Advertisement

A Vector General Nonlinear Schrödinger Equation with \((m+n)\) Components

  • Xianguo Geng
  • Ruomeng LiEmail author
  • Bo Xue
Article
  • 40 Downloads

Abstract

A vector general nonlinear Schrödinger equation with \((m+n)\) components is proposed, which is a new integrable generalization of the vector nonlinear Schrödinger equation and the vector derivative nonlinear Schrödinger equation. Resorting to the Riccati equations associated with the Lax pair and the gauge transformations between the Lax pairs, a general N-fold Darboux transformation of the vector general nonlinear Schrödinger equation with \((m+n)\) components is constructed, which can be reduced directly to the classical N-fold Darboux transformation and the generalized Darboux transformation without taking limits. As an illustrative example, some exact solutions of the two-component general nonlinear Schrödinger equation are obtained by using the general Darboux transformation, including a first-order rogue-wave solution, a fourth-order rogue-wave solution, a breather solution, a breather–rogue-wave interaction, two solitons and the fission of a breather into two solitons. It is a very interesting phenomenon that, for all \(M>0\), there exists a rogue-wave solution for the two-component general nonlinear Schrödinger equation such that the amplitude of the rogue wave is M times higher than its background wave.

Keywords

Vector general nonlinear Schrödinger equation General N-fold Darboux transformation Soliton solutions Breather solutions Rogue-wave solutions 

Mathematics Subject Classification

35Q51 35Q55 35Q53 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11931017, 11871440 and 11971442).

References

  1. Ablowitz, M.J., Segure, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)CrossRefGoogle Scholar
  2. Ablowitz, M.J., Prinari, B., Trubatch, A.D.: Discrete and Continuous Nonlinear Schrödinger Systems. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  3. Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, San Diego (2007)zbMATHGoogle Scholar
  4. Baronio, F., Degasperis, A., Conforti, M., Wabnitz, S.: Solutions of the vector nonlinear Schrödinger equations: evidence for deterministic rogue waves. Phys. Rev. Lett. 109, 044102 (2012)CrossRefGoogle Scholar
  5. Chan, H.N., Chow, K.W.: Rogue waves for an alternative system of coupled Hirota equations: structural robustness and modulation instabilities. Stud. Appl. Math. 139, 78–103 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  6. Chan, H.N., Malomed, B.A., Chow, K.W., Ding, E.: Rogue waves for a system of coupled derivative nonlinear Schrödinger equations. Phys. Rev. E 93, 012217 (2016)MathSciNetCrossRefGoogle Scholar
  7. Chen, J.B., Pelinovsky, D.E.: Rogue periodic waves of the modified KdV equation. Nonlinearity 31, 1955–1980 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  8. Chen, J.B., Pelinovsky, D.E.: Rogue periodic waves of the focusing nonlinear Schrödinger equation. Proc. A 474, 20170814 (2018)MathSciNetzbMATHGoogle Scholar
  9. Geng, X.G.: Lax pair and Darboux transformation solutions of the modified Boussinesq equation. Acta Math. Appl. Sinica. 11, 324–328 (1988)MathSciNetzbMATHGoogle Scholar
  10. Geng, X.G.: Darboux transformation of the discrete Ablowitz–Ladik eigenvalue problem. Acta Math. Sci. (English Ed.) 9, 21–26 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  11. Geng, X.G., Li, R.M.: Darboux transformation of the Drinfeld–Sokolov–Satsuma–Hirota system and exact solutions. Ann. Phys. 361, 215–225 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  12. Geng, X.G., Liu, H.: The nonlinear steepest descent method to long-time asymptotics of the coupled nonlinear Schrödinger equation. J. Nonlinear Sci. 28, 739–763 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  13. Geng, X.G., Lv, Y.Y.: Darboux transformation for an integrable generalization of the nonlinear Schrödinger equation. Nonlinear Dynam. 69, 1621–1630 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  14. Geng, X.G., Tam, H.W.: Darboux transformation and soliton solutions for generalized nonlinear Schrödinger equations. J. Phys. Soc. Jpn. 68, 1508–1512 (1999)zbMATHCrossRefGoogle Scholar
  15. Geng, X.G., Liu, H., Zhu, J.Y.: Initial-boundary value problems for the coupled nonlinear Schrödinger equation on the half-line. Stud. Appl. Math. 135, 310–346 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  16. Guo, B.L., Ling, L.M., Liu, Q.P.: Nonlinear Schrödinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85, 026607 (2012)CrossRefGoogle Scholar
  17. He, J.S., Xu, S.W., Porsezian, K., Cheng, Y., Dinda, P.T.: Rogue wave triggered at a critical frequency of a nonlinear resonant medium. Phys. Rev. E 93, 062201 (2016)MathSciNetCrossRefGoogle Scholar
  18. Hisakado, M., Wadati, M.: Gauge transformations among generalised nonlinear Schrödinger equations. J. Phys. Soc. Jpn. 63, 3962–3966 (1994)zbMATHCrossRefGoogle Scholar
  19. Hoefer, M.A., Chang, J.J., Hamner, C., Engels, P.: Dark–dark solitons and modulational instability in miscible two-component Bose–Einstein condensates. Phys. Rev. A 84, 041605 (2011)CrossRefGoogle Scholar
  20. 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
  21. Kaup, D.J., Newell, A.C.: An exact solution for a derivative nonlinear Schrödinger equation. J. Math. Phys. 9, 789–801 (1978)zbMATHGoogle Scholar
  22. Kevrekidis, P.G., Frantzeskakis, D., Carretero-Gonzalez, R.: Emergent Nonlinear Phenomena in Bose–Einstein Condensates: Theory and Experiment. Springer, Berlin (2009)zbMATHGoogle Scholar
  23. Liu, Q.P.: Darboux transformations for supersymmetric Korteweg-de Vries equations. Lett. Math. Phys. 35, 115–122 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  24. Liu, H., Geng, X.G.: The vector derivative nonlinear Schröodinger equation on the half-line. IMA J. Appl. Math. 83, 148–173 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  25. Ma, L.Y., Zhao, H.Q., Shen, S.F., Ma, W.X.: Abundant exact solutions to the discrete complex mKdV equation by Darboux transformation. Commun. Nonlinear Sci. Numer. Simul. 68, 31–40 (2019)MathSciNetCrossRefGoogle Scholar
  26. Manakov, S.: On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov. Phys. JETP 38, 248–253 (1974)Google Scholar
  27. Meng, G.Q., Gao, Y.T., Yu, X., Shen, Y.J., Qin, Y.: Multi-soliton solutions for the coupled nonlinear Schrödinger-type equations. Nonlinear Dynam. 70, 609–617 (2012)MathSciNetCrossRefGoogle Scholar
  28. Mio, W., Ogino, T., Minami, K., Takeda, S.: Modified nonlinear Schrödinger equation for Alfvén waves propagating along the magnetic field in cold plasmas. J. Phys. Soc. Jpn. 41, 265–271 (1976)zbMATHCrossRefGoogle Scholar
  29. Mjølhus, E.: On the modulational instability of hydromagnetic waves parallel to the magnetic field. J. Plasma Phys. 16, 321–334 (1976)CrossRefGoogle Scholar
  30. Newell, A.C.: Solitons in Mathematics and Physics. SIAM, Philadelphia (1985)zbMATHCrossRefGoogle Scholar
  31. Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., Zakharov, V.E.: Theory of Solitons, the Inverse Scattering Methods. Consultants Bureau, New York (1984)zbMATHGoogle Scholar
  32. Novikov, S.P., Manakov, S.V., Pitaevskii, L.P., Zaharov, V.E.: Theory of Solitons, the Method of the Inverse Problem. Consultants Bureau, New York (1984)Google Scholar
  33. Tsuchida, T.: \(N\)-soliton collision in the Manakov model. Prog. Theor. Phys. 111, 151–182 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  34. Wadati, M., Sogo, K.: Gauge transformations in soliton theory. J. Phys. Soc. Jpn. 52, 394–398 (1983)MathSciNetCrossRefGoogle Scholar
  35. Wazwaz, A.M.: Exact and explicit travelling wave solutions for the nonlinear Drinfeld–Sokolov system. Commun. Nonlinear Sci. Numer. Simul. 11, 311–325 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  36. Wazwaz, A.M.: The Cole-Hopf transformation and multiple soliton solutions for the integrable sixth-order Drinfeld–Sokolov–Satsuma–Hirota equation. Appl. Math. Comput. 207, 248–255 (2009)MathSciNetzbMATHGoogle Scholar
  37. Wei, J., Wang, X., Geng, X.G.: Periodic and rational solutions of the reduced Maxwell–Bloch equations. Commun. Nonlinear Sci. Numer. Simul. 59, 1–14 (2018)MathSciNetCrossRefGoogle Scholar
  38. Yan, Z.Y., Chow, K.W., Malomed, B.A.: Exact stationary wave patterns in three coupled nonlinear Schrödinger/Gross–Pitaevskii equations. Chaos Solitons Fractals 42, 3013–3019 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  39. Yan, D., Chang, J.J., Hamner, C., Kevrekidis, P.G., Engels, P., Achilleos, V., Frantzeskakis, D.J., Carretero-Gonzalez, R., Schmelcher, P.: Multiple dark-bright solitons in atomic Bose–Einstein condensates. Phys. Rev. A 84, 053630 (2011)CrossRefGoogle Scholar
  40. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of wave in nonlinear media. Sov. Phys. JETP 34, 62–69 (1972)MathSciNetGoogle Scholar
  41. Zhang, G.Q., Yan, Z.Y.: The \(n\)-component nonlinear Schrödinger equations: dark-bright mixed \(N\)- and high-order solitons and breathers, and dynamics. Proc. A 474, 20170688 (2018)MathSciNetzbMATHGoogle Scholar
  42. Zhang, G.Q., Yan, Z.Y., Wen, X.Y.: Three-wave resonant interactions: multi-dark–dark–dark solitons, breathers, rogue waves, and their interactions and dynamics. Phys. D 366, 27–42 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  43. Zhao, H.Q., Yuan, J.Y., Zhu, Z.N.: Integrable semi-discrete Kundu-Eckhaus equation: Darboux transformation, breather, rogue wave and continuous limit theory. J. Nonlinear Sci. 28, 43–68 (2018)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouPeople’s Republic of China

Personalised recommendations