Advertisement

Analysis and Mathematical Physics

, Volume 8, Issue 3, pp 415–426 | Cite as

Trace formula and new form of N-soliton to the Gerdjikov–Ivanov equation

  • Hui Nie
  • Junyi Zhu
  • Xianguo Geng
Article

Abstract

The Gerdjikov–Ivanov equation is investigated by the Riemann–Hilbert approach and the technique of regularization. The trace formula and new form of N-soliton solution are given. The dynamics of the stationary solitons and non-stationary solitons are discussed.

Keywords

Gerdjikov–Ivanov equation Riemann–Hilbert approach trace formula soliton 

Mathematics Subject Classification

35Q15 37K15 

Notes

Acknowledgements

Projects 11471295 and 11331008 were supported by the National Natural Science Foundation of China.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)CrossRefMATHGoogle Scholar
  2. 2.
    Ablowitz, M.J., Fokas, A.S.: Complex Variables: Introduction and Applications, 2nd edn. Cambridge University Press, Cambridge (2003)CrossRefMATHGoogle Scholar
  3. 3.
    Beals, R., Coifman, R.R.: Linear spectral problems, non-linear equations and the deltamacr-method. Inverse Probl. 5, 87–130 (1989)CrossRefMATHGoogle Scholar
  4. 4.
    Chen, H.H., Lee, Y.C., Liu, C.S.: Integrability of nonlinear Hamiltonian systems by inverse scattering method. Phys. Scr. 20, 490–492 (1979)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dai, H.H., Fan, E.G.: Variable separation and algebro-geometric solutions of the Gerdjikov–Ivanov equation. Chaos Soliton. Fract. 22, 93–101 (2004)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Doktorov, E.V., Leble, S.B.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Springer, Dordrecht (2007)Google Scholar
  7. 7.
    Faddeev, L.D., Takhtajan, L.A.: Hamiltonian Methods in the Theory of Solitons. Springer, Berlin (1987)CrossRefMATHGoogle Scholar
  8. 8.
    Fan, E.G.: Darboux transformation and soliton-like solutions for the Gerdjikov–Ivanov equation. J. Phys. A Math. Gen. 33, 6925–6933 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fan, E.G.: Integrable evolution systems based on Gerdjikov–Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformation. J. Math. Phys. 41, 7769–7782 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fan, E.G.: Explicit N-fold Darboux transformations and soliton solutions for nonlinear derivative Schrödinger equations. Commun. Theor. Phys. 35, 651–656 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Geng, X.G., Wu, J.P.: Riemann–Hilbert approach and N-soliton solutions for a generalized Sasa–Satsuma equation. Wave Motion 60, 62–72 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gerdjikov, V.S.: Algebraic and analytic aspects of N-wave type tquations. Contemp. Math. 301, 35–68 (2002)CrossRefGoogle Scholar
  13. 13.
    Gerdjikov, V.S.: Basic aspects of soliton theory. In: Mladenov, I.M., Hirshfeld, A.C. (eds.) Geometry, Integrability and Quantization. Sofetex, Sofia (2005)Google Scholar
  14. 14.
    Gerdjikov, V.S., Ivanov, I.: A quadratic pencil of general type and nonlinear evolution equations. II. Hierarchies of Hamiltonian structures. Bulg. J. Phys. 10, 130–143 (1983)MathSciNetGoogle Scholar
  15. 15.
    Guo, L.J., Zhang, Y.S., Xu, S.W., Wu, Z.W., He, J.S.: The higher order rogue wave solutions of the Gerdjikov–Ivanov equation. Phys. Scr. 89, 035501 (2014)CrossRefGoogle Scholar
  16. 16.
    Hou, Y., Fan, E.G., Zhao, P.: Algebro-geometric solutions for the Gerdjikov–Ivanov hierarchy. J. Math. Phys. 54, 073505 (2013)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kakei, S., Kikuchi, T.: Affine Lie group approach to a derivative nonlinear Schrödinger equation and its similarity reduction. Int. Math. Res. Not. 78, 4181–4209 (2004)CrossRefMATHGoogle Scholar
  18. 18.
    Kakei, S., Kikuchi, T.: Solutions of a derivative nonlinear Schrödinger hierarchy and its similarity reduction. Glasg. Math. J. 47, 99–107 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kakei, S., Sasa, N., Satsuma, J.: Bilinearization of a generalized derivative nonlinear Schrödinger equation. J. Phys. Soc. Jpn. 64, 1519–1523 (1995)CrossRefMATHGoogle Scholar
  20. 20.
    Kaup, D.J., Newell, A.C.: An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 19, 798–801 (1978)CrossRefMATHGoogle Scholar
  21. 21.
    Kundu, A.: Exact solutions to higher-order nonlinear equations through gauge transformation. Phys. D 25, 399–406 (1987)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Lenells, J., Fokas, A.S.: On a novel integrable generalization of the nonlinear Schrödinger equation. Nonlinearity 22, 11–27 (2009)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lu, X., Ma, W.X., Yu, J.: A new (2+1)-dimensional integrable system and its algebro-geometric solution. Nonlinear Dyn. 82, 1211–1220 (2015)CrossRefGoogle Scholar
  24. 24.
    Shchesnovich, V.S., Yang, J.K.: General soliton matrices in the Riemann–Hilbert problem for integrable nonlinear equations. J. Math. Phys. 44, 4604–4639 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Xu, S.W., He, J.S.: The rogue wave and breather solution of the Gerdjikov–Ivanov equation. J. Math. Phys. 53, 063507 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Yu, J., He, J.S., Han, J.W.: Two kinds of new integrable decompositions of the Gerdjikov–Ivanov equation. J. Math. Phys. 53, 033510 (2012)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Zakharov, V.E., Manakov, S.V., Novikov, S.P., Pitaevskii, L.P.: Theory of Soliton: The inverse Scattering Technique. Nauka, Moscow (1980)MATHGoogle Scholar
  28. 28.
    Zakharov, V.E., Shabat, A.B.: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. Funct. Anal. Appl. 8, 226–235 (1974)CrossRefMATHGoogle Scholar
  29. 29.
    Zakharov, V.E., Shabat, A.B.: Integration of nonlinear equations of mathematical physics by the method of the inverse scattering. II. Funct. Anal. Appl. 13, 166–174 (1979)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zhu, J.Y., Li, Z.: Dressing method for a generalized focusing NLS equation via local Riemann–Hilbert problem. Acta Phys. Pol. B 42, 1893–1904 (2011)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Yang, J.J., Zhu, J.Y., Wang, L.L.: Dressing by regularization to the Gerdjikov–Ivanov equation and the higher-order soliton. arXiv: 1504.03407v2 (2015)

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouChina

Personalised recommendations