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Bright–dark solitons in the space-shifted nonlocal coupled nonlinear Schrödinger equation

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Abstract

Multiple bright–dark soliton solutions in terms of determinants for the space-shifted nonlocal coupled nonlinear Schrödinger equation are constructed by using the bilinear (Kadomtsev–Petviashvili) KP hierarchy reduction method. It is found that the bright–dark two-soliton only occurs elastic collisions. Upon their amplitudes, the bright two solitons only admit one pattern whose amplitude are equal, and the dark two solitons have three different non-degenerated patterns and two different degenerated patterns. The bright–dark four-soliton is the superposition of the two-soliton pairs and can generate the bound-state solitons. The multiple double-pole bright–dark soliton solutions are derived through a long wave limit of the obtained bright–dark soliton solutions, and their collision dynamics are also investigated.

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References

  1. Kevrekidis, P.G., Frantzeskakis, D.J.: Solitons in coupled nonlinear Schrödinger models: a survey of recent developments. Rev. Phys. 1, 140 (2016)

    Article  Google Scholar 

  2. Bashkin, E.P., Vagov, A.V.: Instability and stratification of a two-component Bose-Einstein condensate in a trapped ultracold gas. Phys. Rev. B 56, 6207 (1997)

    Article  Google Scholar 

  3. Abowitz, M.J., Horikis, T.K.: Interacting nonlinear wave envelopes and rogue wave formation in deep water. Phys. Fluids 27, 012107 (2015)

    Article  MATH  Google Scholar 

  4. Bailung, H., Sharma, S.K., Nakamura, Y.: Observation of peregrine solitons in a multicomponent plasma with negative ions. Phys. Rev. Lett. 107, 255005 (2011)

    Article  Google Scholar 

  5. Akhmediev, N., Ankiewicz, A.: Solitons: Nonlinear Pulses and Beams. Chapman and Hall, London (1997)

    MATH  Google Scholar 

  6. Makhan’kov, V.G., Pashaev, O.K.: Nonlinear Schrödinger equation with noncompact isogroup. Theor. Math. Phys. 53, 979 (1982)

    Article  Google Scholar 

  7. Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110, 064105 (2013)

    Article  Google Scholar 

  8. Chen, K., Deng, X., Lou, S.Y., Zhang, D.J.: Solutions of nonlocal equations reduced from the AKNS hierarchy. Stud. Appl. Math. 141, 113 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feng, B.F., Luo, X.D., Ablowitz, A.J., Musslimani, Z.H.: General soliton solution to a nonlocal nonlinear Schrödinger equation with zero and nonzero boundary conditions. Nonlinearity 31, 5385 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gürses, M., Pekcan, A.: Nonlocal nonlinear Schrödinger equations and their soliton solutions. J. Math. Phys. 59, 051501 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ablowitz, M.J., Musslimani, Z.H.: Integrable nonlocal nonlinear equations. Stud. Appl. Math. 139, 7 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fokas, A.S.: Integrable multidimensional versions of the nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 319 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Rao, J., Zhang, Y., Fokas, A.S., He, J.: Rogue waves of the nonlocal Davey-Stewartson I equation. Nonlinearity 31, 4090–4107 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rao, J., He, J., Mihalache, D., Cheng, Y.: \(PT\)-symmetric nonlocal Davey-Stewartson I equation: general lump-soliton solutions on a background of periodic line waves. Appl. Math. Lett. 106, 106246 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. Yan, Z.: Integrable \(PT\)-symmetric local and nonlocal vector nonlinear Schrödinger equations: a unified two-parameter model. Appl. Math. Lett. 47, 61 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang, B., Yang, J.: Transformations between nonlocal and local integrable equations. Stud. Appl. Math. 140, 178 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yang, B., Chen, Y.: Several reverse-time integrable nonlocal nonlinear equations: Rogue-wave solutions. Chaos 28, 053104 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shi, Y., Zhang, Y.S., Xu, S.W.: Families of nonsingular soliton solutions of a nonlocal Schrödinger-Boussinesq equation. Nonlinear Dyn. 94, 2327–2334 (2018)

    Article  MATH  Google Scholar 

  19. Shi, Y., Shen, S., Zhao, S.: Solutions and connections of nonlocal derivative nonlinear Schrödinger equations. Nonlinear Dyn. 95, 1257–1267 (2019)

    Article  MATH  Google Scholar 

  20. Liu, Y., Li, B.: Dynamics of solitons and breathers on a periodic waves background in the nonlocal Mel’nikov equation. Nonlinear Dyn. 100, 3717–3731 (2020)

    Article  Google Scholar 

  21. Chen, J., Yan, Q., Zhang, H.: Multiple bright soliton solutions of a reverse-space nonlocal nonlinear Schrödinger equation. Appl. Math. Lett. 106, 106375 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ablowitz, M.J., Musslimani, Z.H.: Integrable space-time shifted nonlocal nonlinear equations. Phys. Lett. A 409, 127516 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gürses, M., Pekcan, A.: Soliton solutions of the shifted nonlocal NLS and MKdV equations, arXiv:210614252v2 [nlin.SI]

  24. Liu, S.M., Wang, J., Zhang, D.J.: Solutions to integrable space-time shifted nonlocal equations, arXiv:210704183v1 [nlin.SI]

  25. Stalin, S., Senthilvelan, M., Lakshmanan, M.: Energysharing collisions and the dynamics of degenerate solitons in the nonlocal Manakov system. Nonlinear Dyn. 95, 343 (2019)

    Article  MATH  Google Scholar 

  26. Rao, J., He, J., Kanna, T., Mihalache, D.: Nonlocal M-component nonlinear Schrödinger equations: Bright solitons, energy-sharing collisions, and positons. Phys. Rev. E 102, 032201 (2021)

    Article  Google Scholar 

  27. Zhang, Z., Li, B., Guo, Q.: Construction of higher-order smooth positons and breather positons via Hirota’s bilinear method. Nonlinear Dyn. 105, 2611–2618 (2021)

    Article  Google Scholar 

  28. Gagon, L., Stiévenart, N.: \(N\)-soliton interaction in optical fibers: the multiple-pole case. Opt. Lett. 19, 619 (1994)

    Article  Google Scholar 

  29. Aktosun, T., Demontis, F., van der Mee, C.: Exact solutions to the focusing nonlinear Schrödinger equation. Inverse Probl. 23, 2171–2195 (2007)

    Article  MATH  Google Scholar 

  30. Martines, T.: Generalized inverse scattering transform for the nonlinear Schrödinger equation for bound states with higher multiplicities. Electron. J. Differ. Equ. 179, 1–15 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Tanaka, S.: Non-linear Schrödinger equation and modified Korteweg-de Vries equation; construction of solutions in terms of scattering data. Publ. RIMS, Kyoto Univ. 10, 329–357 (1975)

    Article  MATH  Google Scholar 

  32. Olmedilla, E.: Multiple pole solutions of the non-linear Schrödinger equation. Physica D 25, 330–346 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  33. Schiebold, C.: Asymptotics for the multiple pole solutions of the nonlinear Schrödinger equation. Nonlinearity 30, 2930–2981 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, Y., Tao, X., Yao, T., He, J.: The regularity of the multiple higher-order poles solitons of the NLS equation. Stud. Appl. Math. 145, 812–827 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  35. Bilman, D., Buckingham, R.: Large-Order Asymptotics for Multiple-pole solitons of the focusing nonlinear Schrödinger Equation. J. Noninear Sci. 29, 2185–2229 (2019)

    Article  MATH  Google Scholar 

  36. Lai, D.W.C., Chow, K.W., Nakkeeran, K.: Multiple-pole soliton interactions in optical fibres with higher-order effects. J. Mod. Optic 51, 455–460 (2004)

    Article  Google Scholar 

  37. Zhang, X., Ling, L.: Asymptotic analysis of high-order solitons for the Hirota equation. Phyica D 426, 132982 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  38. Li, M., Zhang, X., Xu, T., Li, L.: Asymptotic analysis and soliton interactions of the multi-pole solutions in the Hirota equation. J. Phys. Soc. Jpn. 89, 054004 (2020)

    Article  Google Scholar 

  39. Wadati, M., Ohkuma, K.: Multiple-pole solutions of the modified Korteweg-de Vries equation. J. Phys. Soc. Jpn. 51, 2029–2035 (1982)

    Article  MathSciNet  Google Scholar 

  40. Zhang, D.J., Zhao, S.L., Sun, Y.Y., Zhou, J.: Solutions to the modified Korteweg-de Vries equation. Rev. Math. Phys. 26, 1430006 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  41. Aktosun, T., Demontis, F., van der Mee, C.: Exact solutions to the sine-Gordon equation. J. Math. Phys. 51, 123521 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. Pöppe, C.: Construction of solutions of the sine-Gordon equation by means of Fredholm determinants. Physica D 9, 103–139 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  43. Tsuru, H., Wadati, M.: The multiple pole solutions of the sine-Gordon equation. J. Phys. Soc. Jpn. 53, 2908–2921 (1984)

    Article  MathSciNet  Google Scholar 

  44. Rao, J., Kanna, T., Sakkaravarthi, K., He, J.: Multiple double-pole bright-bright and bright-dark solitons and energy-exchanging collision in the M-component nonlinear Schrödinger equations. Phys. Rev. E 103, 062214 (2021)

    Article  Google Scholar 

  45. Shchesnovich, V., Yang, J.: Higher-order solitons in the \(N\)-wave system. Stud. Appl. Math. 110, 297–332 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Kuang, Y., Zhu, J.: The higher-order soliton solutions for the coupled Sasa-Satsuma system via the \(\partial \)-dressing method. Appl. Math. Lett. 66, 47–53 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  47. Jimbo, M., Miwa, T.: Solitons and infinite dimensional Lie algebras. Publ. RIMS Kyoto Univ. 19, 943–1001 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  48. Date, E., Kashiwara, M., Jimbo, M., Miwa, T.: Transformation groups for soliton equations, in Nonlinear Integrable Systems–Classical Theory and Quantum Theory. eds. M. Jimbo and T. Miwa ( World Scientific, Singapore, 1983)

  49. Hirota, R.: The direct method in soliton theory. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  50. Ohta, Y., Wang, D.S., Yang, J.: General N-dark-dark solitons in the coupled nonlinear Schrödinger Equations. Stud. Appl. Math. 127, 345–371 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Ohta, Y., Yang, J.: General high-order rogue waves and their dynamics in the nonlinear Schrödinger equation. Proc. R. Soc. A 468, 1716 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by the NSCF of China under Grant Nos. 12071304

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Authors and Affiliations

  1. College of Medicine and Health Science, WuHan Polytechnic University, Wuhan, 430023, Hubei, People’s Republic of China

    Ping Ren

  2. School of Mathematics and Statistics, Hubei University of Science and Technology, Xianning, 437100, Hubei, People’s Republic of China

    Jiguang Rao

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  1. Ping Ren
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Ren, P., Rao, J. Bright–dark solitons in the space-shifted nonlocal coupled nonlinear Schrödinger equation. Nonlinear Dyn 108, 2461–2470 (2022). https://doi.org/10.1007/s11071-022-07269-x

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