Research article

Solitary wave solutions and integrability for generalized nonlocal complex modified Korteweg-de Vries (cmKdV) equations

  • Received: 22 June 2021 Accepted: 25 July 2021 Published: 30 July 2021
  • MSC : 35Q51, 37K40

  • In this paper, the reverse space cmKdV equation, the reverse time cmKdV equation and the reverse space-time cmKdV equation are constructed and each of three types diverse soliton solutions is derived based on the Hirota bilinear method. The Lax integrability of three types of nonlocal equations is studied from local equation by using variable transformations. Based on exact solution formulae of one- and two-soliton solutions of three types of nonlocal cmKdV equation, some figures are used to describe the soliton solutions. According to the dynamical behaviors, it can be found that these solutions possess novel properties which are different from the ones of classical cmKdV equation.

    Citation: Wen-Xin Zhang, Yaqing Liu. Solitary wave solutions and integrability for generalized nonlocal complex modified Korteweg-de Vries (cmKdV) equations[J]. AIMS Mathematics, 2021, 6(10): 11046-11075. doi: 10.3934/math.2021641

    Related Papers:

  • In this paper, the reverse space cmKdV equation, the reverse time cmKdV equation and the reverse space-time cmKdV equation are constructed and each of three types diverse soliton solutions is derived based on the Hirota bilinear method. The Lax integrability of three types of nonlocal equations is studied from local equation by using variable transformations. Based on exact solution formulae of one- and two-soliton solutions of three types of nonlocal cmKdV equation, some figures are used to describe the soliton solutions. According to the dynamical behaviors, it can be found that these solutions possess novel properties which are different from the ones of classical cmKdV equation.



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    [1] M. J. Ablowitz, Z. H. Musslimani, Integrable nonlocal nonlinear Schrödinger equation, Phys. Rev. Lett., 110 (2013), 064105. doi: 10.1103/PhysRevLett.110.064105
    [2] C. M. Bender, S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett., 80 (1998), 5243–5246. doi: 10.1103/PhysRevLett.80.5243
    [3] J. J. Fang, C. Q. Dai, Optical solitons of a time-fractional higher-order nonlinear Schrödinger equation, Optik, 209 (2020), 164574. doi: 10.1016/j.ijleo.2020.164574
    [4] K. Hosseini, M. Matinfar, M. Mirzazadeh, Soliton solutions of high-order Schrödinger equation with different laws of nonlinearities, Regul. Chaotic Dyn., 26 (2021), 105–112. doi: 10.1134/S1560354721010068
    [5] K. Hosseini, K. Sadri, M. Mirzazadeh, S. Salahshour, An integrable (2+1)-dimensional nonlinear Schrödinger system and its optical soliton solutions, Optik, 229 (2021), 166247. doi: 10.1016/j.ijleo.2020.166247
    [6] K. Hosseini, K. Sadri, M. Mirzazadeh, Y. M. Chu, A. Ahmadian, B. A. Pansera, S. Salahshour, A high-order nonlinear Schrödinger equation with the weak non-local nonlinearity and its optical solitons, Results Phys., 23 (2021), 104035. doi: 10.1016/j.rinp.2021.104035
    [7] K. Hosseini, S. Salahshour, M. Mirzazadeh, A. Ahmadian, D. Baleanu, A. Khoshrang, The (2+1)-dimensional Heisenberg ferromagnetic spin chain equation: its solitons and Jacobi elliptic function solutions, Eur. Phys. J. Plus, 136 (2021), 206. doi: 10.1140/epjp/s13360-021-01160-1
    [8] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys., 71 (1999), 463–512. doi: 10.1103/RevModPhys.71.463
    [9] J. Yang, Physically significant nonlocal nonlinear Schrödinger equation and its soliton solutions, Phys. Rev. E, 98 (2018), 042202. doi: 10.1103/PhysRevE.98.042202
    [10] X. Deng, S. Y. Lou, D. J. Zhang, Bilinearisation-reduction approach to the nonlocal discrete nonlinear Schrödinger equations, Appl. Math. Comput., 332 (2018), 477–483.
    [11] L. Y. Peng, Symmetries and reductions of integrable nonlocal partial differential equations, Symmetry, 11 (2019), 884. doi: 10.3390/sym11070884
    [12] N. V. Priya, M. Senthilvelan, G. Rangarajan, M. Lakshmanan, On symmetry preserving and symmetry broken bright, dark and antidark soliton solutions of nonlocal nonlinear Schrödinger equation, Phys. Lett. A, 383 (2019), 15–26. doi: 10.1016/j.physleta.2018.10.011
    [13] K. Hosseini, S. Salahshour, M. Mirzazadeh, Bright and dark solitons of a weakly nonlocal Schrödinger equation involving the parabolic law nonlinearity, Optik, 227 (2021), 166042. doi: 10.1016/j.ijleo.2020.166042
    [14] B. Ren, J. Lin, Soliton molecules, nonlocal symmetry and CRE method of the KdV equation with higher-order corrections, Physica Scrip., 95(2020), 075202. doi: 10.1088/1402-4896/ab8d02
    [15] Z. X. Xu, K. W. Chow, Breathers and rogue waves for a third order nonlocal partial differential equation by a bilinear transformation, Appl. Math. Lett., 56 (2016), 72–77. doi: 10.1016/j.aml.2015.12.016
    [16] S. Y. Lou, Multi-place physics and multi-place nonlocal systems, Commun. Theor. Phys., 72 (2020), 057001. doi: 10.1088/1572-9494/ab770b
    [17] K. Chen, X. Deng, S. Y. Lou, D. J. Zhang, Solutions of nonlocal equations reduced from the AKNS hierarchy, Stud. Appl. Math., 141 (2018), 113–141. doi: 10.1111/sapm.12215
    [18] J. G. Rao, Y. Cheng, K. Porsezian, D. Mihalache, J. S. He, PT-symmetric nonlocal Davey-Stewartson I equation: soliton solutions with nonzero background, Physica D, 401 (2020), 132180. doi: 10.1016/j.physd.2019.132180
    [19] J. G. Rao, J. S. He, D. Mihalache, Y. Cheng, PT-symmetric nonlocal Davey-Stewartson I equation: General lump-soliton solutions on a background of periodic line waves, Appl. Math. Lett., 104 (2020), 106246. doi: 10.1016/j.aml.2020.106246
    [20] F. J. Yu, R. Fan, Nonstandard bilinearization and interaction phenomenon for PT-symmetric coupled nonlocal nonlinear Schrödinger equations, Appl. Math. Lett., 103 (2020), 106209. doi: 10.1016/j.aml.2020.106209
    [21] A. R. Seadawy, R. I. Nuruddeen, K. S. Aboodh, Y. F. Zakariya, On the exponential solutions to three extracts from extended fifth-order KdV equation, J. King Saud Univ. Sci., 32 (2020), 765–769.
    [22] R. I. Nuruddeen, Multiple soliton solutions for the (3+1) conformable space-time fractional modified Korteweg-de Vries equations, J. Ocean Eng. Sci., 3(2018), 11–18.
    [23] C. Park, R. I. Nuruddeen, K.K. Ali, L. Muhammad, M. S. Osman, D. Baleanu, Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg-de Vries equations, Adv. Differ. Equ., 2020 (2020), 627. doi: 10.1186/s13662-020-03087-w
    [24] S. F. Tian, Initial-boundary value problems of the coupled modified Korteweg-de Vries equation on the half-line via the Fokas method, J. Phys. A: Math. Theor., 50 (2017), 395204. doi: 10.1088/1751-8121/aa825b
    [25] S. Y. Lou, F. Huang, Alice-Bob physics: Coherent solutions of nonlocal KdV systems, Sci. Rep., 7 (2017), 869. doi: 10.1038/s41598-017-00844-y
    [26] X. Y. Tang, Z. F. Liang, X. Z. Hao, Nonlinear waves of a nonlocal modified KdV equation in the atmospheric and oceanic dynamical system, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 62. doi: 10.1016/j.cnsns.2017.12.016
    [27] M. J. Ablowitz, Z. H. Musslimani, Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation, Nonlinearity, 29 (2016), 915–946. doi: 10.1088/0951-7715/29/3/915
    [28] M. J. Ablowitz, Z. H. Musslimani, Integrable nonlocal nonlinear equations, Stud. Appl. Math., 139 (2016), 7–59.
    [29] B. Yang, J. Yang, Transformations between nonlocal and local integrable equations, Stud. Appl. Math., 140 (2018), 178–201. doi: 10.1111/sapm.12195
    [30] L. Y. Ma, S. F. Shen, Z. N. Zhu, Integrable nonlocal complex mKdV equation: soliton solution and gauge equivalence, arXiv: 1612.06723v1 [nlin.SI] 20 Dec 2016.
    [31] L. Li, C. Duan, F. Yu, An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation, Phys. Lett. A, 383 (2019), 1578–1582. doi: 10.1016/j.physleta.2019.02.031
    [32] M. G$\ddot{\mathrm{u}}$rses, A. Pekcan, Nonlocal modified KdV equations and their soliton solutions by Hirota method, Commun. Nonlinear Sci. Numer. Simulat., 67 (2019), 427–448. doi: 10.1016/j.cnsns.2018.07.013
    [33] F. J. He, E. G. Fan, J. Xu, Long-Time asymptotics for the nonlocal MKdV equation, Commun. Theor. Phys., 71 (2019), 475–488. doi: 10.1088/0253-6102/71/5/475
    [34] G. Zhang, Z. Yan, Inverse scattering transforms and soliton solutions of focusing and defocusing nonlocal mKdV equations with non-zero boundary conditions, Physica D, 402 (2020), 132170. doi: 10.1016/j.physd.2019.132170
    [35] J. L. Ji, Z. N. Zhu, Soliton solutions of an integrable nonlocal modified Korteweg-deVries equation through inverse scattering transform, J. Math. Anal. Appl., 453 (2017), 973–984. doi: 10.1016/j.jmaa.2017.04.042
    [36] J. Yang, Physically significant nonlocal nonlinear Schrödinger equation and its soliton solutions, Phys. Rev. E, 98 (2018), 042202. doi: 10.1103/PhysRevE.98.042202
    [37] L. M. Ling, W. X. Ma, Inverse scattering and soliton solutions of nonlocal complex reverse-spacetime modified Korteweg-de Vries hierarchies, Symmetry, 13 (2021), 512. doi: 10.3390/sym13030512
    [38] W. X. Ma, Riemann-Hilbert problems and soliton solutions of nonlocal real reverse-spacetime mKdV equations, J. Math. Anal. Appl., 498 (2021), 124980. doi: 10.1016/j.jmaa.2021.124980
    [39] R. Hirota, The Direct Method in Soliton Theory, New York: Cambridge University Press, 2004.
    [40] J. Zhuang, Y. Liu, P. Zhuang, Variety interaction solutions comprising lump solitons for the (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada equation, AIMS Math., 6 (2021), 5370–5386. doi: 10.3934/math.2021316
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