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Bright soliton solutions to a nonlocal nonlinear Schrödinger equation of reverse-time type

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Abstract

Two types of multiple bright soliton solutions for a focusing nonlocal reverse-time nonlinear Schrödinger equation are derived by using the Hirota’s bilinear method and the Kadomtsev–Petviashvili hierarchy reduction. Compared with the case of the local Manakov system, both types of soliton solutions are restricted to ones with even numbers. For the first type of solution, the fundamental paired soliton exhibits two paralleled line solitons along with the time axis and the special breathing soliton. For the second type of solution, the fundamental paired soliton only allows head-on collisions with the same velocity, in which the collision with spatial interferences and the degenerate soliton with position shifts can be realized with certain parameters. General higher-order soliton solutions of two types describe the superposition of the fundamental paired soliton, which show a few interesting dynamical behaviors such as the superposed breathing solitons, the interaction of two paired soliton bound states, the interaction of two degenerate solitons with position shifts and the breathing soliton with position shifts.

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References

  1. Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243 (1998)

    MathSciNet  MATH  Google Scholar 

  2. Konotop, V.V., Yang, J., Zezyulin, D.A.: Nonlinear waves in PT-symmetric systems. Rev. Modern Phys. 88, 035002 (2016)

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Ablowitz, M.J., Musslimani, Z.H.: Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29, 915 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Ablowitz, M.J., Luo, X.D., Musslimani, Z.H.: Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions. J. Math. Phys. 59, 011501 (2018)

    MathSciNet  MATH  Google Scholar 

  7. 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)

    MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. 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)

    MathSciNet  MATH  Google Scholar 

  10. Yang, B., Chen, Y.: Dynamics of high-order solitons in the nonlocal nonlinear Schrödinger equations. Nonlinear Dyn. 94, 489 (2018)

    MATH  Google Scholar 

  11. Wen, X.Y., Yan, Z.Y., Yang, Y.Q.: Dynamics of higher-order rational solitons for the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potential. Chaos 26, 063123 (2016)

    MathSciNet  Google Scholar 

  12. Zhang, G.Q., Yan, Z.Y., Chen, Y.: Novel higher-order rational solitons and dynamics of the defocusing integrable nonlocal nonlinear Schrödinger equation via the determinants. Appl. Math. Lett. 69, 113 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Yang, J.K.: General \(N\)-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations. Phys. Lett. A 383, 328 (2019)

    MathSciNet  Google Scholar 

  14. Yang, B., Yang, J.K.: Rogue waves in the nonlocal PT-symmetric nonlinear Schrödinger equation. Lett. Math. Phys. 109, 945 (2019)

    MathSciNet  MATH  Google Scholar 

  15. Xu, T., Lan, S., Li, M., Li, L.L., Zhang, G.W.: Mixed soliton solutions of the defocusing nonlocal nonlinear Schrödinger equation. Physica D 390, 47 (2019)

    MathSciNet  Google Scholar 

  16. Yang, B., Yang, J.K.: Transformations between nonlocal and local integrable equations. Stud. Appl. Math. 140, 178 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Lou, S.Y., Huang, F.: Alice–Bob physics: coherent solutions of nonlocal KdV systems. Sci. Rept. 7, 869 (2017)

    Google Scholar 

  18. Lou, S.Y.: Alice–Bob systems, \(P_s-T_d-C\) symmetry invariant and symmetry breaking soliton solutions. J. Math. Phys. 59, 083507 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Li, C.C., Lou, S.Y., Jia, M.: Coherent structure of Alice–Bob modified Korteweg de–Vries equation. Nonlinear Dyn. 93, 1799 (2018)

    MATH  Google 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. Number. Simul. 42, 699 (2017)

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. Zhou, Z.X.: Darboux transformations and global explicit solutions for nonlocal Davey–Stewartson I equation. Stud. Appl. Math. 141, 186 (2018)

    MathSciNet  MATH  Google Scholar 

  24. Yang, B., Chen, Y.: Reductions of Darboux transformations for the PT-symmetric nonlocal Davey–Stewartson equations. Appl. Math. Lett. 82, 43 (2018)

    MathSciNet  MATH  Google Scholar 

  25. Yang, B., Chen, Y.: Dynamics of rogue waves in the partially PT-symmetric nonlocal Davey–Stewartson systems. Commun. Nonlinear Sci. Numer. Simulat. 69, 287 (2019)

    MathSciNet  Google Scholar 

  26. Dai, C.Q., Wang, Y.Y.: Spatiotemporal localizations in \((3+1)\)-dimensional PT-symmetric and strongly nonlocal nonlinear media. Nonlinear Dyn. 83, 2453 (2016)

    MathSciNet  Google Scholar 

  27. Yan, Z.Y.: Integrable PT-symmetric local and nonlocal vector nonlinear Schrödinger equations: a unified twoparameter model. Appl. Math. Lett. 47, 61 (2015)

    MathSciNet  Google Scholar 

  28. Yan, Z.Y.: A novel hierarchy of two-family-parameter equations: local, nonlocal, and mixed-local–nonlocal vector nonlinear Schrödinger equations. Appl. Math. Lett. 79, 123 (2018)

    MathSciNet  Google Scholar 

  29. Sinha, D., Ghosh, P.K.: Integrable nonlocal vector nonlinear Schrödinger equation with self-induced parity-time-symmetric potential. Phys. Lett. A 381, 124 (2017)

    MathSciNet  MATH  Google Scholar 

  30. Stalin, S., Senthilvelan, M., Lakshmanan, M.: Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: I symmetry preserving and symmetry breaking solutions. Nonlinear Dyn. 95, 343 (2019)

    Google Scholar 

  31. Stalin, S., Senthilvelan, M., Lakshmanan, M.: Energy-sharing collisions and the dynamics of degenerate solitons in the nonlocal Manakov system. Nonlinear Dyn. 95, 1767 (2019)

    MATH  Google Scholar 

  32. Ablowitz, M.J., Musslimani, Z.H.: Integrable discrete PT symmetric model. Phys. Rev. E 90, 032912 (2014)

    Google Scholar 

  33. Sarma, A.K., Miri, M.A., Musslimani, Z.H.: Continuous and discrete Schrödinger systems with parity-time-symmetric nonlinearities. Phys. Rev. E 89, 052918 (2014)

    Google Scholar 

  34. Grahovski, G.G., Mohammed, A.J., Susanto, H.: Nonlocal reductions of the Ablowitz–Ladik equation. Theor. Math. Phys. 197, 1412 (2018)

    MathSciNet  MATH  Google Scholar 

  35. Xu, T., Li, H.J., Zhang, H.J., Li, M., Lan, S.: Darboux transformation and analytic solutions of the discrete PT-symmetric nonlocal nonlinear Schrödinger equation. Appl. Math. Lett. 63, 88 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Hanif, Y., Saleem, U.: Broken and unbroken PT-symmetric solutions of semi-discrete nonlocal nonlinear Schrödinger equation. Nonlinear Dyn. 98, 233 (2019)

    MATH  Google Scholar 

  37. Deng, X., Lou, S.Y., Zhang, D.J.: Bilinearisation-reduction approach to the nonlocal discrete nonlinear Schrödinger equations. Appl. Math. Comput. 332, 477 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Hu, Y.H., Chen, J.C.: Solutions to nonlocal integrable discrete nonlinear Schrödinger equations via reduction. Chin. Phys. Lett. 35, 110201 (2018)

    Google Scholar 

  39. Chen, J.C., Feng, B.F., Jin, Y.Y.: Gram determinant solutions to nonlocal integrable discrete nonlinear Schrödinger equations via the pair reduction. Wave Motion 93, 102487 (2019)

    Google Scholar 

  40. Yang, J.K.: Physically significant nonlocal nonlinear Schrödinger equation and its soliton solutions. Phys. Rev. E 98, 042202 (2018)

    MathSciNet  Google Scholar 

  41. Lou, S.Y.: Prohibitions caused by nonlocality for nonlocal Boussinesq–KdV type systems. Stud. Appl. Math. 143, 123 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Li, H., Lou, S.Y.: Multiple soliton solutions of Alice–Bob Boussinesq equations. Chin. Phys. Lett. 36, 050501 (2019)

    Google Scholar 

  43. Feng, B.F.: General \(N\)-soliton solution to a vector nonlinear Schrödinger equation. J. Phys. A: Math. Theor. 47, 355203 (2014)

    MATH  Google Scholar 

  44. Gilson, C., Hietarinta, J., Nimmo, J., Ohta, Y.: Sasa–Satsuma higher-order nonlinear Schrödinger equation and its bilinearization and multisoliton solutions. Phys. Rev. E 68, 016614 (2003)

    MathSciNet  Google Scholar 

  45. Zhao, L.C., Ling, L., Yang, Z.Y., Liu, J.: Properties of the temporal–spatial interference pattern during soliton interaction. Nonlinear Dyn. 83, 659 (2016)

    MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by the NNSF of China (Nos. 11705077 and 11775104), the General Scientific Research Project of Zhejiang Education Department( No. Y201738648).

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  1. Department of Mathematics and Institute of Nonlinear Analysis, Lishui University, Lishui, 323000, China

    Junchao Chen & Qixiu Yan

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  1. Junchao Chen
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  2. Qixiu Yan
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Correspondence to Junchao Chen.

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Chen, J., Yan, Q. Bright soliton solutions to a nonlocal nonlinear Schrödinger equation of reverse-time type. Nonlinear Dyn 100, 2807–2816 (2020). https://doi.org/10.1007/s11071-020-05673-9

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  • DOI: https://doi.org/10.1007/s11071-020-05673-9

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