/20200022

NONLOCAL SYMMETRIES AND EXACT SOLUTIONS OF A VARIABLE COEFFICIENT AKNS SYSTEM

  • Corresponding author: Email address:xinxiangpeng@lcu.edu.cn.(X. Xin) 
  • Fund Project: This work is supported by the National Natural Science Foundation of China (Nos.11505090, 1171041), the doctorial foundation of Liaocheng University under Grant No.318051413, the Natural Science Foundation of Shaanxi Province under Grant No.2014JM2-1009, and the Science and Technology Innovation Foundation of Xioan under Grant No.2017CGWL06
  • In this paper, nonlocal symmetries of variable coefficient Ablowitz-Kaup-Newell-Segur(AKNS) system are studied for the first time. In order to construct some new analytic solutions, a new variable is introduced, which can transform nonlocal symmetries into Lie point symmetries. Furthermore, using the Lie point symmetries of closed system, we give out two types of symmetry reductions and some analytic solutions. For some interesting solutions, such as interaction solutions among solitons and other complicated waves, we give corresponding images to describe their dynamic behavior.
    MSC: 76M60, 17B80, 83C15
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  • [1] G. W. Bluman and S. Kumei, Symmetries and Differential Equations, Springer-Verlag, New York, 1989.

    Google Scholar

    [2] G. W. Bluman, A. F. Cheviakov and S.C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer, New York, 2010.

    Google Scholar

    [3] G. W. Bluman and A. F. Cheviakov, Framework for potential systems and nonlocal symmetries: Algorithmic approach, J. Math. Phys., 2005, 46, 123506. doi: 10.1063/1.2142834

    CrossRef Google Scholar

    [4] C. Chun and B. Neta, Comparative study of methods of various orders for finding simple roots of nonlinear equations, J. Appl. Anal. Comput., 2019, 9(2), 400-427.

    Google Scholar

    [5] S. Chen, B. Tian, Y. Sun and C. Zhang, Generalized Darboux Transformations, Rogue Waves, and Modulation Instability for the Coherently Coupled Nonlinear Schrödinger Equations in Nonlinear Optics, Ann. Phys. (Berlin), 2019, 531(8), 1900011. doi: 10.1002/andp.201900011

    CrossRef Google Scholar

    [6] Z. Cheng and Z. Bi, Study on a kind of p-Laplacian neutral differential equation with multiple variable coefficients, J. Appl. Anal. Comput., 2019, 9(2), 501-525.

    Google Scholar

    [7] X. Du, B. Tian, Q. Qu, Y. Yuan and X. Zhao, Lie group analysis, solitons, self-adjointness and conservation laws of the modified Zakharov-Kuznetsov equation in an electron-positron-ion magnetoplasma, Chaos Solitons Fract., 2020, 134, 109709. doi: 10.1016/j.chaos.2020.109709

    CrossRef Google Scholar

    [8] Z. Du, B. Tian, H. Chai and X. Zhao, Dark-bright semi-rational solitons and breathers for a higher-order coupled nonlinear Schrödinger system in an optical fiber, Appl. Math. Lett., 2020, 102, 106110. doi: 10.1016/j.aml.2019.106110

    CrossRef Google Scholar

    [9] A. Deliceoglu and D. Bozkurt, Structural bifurcation of divergence-free vector fields near non-simple degenerate points with symmetry, J. Appl. Anal. Comput., 2019, 9(2), 718-738.

    Google Scholar

    [10] F. Galas, New nonlocal symmetries with pseudopotentials, J. Phys. A: Math. Gen., 1992, 25, L981. doi: 10.1088/0305-4470/25/15/014

    CrossRef Google Scholar

    [11] X. Gao, Mathematical view with observational/experimental consideration on certain (2+1)-dimensional waves in the cosmic/laboratory dusty plasmas, Appl. Math. Lett., 2019, 91, 165-172. doi: 10.1016/j.aml.2018.11.020

    CrossRef Google Scholar

    [12] X. Gao, Y. Guo and W. Shan, Water-wave symbolic computation for the Earth, Enceladus and Titan: The higher-order Boussinesq-Burgers system, auto- and non-auto-Bäcklund transformations, Appl. Math. Lett., 2020, 104, 106170. doi: 10.1016/j.aml.2019.106170

    CrossRef Google Scholar

    [13] C. Hu, B. Tian, H. Yin, C. Zhang and Z. Zhang, Dark breather waves, dark lump waves and lump wave-soliton interactions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in a fluid, Comput. Math. Appl., 2019, 78, 166-177. doi: 10.1016/j.camwa.2019.02.026

    CrossRef Google Scholar

    [14] X. Hu, S. Lou and Y. Chen, Explicit solutions from eigenfunction symmetry of the Korteweg-de Vries equation, Phys. Rev. E, 2012, 85, 85056607-1.

    Google Scholar

    [15] J. Han and L. Yan, A time fractional functional differential equation driven by the fractional Brownian motion, J. Appl. Anal. Comput., 2019, 9(2), 547-567.

    Google Scholar

    [16] Q. Huang, Y. Gao, and Y. Feng, Lax pair, infinitely-many conservation laws and soliton solutions for a set of the time-dependent Whitham-Broer-Kaup equations for the shallow water, Waves in Random and Complex Media 2019, 29(1), 19-33.

    Google Scholar

    [17] N. H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Boston, MA: Reidel, 1985.

    Google Scholar

    [18] H. Khan, C. Tunc and A. Khan, Stability results and existence theorems for nonlinear delay-fractional differential equations with $\phi{^*_p}$-operator, J. Appl. Anal. Comput., 2020, 10(2), 584-597.

    Google Scholar

    [19] B. Zeng, J. Yang and B. Ren, Exact solutions and residual symmetries of the Ablowitz-Kaup-Newell-Segur system, Chin. Phys. B, 2015, 24(1), 010202. doi: 10.1088/1674-1056/24/1/010202

    CrossRef Google Scholar

    [20] S. Lie, über die Integration durch bestimmte Integrale von einer Classe linearer partieller Differentialgleichungen, Arch. Math. 1881, 6, 328-368.

    Google Scholar

    [21] S. Lou, X. Hu and Y. Chen, Nonlocal symmetries related to Bäcklund transformation and their applications, J. Phys. A: Math. Theor., 2012, 45, 155209. doi: 10.1088/1751-8113/45/15/155209

    CrossRef Google Scholar

    [22] Q. Miao, X. Xin and Y. Chen, Nonlocal symmetries and explicit solutions of the AKNS system, Appl. Math. Lett., 2014, 28, 7-13. doi: 10.1016/j.aml.2013.09.002

    CrossRef Google Scholar

    [23] P. J. Olver, Applications of Lie Groups to Differential Equations, Berlin: Springer, 1986.

    Google Scholar

    [24] L. V. Ovsiannikov, Group Analysis of Differential Equations, New York: Academic, 1982.

    Google Scholar

    [25] C. Qin, S. Tian, L. Zou, et al, Lie symmetry analysis, conservation laws and exact solutions of fourth-order time fractional Burgers equation, J. Appl. Anal. Comput., 2018, 8(6), 1727-1746.

    Google Scholar

    [26] W. Qian, Y. Li and X. Yang, The Isoenergetic KAM-Type Theorem at Resonant Case for Nearly Integrable Hamiltonian Systems, J. Appl. Anal. Comput., 2019, 9(5), 1616-1638.

    Google Scholar

    [27] S. Sui and B. Li, Bifurcation of limit cycles from the global center of a class of integrable non-Hamilton systems, J. Appl. Anal. Comput., 2018, 8(5), 1441-1451.

    Google Scholar

    [28] M. Wang, B. Tian, Y. Sun and Z. Zhang, Lump, mixed lump-stripe and rogue wave-stripe solutions of a (3+1)-dimensional nonlinear wave equation for a liquid with gas bubbles, Comput. Math. Appl., 2020, 79, 576-587. doi: 10.1016/j.camwa.2019.07.006

    CrossRef Google Scholar

    [29] X. Xin and X. Liu, Interaction Solutions for (1+1)-Dimensional Higher-Order Broer-Kaup System, Commun. Theor. Phys., 2016, 66(5), 479-482. doi: 10.1088/0253-6102/66/5/479

    CrossRef Google Scholar

    [30] X. Xin, Y. Liu and X. Liu, Nonlocal symmetries, exact solutions and conservation laws of the coupled Hirota equations, Appl. Math. Lett., 2016, 55, 63-71. doi: 10.1016/j.aml.2015.11.009

    CrossRef Google Scholar

    [31] Y. Xia, X. Xin and S. Zhang, Residual symmetry, interaction solutions, and conservation laws of the (2+1)-dimensional dispersive long-wave system, Chin. Phys. B, 2017, 26(3), 030202. doi: 10.1088/1674-1056/26/3/030202

    CrossRef Google Scholar

    [32] X. Xin, L. Zhang, Y. Xia, et al. Nonlocal symmetries and exact solutions of the (2+1)-dimensional generalized variable coefficient shallow water wave equation, Appl. Math. Lett., 2019, 94, 112-119. doi: 10.1016/j.aml.2019.02.028

    CrossRef Google Scholar

    [33] X. Xin, H. Liu, L. Zhang, et al. High order nonlocal symmetries and exact interaction solutions of the variable coefficient KdV equation, Appl. Math. Lett., 2019, 88, 132-140. doi: 10.1016/j.aml.2018.08.023

    CrossRef Google Scholar

    [34] H. Yin, B. Tian and X. Zhao, Chaotic breathers and breather fission/fusion for a vector nonlinear Schrödinger equation in a birefringent optical fiber or wavelength division multiplexed system, Appl. Math. Comput., 2020, 368, 124768.

    Google Scholar

    [35] X. Zheng and L. Wei, Symmetry analysis conservation laws of a time fractional fifth-order Sawada-Kotera equation, J. Appl. Anal. Comput., 2017, 7(4), 1275-1284.

    Google Scholar

    [36] Z. Zhao and B. Han, On Symmetry Analysis and Conservation Laws of the AKNS System, Z. Naturforsch., 2016, 71(8)a, 741-750.

    Google Scholar

    [37] C. Zhang, B. Tian, Q. Qu, L. Liu and H. Tian, Vector bright solitons and their interactions of the couple Fokas-Lenells system in a birefringent optical fiber, Z. Angew. Math. Phys., 2020, 71(1), 1-19.

    Google Scholar

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