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Nonlinear Dynamics

, Volume 94, Issue 4, pp 2749–2761 | Cite as

Modulation instability analysis and soliton solutions of an integrable coupled nonlinear Schrödinger system

  • Ding Guo
  • Shou-Fu Tian
  • Tian-Tian Zhang
  • Jin Li
Original Paper
  • 121 Downloads

Abstract

In this paper, an integrable coupled nonlinear Schrödinger system is investigated, which is derived from the integrable Kadomtsev–Petviashvili system, and can be used to describe the stability of soliton in a nonlinear media with weak dispersion. By using the Hirota bilinear method, we derive the exact bilinear formalism and soliton solutions of the system, respectively. Furthermore, we also obtain the linear stability analysis via analyzing the stability condition of the system. Finally, we discuss two kinds of interaction phenomena between solitary wave solutions. It is hoped that our results can be used to enrich the dynamical of the nonlinear Schrödinger system.

Keywords

An integrable coupled nonlinear Schrödinger system Integrable Kadomtsev–Petviashvili system Hirota bilinear method Stability analysis Resonant interactions 

Notes

Acknowledgements

We express our sincere thanks to the Editor and Reviewers for their valuable comments. This work was supported by the Jiangsu Province Natural Science Foundation of China under Grant No. BK20151139. This work was supported by the Postgraduate Research & Practice Program of Education & Teaching Reform of CUMT under Grant No. YJSJG_2018_036, the Qinglan Engineering project of Jiangsu Universities, the National Natural Science Foundation of China under Grant No. 11301527, the Fundamental Research Fund for the Central Universities under the Grant No. 2017XKQY101 and the General Financial Grant from the China Postdoctoral Science Foundation under Grant Nos. 2015M570498 and 2017T100413.

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.: Soliton, Nonlinear Evolution Equation and Inverse Scattering. Cambridge University Press, New York (1991)CrossRefGoogle Scholar
  2. 2.
    Korteweg, P.G., Vries, G.: On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag. 39, 422–443 (1895)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fan, E.G.: Extended tanh-function method and its applications to nonlinear equations. Phys. Lett. A 277, 212–218 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Wazwaz, A.M.: Partial Differential Equations: Methods and Applications. Balkema Publishers, Rotterdam (2002)zbMATHGoogle Scholar
  5. 5.
    Wazwaz, A.M.: Gaussian solitary wave solutions for nonlinear evolution equations with logarithmic nonlinearities. Nonlinear Dyn. 83, 591–596 (2016)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wazwaz, A.M., Xu, G.Q.: Negative-order modified KdV equations: multiple soliton and multiple singular soliton solutions. Math. Methods Appl. Sci. 39(4), 661–667 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differ. Equ. 264(4), 2633–2659 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Yan, X.W., Tian, S.F., Dong, M.J., Zhou, L., Zhang, T.T.: Characteristics of solitary wave, homoclinic breather wave and rogue wave solutions in a (2+1)-dimensional generalized breaking soliton equation. Comput. Math. Appl. 76(1), 179–186 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Tian, S.F.: Asymptotic behavior of a weakly dissipative modified two-component Dullin–Gottwald–Holm system. Appl. Math. Lett. 83, 65–72 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Tu, J.M., Tian, S.F., Xu, M.J., Song, X.Q., Zhang, T.T.: Bäcklund transformation, infinite conservation laws and periodic wave solutions of a generalized (3+1)-dimensional nonlinear wave in liquid with gas bubbles. Nonlinear Dyn. 83, 1199–1215 (2016)CrossRefGoogle Scholar
  11. 11.
    Qin, C.Y., Tian, S.F., Wang, X.B., Zhang, T.T., Li, J.: Rogue waves, bright-dark solitons and traveling wave solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation. Comput. Math. Appl. 75(12), 4221–4231 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Wang, X.B., Tian, S.F., Zhang, T.T.: Characteristics of the breather and rogue waves in a (2+1)-dimensional nonlinear Schrödinger equation. Proc. Am. Math. Soc. 146(8), 3353–3365 (2018)CrossRefGoogle Scholar
  13. 13.
    Nakamura, Y., Hirota, R.: Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–4 (1971)CrossRefGoogle Scholar
  14. 14.
    Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  15. 15.
    Zheng, C.L., Fang, J.P.: New exact solutions and fractal patterns of generalized Broer–Kaup system via a mapping approach. Chaos Solitons Fractals 27, 1321–7 (2006)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chen, J.C., Chen, Y., Feng, B.F., Maruno, K.: Rational solutions to two- and one-dimensional multicomponent Yajima–Oikawa systems. Phys. Lett. A 379, 1510 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Dai, C.Q., Liu, J., Fan, Y., Yu, D.G.: Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality. Nonlinear Dyn. 88(2), 1373–1383 (2017)CrossRefGoogle Scholar
  18. 18.
    Yan, X.W., Tian, S.F., Dong, M.J., Zou, L.: Bäcklund transformation, rogue wave solutions and interaction phenomena for a (3+1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation. Nonlinear Dyn. 92(2), 709–720 (2018)CrossRefGoogle Scholar
  19. 19.
    Yan, X.W., Tian, S.F., Dong, M.J., Wang, X.B., Zhang, T.T.: Nonlocal symmetries, conservation laws and interaction solutions of the generalised dispersive modified Benjamin–Bona–Mahony equation. Z. Naturforsch. A 73(5), 399–405 (2018)Google Scholar
  20. 20.
    Dong, M.J., Tian, S.F., Yan, X.W., Zou, L.: Solitary waves, homoclinic breather waves and rogue waves of the (3+1)-dimensional Hirota bilinear equation. Comput. Math. Appl. 75(3), 957–964 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tian, S.F., Zhou, S.W., Jiang, W.Y., Zhang, H.Q.: Analytic solutions, Darboux transformation operators and supersymmetry for a generalized one-dimensional time-dependent Schrödinger equation. Appl. Math. Comput. 218(13), 7308–7321 (2012)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Wen, X.Y.: Construction of new exact rational form non-travelling wave solutions to the (2+1)-dimensional generalized Broer–Kaup system. Appl. Math. Comput. 217, 1367–75 (2010)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lv, X., Ma, W.X., Zhou, Y., Khalique, C.M.: Rational solutions to an extended Kadomtsev–Petviashvili-like equation with symbolic computation. Comput. Math. Appl. 71, 1560–1567 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Boiti, M., Martina, L., Pashaev, O.K., Pempineli, F.: Dynamics of multidimensional solitons. Phys. Lett. A 160, 55 (1991)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wang, D.S., Wang, X.L.: Long-time asymptotics and the bright N-soliton solutions of the Kundu–Eckhaus equation via the Riemann–Hilbert approach. Nonlinear Anal. 41, 334–361 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wang, D.S., Shi, Y.R., Feng, W.X., Wen, L.: Dynamical and energetic instabilities of \(F=2\) spinor Bose–Einstein condensates in an optical lattice. Phys. D 351–352, 30–41 (2017)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Dai, C.Q., Zhou, G.Q., Chen, R.P., Lai, X.J., Zheng, J.: Vector multipole and vortex solitons in two-dimensional Kerr media. Nonlinear Dyn. 88, 2629–2635 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Dai, C.Q., Wang, Y.Y., Fan, Y., Yu, D.G.: Reconstruction of stability for Gaussian spatial solitons in quintic-septimal nonlinear materials under-symmetric potentials. Nonlinear Dyn. 92, 1351–1358 (2018)CrossRefGoogle Scholar
  29. 29.
    Yu, F.J.: Dynamics of nonautonomous discrete rogue wave solutions for an Ablowitz–Musslimani equation with PT-symmetric potential. Chaos 27, 023108 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wang, L., Zhang, J.H., Wang, Z.Q., Liu, C., Li, M., Qi, F.H., Guo, R.: Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation. Phys. Rev. E 93, 012214 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Guo, R., Hao, H.Q.: Breathers and localized solitons for the Hirota–Maxwell–Bloch system on constant backgrounds in erbium doped fibers. Ann. Phys. 344, 10–16 (2014)CrossRefGoogle Scholar
  32. 32.
    Wang, X.B., Tian, S.F., Feng, L.L., Zhang, T.T.: On quasi-periodic waves and rogue waves to the (4+1)-dimensional nonlinear Fokas equation. J. Math. Phys. 59, 073505 (2018)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Feng, L.L., Tian, S.F., Wang, X.B., Zhang, T.T.: Rogue waves, homoclinic breather waves and soliton waves for the (2+1)-dimensional B-type Kadomtsev–Petviashvili equation. Appl. Math. Lett. 65, 90–97 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Wang, X.B., Tian, S.F., Qin, C.Y., Zhang, T.T.: Characteristics of the breathers, rogue waves and solitary waves in a generalized (2+1)-dimensional Boussinesq equation. EPL 115, 10002 (2016)CrossRefGoogle Scholar
  35. 35.
    Wang, X.B., Tian, S.F., Yan, H., Zhang, T.T.: On the solitary waves, breather waves and rogue waves to a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation. Comput. Math. Appl. 74, 556–563 (2017)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Feng, L.L., Zhang, T.T.: Breather wave, rogue wave and solitary wave solutions of a coupled nonlinear Schrödinger equation. Appl. Math. Lett. 78, 133–140 (2018)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Wang, X.B., Tian, S.F., Qin, C.Y., Zhang, T.T.: Dynamics of the breathers, rogue waves and solitary waves in the (2+1)-dimensional Ito equation. Appl. Math. Lett. 68, 40–47 (2017)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Wang, X.B., Tian, S.F., Qin, C.Y., Zhang, T.T.: Characteristics of the solitary waves and rogue waves with interaction phenomena in a generalized (3+1)-dimensional Kadomtsev–Petviashvili equation. Appl. Math. Lett 72, 58–64 (2017)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Qin, C.Y., Tian, S.F., Wang, X.B., Zhang, T.T.: On breather waves, rogue waves and solitary waves to a generalized (2+1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation. Commun. Nonlinear Sci. Numer. Simul. 62, 378–385 (2018)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Kadomtsev, B.B., Petviashvili, V.I.: On the stability of solitary waves in weakly dispersive media. Sov. Phys. Dokl. 15, 539–541 (1970)zbMATHGoogle Scholar
  41. 41.
    Maccari, A.: The Kadomtsev–Petviashvili equation as a source of integrable model equations. J. Math. Phys. 37, 6207 (1996)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Meng, G.Q., Gao, Y.T., Yu, X., Shen, Y.J., Qin, Y.: Multi-soliton solutions for the coupled nonlinear Schrödinger-type equations. Nonlinear Dyn. 70, 609 (2012)CrossRefGoogle Scholar
  43. 43.
    Ma, W.X., Qin, Z.Y., La, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84, 923–31 (2016)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Ma, W.X., You, Y.: Solving the Korteweg–de Vries equation by its bilinear form: Wronskian solutions. Trans. Am. Math. Soc. 357, 1753–1778 (2005)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Ma, W.X., Li, C.X., He, J.: A second Wronskian formulation of the Boussinesq equation. Nonlinear Anal. 70, 4245–4258 (2009)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Tian, S.F., Zhang, H.Q.: On the integrability of a generalized variable-coefficient forced Korteweg–de Vries equation in fluids. Stud. Appl. Math. 132, 212 (2014)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Tian, S.F., Zhang, H.Q.: On the integrability of a generalized variable-coefficient Kadomtsev–Petviashvili equation. J. Phys. A Math. Theor. 45, 055203 (2012)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Xu, M.J., Tian, S.F., Tu, J.M., Ma, P.L., Zhang, T.T.: On quasiperiodic wave solutions and integrability to a generalized (2+1)-dimensional Korteweg–de Vries equation. Nonlinear Dyn. 82, 2031–2049 (2015)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Tu, J.M., Tian, S.F., Xu, M.J., Ma, P.L., Zhang, T.T.: On periodic wave solutions with asymptotic behaviors to a (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation in fluid dynamics. Comput. Math. Appl. 72, 2486–2504 (2016)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Wang, X.B., Zhang, T.T., Dong, M.J.: Dynamics of the breathers and rogue waves in the higher-order nonlinear Schrödinger equation. Appl. Math. Lett. 86, 298–304 (2018)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Wang, X.B., Tian, S.F., Feng, L.L., Yan, H., Zhang, T.T.: Quasiperiodic waves, solitary waves and asymptotic properties for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation. Nonlinear Dyn. 88, 2265–2279 (2017)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Wazwaz, A.M.: Multiple-soliton solutions for the fifth-order Caudrey–Dodd–Gibbon equation. Appl. Math. Comput. 197, 719–724 (2008)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Wazwaz, A.M.: Multiple soliton solutions for (2+1)-dimensional Sawada–Kotera and Caudrey–Dodd–Gibbon equations. Math. Methods Appl. Sci. 34, 1580–1586 (2011)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Wazwaz, A.M., El-Tantawy, S.A.: Solving the (3+1)-dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota’s method. Nonlinear. Dyn 88, 3017–3021 (2017)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Tian, S.F.: The mixed coupled nonlinear Schrödinger equation on the half-line via the Fokas method. Proc. R. Soc. Lond. A 472, 20160588 (2016)CrossRefGoogle Scholar
  56. 56.
    Tian, S.F.: Initial-boundary value problems for the coupled modified Korteweg–de Vries equation on the interval. Commun. Pure Appl. Anal. 173, 923–957 (2018)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Tian, S.F., Zhang, T.T.: Long-time asymptotic behavior for the Gerdjikov–Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition. Proc. Am. Math. Soc. 146(4), 1713–1729 (2018)CrossRefGoogle Scholar
  58. 58.
    Tian, S.F.: Initial-boundary value problems of the coupled modified Korteweg–de Vries equation on the half-line via the Fokas method. J. Phys. A Math. Theor. 50, 395204 (2017)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Tu, J.M., Tian, S.F., Xu, M.J., Zhang, T.T.: On Lie symmetries, optimal systems and explicit solutions to the Kudryashov–Sinelshchikov equation. Appl. Math. Comput. 275, 345–352 (2016)MathSciNetGoogle Scholar
  60. 60.
    Tian, S.F., Zhang, Y.F., Feng, B.L., Zhang, H.Q.: On the Lie algebras, generalized symmetries and Darboux transformations of the fifth-order evolution equations in shallow water. Chin. Ann. Math. B 36(4), 543–560 (2015)MathSciNetCrossRefGoogle Scholar
  61. 61.
    Tian, S.F.: Initial-boundary value problems for the general coupled nonlinear Schrödinger equations on the interval via the Fokas method. J. Differ. Equ. 262, 506–558 (2017)CrossRefGoogle Scholar
  62. 62.
    Feng, L.L., Tian, S.F., Zhang, T.T., Zhou, J.: Nonlocal symmetries, consistent riccati expansion, and analytical solutions of the variant Boussinesq system. Z. Naturforsch. A 72(7), 655–663 (2017)CrossRefGoogle Scholar
  63. 63.
    Wang, X.B., Tian, S.F., Qin, C.Y., Zhang, T.T.: Lie symmetry analysis, analytical solutions, and conservation laws of the generalised Whitham–Broer–Kaup–Like equations. Z. Naturforsch. A 72(3), 269–279 (2017)CrossRefGoogle Scholar
  64. 64.
    Feng, L.L., Tian, S.F., Zhang, T.T.: Nonlocal symmetries and consistent riccati expansions of the (2+ 1)-dimensional dispersive long wave equation. Z. Naturforsch. A 72(5), 425–431 (2017)CrossRefGoogle Scholar
  65. 65.
    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)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Li, M., Xu, T., Meng, D.: Rational solitons in the parity-time-symmetric nonlocal nonlinear Schrödinger model. J. Phys. Soc. Jpn. 85, 124001 (2016)CrossRefGoogle Scholar
  67. 67.
    Li, M., Xu, T., Wang, L., Qi, F.H.: Nonautonomous solitons and interactions for a variable-coefficient resonant nonlinear Schrödinger equation. Appl. Math. Lett. 60, 8–13 (2016)MathSciNetCrossRefGoogle Scholar
  68. 68.
    Li, M., Shui, J.J., Xu, T.: Generation mechanism of rogue waves for the discrete nonlinear Schrödinger equation. Appl. Math. Lett. 83, 110 (2018)MathSciNetCrossRefGoogle Scholar
  69. 69.
    Tian, S.F., Zhang, H.Q.: Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations. J. Math. Anal. Appl. 371, 585–608 (2010)MathSciNetCrossRefGoogle Scholar
  70. 70.
    Tu, J.M., Tian, S.F., Xu, M.J., Zhang, T.T.: Quasi-periodic waves and solitary waves to a generalized KdV–Caudrey–Dodd–Gibbon equation from fluid dynamics. Taiwan. J. Math. 20, 823–848 (2016)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Tian, S.F., Zhang, H.Q.: Riemann theta functions periodic wave solutions and rational characteristics for the (1+ 1)-dimensional and (2+1)-dimensional Ito equation. Chaos Solitons Fractals 47, 27 (2013)MathSciNetCrossRefGoogle Scholar
  72. 72.
    Xu, M.J., Tian, S.F., Tu, J.M., Zhang, T.T.: Bäcklund transformation, infinite conservation laws and periodic wave solutions to a generalized (2+1)-dimensional Boussinesq equation. Nonlinear Anal. Real World Appl. 31, 388–408 (2016)MathSciNetCrossRefGoogle Scholar
  73. 73.
    Tian, S.F., Zhang, H.Q.: A kind of explicit Riemann theta functions periodic wave solutions for discrete soliton equations. Commun. Nonlinear Sci. Numer. Simul. 16, 173–186 (2010)MathSciNetCrossRefGoogle Scholar

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.School of Mathematics and Institute of Mathematical PhysicsChina University of Mining and TechnologyXuzhouPeople’s Republic of China
  2. 2.Department of EngineeringUniversity of CambridgeCambridgeUK

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