Solutions and connections of nonlocal derivative nonlinear Schrödinger equations

  • Ying ShiEmail author
  • Shou-Feng Shen
  • Song-Lin Zhao
Original Paper


All possible nonlocal versions of the derivative nonlinear Schrödinger equations are derived by the nonlocal reduction from the Chen–Lee–Liu equation, the Kaup–Newell equation and the Gerdjikov–Ivanov equation which are gauge equivalent to each other. Their solutions are obtained by composing constraint conditions on the double Wronskian solution of the Chen–Lee–Liu equation and the nonlocal analogues of the gauge transformations among them. Through the Jordan decomposition theorem, those solutions of the reduced equations from the Chen–Lee–Liu equation can be written as canonical form within real field.


Nonlocal derivative nonlinear Schrödinger equations Nonlocal reduction Double Wronskian Canonical form 



This work is supported by the National Natural Science Foundation of China (NSFC) grant (Grant Number 11501510) and the Natural Science Foundation of Zhejiang Province (No. LY17A010024).

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


  1. 1.
    Olver, P.J., Sattinger, D.H.: Solitons in Physics, Mathematics, and Nonlinear Optics. Springer, New York (1990)CrossRefGoogle Scholar
  2. 2.
    Zakharov, V.E.: What is Integrability?. Springer, Berlin (1991)CrossRefGoogle Scholar
  3. 3.
    Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadephia (1981)CrossRefGoogle Scholar
  4. 4.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110(6), 064105 (2013)CrossRefGoogle Scholar
  5. 5.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable discrete PT-symmetric model. Phys. Rev. E 90, 032912 (2014)CrossRefGoogle Scholar
  6. 6.
    Konotop, V.V., Yang, J.K., Zeyulin, D.A.: Nonlinear waves in PT-symmetric systems. Rev. Mod. Phys. 88(3), 035002 (2016)CrossRefGoogle Scholar
  7. 7.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139(1), 7–59 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Yang, B., Yang, J.K.: Transformations between nonlocal and local integrable equations. Stud. Appl. Math. 140(2), 178–201 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhou, Z.X.: Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 62, 480–488 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Zhou, Z.X.: Darboux transformations and global explicit solutions for nonlocal Davey–Stewartson I equation, arXiv:1612.05689 (2016)
  11. 11.
    Lou, S.Y., Huang, F.: Alice–Bob physics: coherent solutions of nonlocal KdV systems. Sci. Rep. 7, 869 (2017)CrossRefGoogle Scholar
  12. 12.
    Ablowitz, M.J., Musslimani, Z.H.: Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29(3), 915–946 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Tang, X.Y., Liang, Z.F.: A general nonlocal nonlinear Schrödinger equation with shifted parity, charge-conjugate and delayed time reversal. Nonlinear Dyn. 92, 815–825 (2018)CrossRefGoogle Scholar
  14. 14.
    Yang, J.K,: General N-solitons and their dynamics in several nonlocal nonlinear Schrödinger equation, arXiv:1712.01181 [nlin.SI] (2017)
  15. 15.
    Feng, B.F., Luo, X.D., Ablowitz, M.J., Musslimani, Z.H.: Genenral soliton solutions to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions, arXiv:1712.09172 (2017)
  16. 16.
    Liu, W., Li, X.L.: General soliton solutions to a (2+1)-dimensional nonlocal nonlinear Schrodinger equation with zero and nonzero boundary conditions. Nonlinear Dyn. 93, 721–731 (2018)CrossRefGoogle Scholar
  17. 17.
    Wazwaz, A.M.: On the nonlocal Boussinesq equation: multiple-soliton solutions. Appl. Math. Lett. 26, 1094–1098 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Shi, Y., Zhang, Y.S., Xu, S.W.: Families of nonsingular soliton solutions of a nonlocal Schrödinger-Boussinesq equation. Nonlinear Dyn.
  19. 19.
    Rao, J.G., Cheng, Y., He, J.S.: Rational and semirational solutions of the nonlocal Davey–Stewartson equations. Stud. Appl. Math. 139, 568–598 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Rao, J.G., Zhang, Y.S., Fokas, A.S., He, J.S.: Rogue waves of the nonlocal Davey–Stewartson I equation. Nonlinearity 31, 4090 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Chen, K., Deng, X., Lou, S.Y., Zhang, D.J.: Solutions of local and nonlocal equations reduced from the AKNS hierarchy. Stud. Appl. Math. 141, 113–141 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ablowitz, M.J., Ladik, J.F.: Nonlinear differential-difference equations. J. Math. Phys. 16(3), 598–603 (1975)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ablowitz, M.J., Ladik, J.F.: Nonlinear differential-difference equations and Fourier analysis. J. Math. Phys. 17(6), 1011–1018 (1976)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Chen, K., Zhang, D.J.: Solutions of the nonlocal nonlinear Schrödinger hierarchy via reduction. Appl. Math. Lett. 75, 82–88 (2017)CrossRefGoogle Scholar
  25. 25.
    Chen, K., Liu, S.M., Zhang, D.J.: Covariant hodograph transformations between nonlocal short pulse models and AKNS(1) system. Appl. Math. Lett. 88, 230–236 (2019)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Deng, X., Lou, S.Y., Zhang, D.J.: Bilinearisation-reduction approach to the nonlocal discrete nonlinear Schrödinger equations. Appl. Math. Comput. 332, 477–483 (2018)MathSciNetGoogle Scholar
  27. 27.
    Chen, H.H., Lee, Y.C., Liu, C.S.: Integrability of nonlinear Hamiltonian systems by inverse scattering method. Phys. Scr. 20, 490–492 (1979)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Zhang, F.Z.: Matrix theory: basic results and techniques, 2nd edn. Springer, New York (2011)CrossRefGoogle Scholar
  29. 29.
    Kaup, D.J., Newell, A.C.: An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 19(4), 798–801 (1978)CrossRefGoogle Scholar
  30. 30.
    Zhai, W., Chen, D.Y.: Rational solutions of the general nonlinear Schrödinger equation with derivative. Phys. Lett. A 372(23), 4217–4221 (2008)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Wadati, M., Sogo, K.: Gauge transformations in Soliton theory. J. Phys. Soc. Jpn. 52(2), 394–398 (1983)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yajima, T.: Derivative nonlinear Schrödinger type equations with multipe components and their solutions. J. Phys. Soc. Jpn. 64(6), 1901–1909 (1995)CrossRefGoogle Scholar
  33. 33.
    Gerdzhikov, V.S., Ivanov, M.I.: A quadratic pencil of general type and nonlinear evolution equations. II. Hierarchies of Hamiltonian structures (Russian. English summary). Bulg. J. Phys. 10(2), 130–143 (1983)zbMATHGoogle Scholar
  34. 34.
    Kundu, A.: Landau–Lifshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger type equations. J. Math. Phys. 25(12), 3433–3438 (1984)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Tsuchida, T.: Integrable discretizations of derivative nonlinear Schrödinger equations. J. Phys. A Math. Gen. 35, 7827–7847 (2002)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Nonlinear-evolution equations of physical significance. Phys. Rev. Lett. 31, 125–127 (1973)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53(4), 249–315 (1974)MathSciNetCrossRefGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of ScienceZhejiang University of Science and TechnologyHangzhouPeople’s Republic of China
  2. 2.Department of Applied MathematicsZhejiang University of TechnologyHangzhouPeople’s Republic of China

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