Integrable Nonlocal Reductions

  • Metin GürsesEmail author
  • Aslı Pekcan
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 266)


We present some nonlocal integrable systems by using the Ablowitz–Musslimani nonlocal reductions. We first present all possible nonlocal reductions of nonlinear Schrödinger (NLS) and modified Korteweg–de Vries (mKdV) systems. We give soliton solutions of these nonlocal equations by using the Hirota method. We extend the nonlocal NLS equation to nonlocal Fordy–Kulish equations by utilizing the nonlocal reduction to the Fordy–Kulish system on symmetric spaces. We also consider the super AKNS system and then show that Ablowitz–Musslimani nonlocal reduction can be extended to super integrable equations. We obtain new nonlocal equations namely nonlocal super NLS and nonlocal super mKdV equations.


Ablowitz–Musslimani type reductions Nonlocal NLS and mKdV equations Hirota bilinear method Soliton solutions Nonlocal Fordy–Kulish system Nonlocal super integrable NLS and mKdV equations 



This work is partially supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK).


  1. 1.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)Google Scholar
  2. 2.
    Ablowitz, M.J., Musslimani, Z.H.: Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 915–946 (2016)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139(1), 7–59 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    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
  5. 5.
    Caudrelier, V.: Interplay between the inverse scattering method and Fokas’s unified transform with an application. Stud. App. Math. 140, 3–26 (2017). arXiv:1704.05306v4 [math-ph]MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, K., Deng, X., Lou, S., Zhang, D.: Solutions of nonlocal equations reduced from the AKNS hierarchy. Stud. App. Math. (2018) (to appear). arXiv:1710.10479 [nlin.SI]
  7. 7.
    Feng, B.F., Luo, X.D., Ablowitz, M.J., Musslimani, Z.H.: General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions. arXiv:1712.09172 [nlin.SI]
  8. 8.
    Fokas, A.S.: Integrable multidimensional versions of the nonlocal Schrödinger equation. Nonlinearity 29, 319–324 (2016)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Fordy, A.P.: Derivative nonlinear Schrödinger equations and hermitian symmetric spaces. J. Phys. A Math. Gen. 17, 1235–1245 (1984)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fordy, A.P., Kulish, P.P.: Nonlinear Schrödinger equations and simple Lie algebras. Commun. Math. Phys. 89, 427–443 (1983)Google Scholar
  11. 11.
    Gerdjikov, V.S.: On nonlocal models of Kulish-Sklyanin type and generalized fourier transforms. Stud. Comp. Int. 681, 37–52 (2017). arXiv:1703.03705 [nlin.SI]
  12. 12.
    Gerdjikov, V.S., Saxena, A.: Complete integrability of nonlocal nonlinear Schrödinger equation. J. Math. Phys. 58(1), 013502 (2017). arXiv:1510.00480 [nlin.SI]MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gerdjikov, V.S., Grahovski, D.G., Ivanov, R.I.: On the N-wave equations with PT symmetry. Theor. Math. Phys. 188(3), 1305–1321 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gerdjikov, V.S., Grahovski, D.G., Ivanov, R.I.: On the integrable wave interactions and Lax pairs on symmetric spaces. Wave Motion 71, 53–70 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gürses, M.: Nonlocal Fordy-Kulish equations on symmetric spaces. Phys. Lett. A 381, 1791–1794 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Gürses, M., Og̃uz, Ö.: A super AKNS scheme. Phys. Lett. A 108(9), 437–440 (1985)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gürses, M., Og̃uz, Ö.: A super soliton connection. Lett. Math. Phys. 11, 235–246 (1986)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gürses, M., Og̃uz, Ö., Salihog̃lu, S.: Nonlinear partial differential equations on homogeneous spaces. Int. J. Mod. Phys. A 5, 1801–1817 (1990)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Gürses, M., Pekcan, A.: Nonlocal nonlinear Schrödinger equations and their soliton solutions. J. Math. Phys. 59, 051501 (2018). arXiv:1707.07610v1 [nlin.SI]MathSciNetCrossRefGoogle Scholar
  20. 20.
    Gürses, M., Pekcan, A.: Nonlocal nonlinear modified KdV equations and their soliton solutions. Commun. Nonlinear Sci. Numer. Simul. 67, 427–448 (2019). arXiv:1711.01588 [nlin.SI]
  21. 21.
    Huang, X., King, L.: Soliton solutions for the nonlocal nonlinear Schrödinger equation. Eur. Phys. J. Plus 131, 148 (2016)Google Scholar
  22. 22.
    Iwao, M., Hirota, R.: Soliton solutions of a coupled modified KdV equations. J. Phys. Soc. Jpn. 66(3), 577–588 (1997)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ji, J.L., Zhu, Z.N.: On a nonlocal modified Korteweg-de Vries equation: integrability, Darboux transformation and soliton solutions. Commun. Nonlinear Sci. Numer. Simul. 42, 699–708 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ji, J.L., Zhu, Z.N.: Soliton solutions of an integrable nonlocal modified Korteweg-de Vries equation through inverse scattering transform. J. Math. Anal. Appl. 453, 973–984 (2017). arXiv:1603.03994 [nlin.SI]MathSciNetCrossRefGoogle Scholar
  25. 25.
    Khare, A., Saxena, A.: Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations. J. Math. Phys. 56, 032104 (2015)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kupperschmidt, B.A.: A super Korteweg-de Vries equation: an integrable system. Phys. Lett. 102A, 213 (1983)Google Scholar
  27. 27.
    Kupperschmidt, B.A.: Bosons and Fermions interacting integrably with the Korteweg-de Vries field. J. Phys. A Math. Gen. 17, L869 (1984)Google Scholar
  28. 28.
    Li, M., Xu, T.: Dark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. Phys. Rev. E 91, 033202 (2015)Google Scholar
  29. 29.
    Ma, L.Y., Shen, S.F., Zhu, Z.N.: Integrable nonlocal complex mKdV equation: soliton solution and Gauge equivalence. arXiv:1612.06723 [nlin.SI]
  30. 30.
    Sakkaravarthi, K., Kanna, T.: Bright solitons in coherently coupled nonlinear Schrödinger equations with alternate signs of nonlinearities. J. Math. Phys. 54, 013701 (2013)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Sinha, D., Ghosh, P.K.: Integrable nonlocal vector nonlinear Schrödinger equation with self-induced parity-time symmetric potential. Phys. Lett. A 381, 124–128 (2017)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Wen, X.Y., Yan, Z., Yang, Y.: Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. Chaos 26, 063123 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Yang, J.: General N-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations. arXiv:1712.01181 [nlin.SI]
  34. 34.
    Yang, B., Yang, J.: Transformations between nonlocal and local integrable equations. Stud. App. Math. 140, 178–201 (2017). arXiv:1705.00332v1 [nlin.PS]MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Faculty of Science, Department of MathematicsBilkent UniversityAnkaraTurkey
  2. 2.Faculty of Science, Department of MathematicsHacettepe UniversityAnkaraTurkey

Personalised recommendations