Advertisement

Nonlinear Dynamics

, Volume 93, Issue 4, pp 1799–1808 | Cite as

Coherent structure of Alice–Bob modified Korteweg de-Vries equation

  • Congcong Li
  • S. Y. Lou
  • Man Jia
Original Paper
  • 139 Downloads

Abstract

To describe two-place events, Alice–Bob systems have been established by means of the shifted parity and delayed time reversal in the preprint arXiv:1603.03975v2 [nlin.SI], (2016). In this paper, we mainly study exact solutions of the integrable Alice–Bob modified Korteweg de-Vries (AB-mKdV) system. The general Nth Darboux transformation for the AB-mKdV equation is constructed. By using the Darboux transformation, some types of shifted parity and time reversal symmetry breaking solutions including one-soliton, two-soliton, and rogue wave solutions are explicitly obtained. In addition to the similar solutions of the mKdV equation (group invariant solutions), there are abundant new localized structures for the AB-mKdV systems.

Keywords

Nonlinear nonlocal partial differential equations Darboux transformations Exact solutions solitons Rogue waves 

Notes

Acknowledgements

The authors are grateful to thank Professors D. J. Zhang, Z. N. Zhu, Q. P. Liu, X. B. Hu, and Y. Chen for their helpful discussions. The work was sponsored by the Global Change Research Program of China (No. 2015CB953904), Shanghai Knowledge Service Platform for Trustworthy Internet of Things (No. ZF1213), the National Natural Science Foundations of China (No. 11435005), and K. C. Wong Magna Fund in Ningbo University.

References

  1. 1.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)CrossRefGoogle Scholar
  2. 2.
    Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70, 947–1018 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Markum, H., Pullirsch, R., Wettig, T.: Non-Hermitian random matrix theory and lattice QCD with chemical potential. Phys. Rev. Lett. 83, 484–487 (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Lin, Z., Schindler, J., Ellis, F.M., Kottos, T.: Experimental observation of the dual behavior of PT-symmetric scattering. Phys. Rev. A 85, 050101 (2012)CrossRefGoogle Scholar
  5. 5.
    Ruter, C.E., Makris, K.G., EI-Ganainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Observation of paritytime symmetry in optics. Nat. Phys. 6, 192–195 (2010)CrossRefGoogle Scholar
  6. 6.
    Musslimani, Z.H., Makris, K.G., EI-Ganainy, R., Christodoulides, D.N.: Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008)CrossRefGoogle Scholar
  7. 7.
    Dalfovo, F., Giorgini, S., Pitaevskii, L.P., Stringari, S.: Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463–512 (1999)CrossRefGoogle Scholar
  8. 8.
    Lou, S.Y.: Alice-Bob systems, \(P_{s}\)-\(T_{d}\)-\(C\) principles and multi-soliton solutions, arXiv:1603.03975v2 [nlin.SI], (2016)
  9. 9.
    Jia, M., Lou, S.Y.: An AB–KdV equation: exact solutions, symmetry reductions and Bäcklund transformations, arXiv:1612.00546v1 [nlin.SI], (2016)
  10. 10.
    Ablowitz, M.J., Musslimani, Z.H.: Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 915–946 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    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 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ablowitz, M.J., Musslimani, Z.H.: Integrable discrete PT symmetric model. Phys. Rev. E 90, 032912 (2014)CrossRefGoogle Scholar
  13. 13.
    Song, C.Q., Xiao, D.M., Zhu, Z.N.: Solitons and dynamics for a general integrable nonlocal coupled nonlinear Schrödinger equation. Commun. Nonlin. Sci. Numer. Simulat. 45, 13–28 (2017)CrossRefGoogle Scholar
  14. 14.
    Dimakos, M., Fokas, A.S.: Davey–Stewartson type equations in 4+2 and 3+1 possessing soliton solutions. J. Math. Phys. 54, 081504 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fokas, A.S.: Integrable nonlinear evolution partial differential equations in 4+2 and 3+1 dimensions. Phys. Rev. Lett. 96, 190201 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fokas, A.S.: Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 319–324 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yan, Z.Y.: Integrable PT-symmetric local and nonlocal vector nonlinear Schrodinger equations: a unified two-parameter model. Appl. Math. Lett. 47, 61–68 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Yan, Z.Y.: Nonlocal general vector nonlinear Schrodinger equations: PT symmetribility, and solutions. Appl. Math. Lett. 62, 101–109 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lou, S.Y., Qiao, Z.J.: Alice–Bob peakon systems. Chin. Phys. Lett. 34, 100201 (2007)CrossRefGoogle Scholar
  20. 20.
    Lou, S.Y., Huang, F.: Alice–Bob Physics, coherent solutions of nonlocal KdV systems, arXiv:1606.03154v1 [nlin.SI], (2016); Sci. Rep. 7, 869 (2017)
  21. 21.
    Lou, S.Y.: Consistent riccati expansion for integrable systems. Stud. Appl. Math. 134, 372–402 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gao, Y., Tang, X.Y.: A coupled variable coefficient modified KdV equation arising from a two-layer fluid system. Commun. Theor. Phys. 48, 961–970 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Ji, J.L., Zhu, Z.N.: On a nonlocal modified Korteweg–de Vries equation: integrability, Darboux transformation and soliton solutions. Commun. Nonlin. Sci. Numer. Simulat. 42, 699–708 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Matveev, V.B., Salle, M.A.: Darboux Transformations and Solitons. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  25. 25.
    Gu, C.H., Hu, H.S., Zhou, Z.X.: Darboux Transformation in Soliton Theory and its Geometric Applications. Shanghai Sci-Tech Publishing House, Shanghai (2005)Google Scholar
  26. 26.
    Ma, W.X.: Wronskian solutions to integrable equations. Discret. Contin. Dyn. Syst, Suppl 506–515 (2009)Google Scholar
  27. 27.
    Wazwaz, A.-M.: Multiple soliton solutions and multiple complex solitons for two distinct Boussinesq equations. Nonlin. Dyn. 85, 731–737 (2016)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wazwaz, A.-M., El-Tantawy, S.A.: New (3+1)-dimensional equations of Burgers type and Sharma Tasso Olver type: multiple soliton solutions. Nonlin. Dyn. 87, 2457–2461 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Center for Nonlinear Science and Department of PhysicsNingbo UniversityNingboChina
  2. 2.Shanghai Key Laboratory of Trustworthy ComputingEast China Normal UniversityShanghaiChina

Personalised recommendations