Nonlinear Dynamics

, Volume 92, Issue 3, pp 815–825 | Cite as

A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal

  • Xiao-Yan Tang
  • Zu-Feng Liang
Original Paper


A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal is derived from the nonlinear inviscid dissipative and equivalent barotropic vorticity equation in a \(\beta \)-plane. The modulational instability (MI) of the obtained system is studied, which reveals a number of possibilities for the MI regions due to the generalized dispersion relation that relates the frequency and wavenumber of the modulating perturbations. Exact periodic solutions in terms of Jacobi elliptic functions are obtained, which, in the limit of the modulus approaches unity, reduce to soliton, kink solutions and their linear superpositions. Representative profiles of different nonlinear wave excitations are displayed graphically. These solutions can be used to model different blocking events in climate disasters. As an illustration, a special approximate solution is given to describe a kind of two correlated dipole blocking events.


Nonlocal NLS equation Shifted parity Delayed time reversal Modulational instability Periodic waves 



The authors acknowledge the financial support by the National Natural Science Foundation of China (Nos. 11675055 and 11475052) and Shanghai Knowledge Service Platform for Trustworthy Internet of Things (No. ZF1213).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)CrossRefGoogle Scholar
  2. 2.
    Pertsch, T., Peschel, U., Kobelke, J., Schuster, K., Bartelt, H., Nolte, S., Tunnermann, A., Lederer, F.: Nonlinearity and disorder in fiber arrays. Phys. Rev. Lett. 93, 053901 (2004)CrossRefGoogle Scholar
  3. 3.
    Ablowitz, M.J., Musslimani, Z.H.: Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 915–946 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fokas, A.S.: Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 319–324 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Liu, Y.B., Mihalache, D., He, J.S.: Families of rational solutions of the y-nonlocal Davey–Stewartson II equation. Nonlinear Dyn. 90(4), 2445–2455 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Liu, Y.K., Li, B.: Rogue waves in the (\(2+1\))-dimensional nonlinear Schrödinger equation with a parity-time-symmetric potential. Chin. Phys. Lett. 34, 010202 (2017)CrossRefGoogle Scholar
  7. 7.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable discrete PT symmetric model. Phys. Rev. E 90, 032912 (2014)CrossRefGoogle Scholar
  8. 8.
    Ma, L.Y., Zhu, Z.N.: N-soliton solution for an integrable nonlocal discrete focusing nonlinear Schrödinger equation. Appl. Math. Lett. 59, 115–121 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zhang, Y., Liu, Y.P., Tang, X.Y.: A general integrable three-component coupled nonlocal nonlinear Schrodinger equation. Nonlinear Dyn. 89(4), 2729–2738 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lou, S.Y., Huang, F.: Alice–Bob physics: coherent solutions of nonlocal KdV systems. Sci. Rep. 7, 869 (2017)CrossRefGoogle Scholar
  11. 11.
    Tang, X.Y., Zhao, J., Huang, F., Lou, S.Y.: Monopole blocking governed by a modified KdV type equation. Stud. Appl. Math. 122, 295–304 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139, 7–59 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jia, M., Gao, Y., Huang, F., Lou, S.Y., Sun, J.L., Tang, X.Y.: Vortices and vortex sources of multiple vortex interaction systems. Nonlinear Anal. Real Word Appl. 13, 2079 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Dudley, J.M., Genty, G., Dias, F., Kibler, B., Akhmediev, N.: Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation. Opt. Express 23, 21497 (2009)CrossRefGoogle Scholar
  15. 15.
    Solli, D.R., Ropers, C., Koonath, P., Jalali, B.: Optical rogue waves. Nature 450, 1054–1057 (2007)CrossRefGoogle Scholar
  16. 16.
    Khare, A., Saxena, A.: Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations. J. Math. Phys. 56, 032104 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Luo, D.H., Li, J.P.: Interaction between a slowly moving planetary-scale dipole envelop Rossby soliton and a wavenumber-two topography in a forced higher order nonlinear Schrödinger equation. Adv. Atmos. Sci. 19, 239–256 (2001)Google Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Shanghai Key Laboratory of Trustworthy ComputingEast China Normal UniversityShanghaiChina
  2. 2.Department of PhysicsHangzhou Normal UniversityHangzhouChina

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