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On general solitons in the parity-time-symmetric defocusing nonlinear Schrödinger equation

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Abstract

Via the Hirota bilinear method combined with the Kadomtsev–Petviashvili (KP) hierarchy reduction method, two types of solitons, namely general solitons on a background of periodic waves and rational soliton solutions, to the parity-time-symmetric nonlocal nonlinear Schrödinger equation with the defocusing-type nonlinearity for nonzero boundary condition are investigated. These two types of soliton solutions are constructed by constraining the tau functions of single-component KP hierarchy satisfying the dimension reduction and the nonlocal symmetry and the complex conjugated conditions. For the solitons on a background of periodic waves, we first construct the periodic solution to provide the background of periodic waves and then combine the periodic solution with general soliton solution, which generate solitons on a background of periodic waves. For the rational solitons, we mainly investigate the dynamical behaviours of interactions between several individual rational dark–antidark solitons, antidark–antidark solitons, and antidark–dark solitons, which are elastic collisions with no phase shift. The degenerated soliton cases are also discussed, in which only one-antidark soliton survives.

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References

  1. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. 23, 142 (1973)

    Article  Google Scholar 

  2. Hasegawa, A., Tappert, F.: Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion. Appl. Phys. Lett. 23, 171 (1973)

    Article  Google Scholar 

  3. Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, New York (1995)

    MATH  Google Scholar 

  4. Hasegawa, A., Kodama, Y.: Solitons in Optical Communications. Clarendon Press, Oxford (1995)

    MATH  Google Scholar 

  5. Kivshar, Y.S., Agrawal, G.P.: Optical Solitons: From Fibers to Photonic Crystals. Academic Press, New York (2003)

    Google Scholar 

  6. Malomed, B.A., Mihalache, D., Wise, F., Torner, L.: Spatiotemporal optical solitons. J. Opt. B 7, R53 (2005)

    Article  Google Scholar 

  7. Mihalache, D.: Multidimensional localized structures in optical and matter-wave media: a topical survey of recent literature. Rom. Rep. Phys. 69, 403 (2017)

    Google Scholar 

  8. Malomed, B.A., Mihalache, D.: Nonlinear waves in optical and matter-wave media: a topical survey of recent theoretical and experimental results. Rom. J. Phys. 64, 106 (2019)

    Google Scholar 

  9. Dalfovo, F., Giorgini, S., Stringari, L.P.: Theory of Bose–Einstein condensation in trapped gases. Rev. Mod. Phys. 71, 463 (1999)

    Article  Google Scholar 

  10. Bagnato, V.S., Frantzeskakis, D.J., Kevrekidis, P.G., Malomed, B.A., Mihalache, D.: Bose–Einstein condensation: twenty years after. Rom. Rep. Phys. 67, 5 (2015)

    Google Scholar 

  11. Kartashov, Y.V., Astrakharchik, G.E., Malomed, B.A., Torner, L.: Frontiers in multidimensional self-trapping of nonlinear fields and matter. Nat. Rev. Phys. 1, 185 (2019)

    Article  Google Scholar 

  12. Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 4, 190 (1968)

    Google Scholar 

  13. Benney, D.J., Roskes, G.K.: Wave instabilities. Stud. Appl. Math. 48, 377 (1969)

    Article  MATH  Google Scholar 

  14. Zakharov, V.E.: Collapse of Langmuir waves. Sov. Phys. JETP 35, 908 (1972)

    Google Scholar 

  15. Zakharov, V.E., Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62 (1972)

    MathSciNet  Google Scholar 

  16. Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: Nonlinear-evolution equations of physical significance. Phys. Rev. Lett. 31, 125 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ablowitz, M.J., Kaup, D.J., Newell, A.C., Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  18. Peregrine, D.H.: Water waves, nonlinear Schrodinger equations and their solutions. J. Aust. Math. Soc. B 25, 16 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kibler, B., Fatome, J., Finot, C., Millot, G., Dias, F., Genty, G., Akhmediev, N., Dudley, J.M.: The Peregrine soliton in nonlinear fibre optics. Nat. Phys. 6, 790 (2010)

    Article  Google Scholar 

  20. Onorato, M., Residori, S., Bortolozzo, U., Montina, A., Arecchi, F.T.: Rogue waves and their generating mechanisms in different physical contexts. Phys. Rep. 528, 47 (2013)

    Article  MathSciNet  Google Scholar 

  21. Chen, S., Baronio, F., Soto-Crespo, J.M., Grelu, P., Mihalache, D.: Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems. J. Phys. A Math. Theor. 50, 463001 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. 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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ablowitz, M.J., Feng, B.F., Luo, X.D., Musslimani, Z.H.: Reverse space-time nonlocal sine-Gordon/Sinh-Gordon equations with nonzero boundary conditions. Stud. Appl. Math. 141, 267 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ablowitz, M.J., Feng, B.F., Luo, X.D., Musslimani, Z.H.: Inverse scattering transform for the nonlocal reverse space-time nonlinear Schrödinger equation. Theor. Math. Phys. 196, 1241 (2018)

    Article  MATH  Google Scholar 

  28. Rybalko, Y., Shepelsky, D.: Long-time asymptotics for the integrable nonlocal nonlinear Schrödinger equation. J. Math. Phys. 60, 031504 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  30. 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)

    Article  MathSciNet  Google Scholar 

  31. Li, M., Xu, T., Meng, D.X.: Rational solitons in the parity-time-symmetric nonlocal nonlinear Schrödinger model. J. Phys. Soc. Jpn. 85, 124001 (2016)

    Article  Google Scholar 

  32. Zhang, Y.S., Qiu, D.Q., Cheng, Y., He, J.S.: Rational solution of the nonlocal nonlinear Schrödinger equation and its application in optics. Rom. J. Phys. 62, 108 (2017)

    Google Scholar 

  33. Gupta, S.K., Sarma, A.K.: Peregrine rogue wave dynamics in the continuous nonlinear Schrödinger system with parity-time symmetric Kerr nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 36, 141 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Gupta, S.K.: A string of Peregrine rogue waves in the nonlocal nonlinear Schrödinger equation with parity-time symmetric self-induced potential. Opt. Commun. 411, 1 (2018)

    Article  Google Scholar 

  35. Yang, B., Yang, J.: General rogue waves in the nonlocal PT-symmetric nonlinear Schrödinger equation (2017).arXiv:1711.05930

  36. Yang, J.: General N-solitons and their dynamics in several nonlocal nonlinear Schrödinger equations. Phys. Rev. E 383, 328 (2019)

    Google Scholar 

  37. 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)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  40. Khare, A., Saxena, A.: Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations. J. Math. Phys. 56, 032104 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  41. Chen, K., Zhang, D.J.: Solutions of the nonlocal nonlinear Schrödinger hierarchy via reduction. Appl. Math. Lett. 75, 82 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  42. 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)

    Article  MathSciNet  MATH  Google Scholar 

  43. Huang, X., Ling, L.M.: Soliton solutions for the nonlocal nonlinear Schrödinger equation. Eur. Phys. J. Plus 131, 148 (2016)

    Article  Google Scholar 

  44. Rao, J.G., Cheng, Y., Porsezian, K., Mihalache, D., He, J.S.: \(PT\)-symmetric nonlocal Davey-Stewartson I equation: soliton solutions with nonzero background. Phys. D 401, 132180 (2020)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  46. Gadzhimuradov, T.A., Agalarov, A.M.: Towards a gauge-equivalent magnetic structure of the nonlocal nonlinear Schrödinger equation. Phys. Rev. A 93, 062124 (2016)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  50. Gerdjikov, V.S., Grahovski, G.G., Ivanov, R.I.: The N-wave equations with \(PT\) symmetry. Theor. Math. Phys. 188, 1305 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  51. Xu, Z.X., Chow, K.W.: Breathers and rogue waves for a third order nonlocal partial differential equation by a bilinear transformation. Appl. Math. Lett. 56, 72 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  52. 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 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  53. Lou, S.Y., Huang, F.: Alice-Bob physics: coherent solutions of nonlocal KdV systems. Sci. Rep. 7, 869 (2017)

    Article  Google Scholar 

  54. Zhou, Z.X.: Darboux transformations and global solutions for a nonlocal derivative nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 62, 480 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  55. Ji, J.L., Zhu, Z.N.: Soliton solutions of an integrable nonlocal modified Korteweg-de Vries equation through inverse scattering transform. J. Math. Anal. Appl. 453, 973 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  56. Rao, J.G., Cheng, Y., He, J.S.: Rational and semi-rational solutions of the nonlocal Davey–Stewartson equations. Stud. Appl. Math. 139, 568 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  57. Ma, L.Y., Shen, S.F., Zhu, Z.N.: Soliton solution and gauge equivalence for an integrable nonlocal complex modified Korteweg-de Vries equation. J. Math. Phys. 58, 103501 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  59. Gürses, M.: Nonlocal Fordy–Kulish equations on symmetric spaces. Phys. Lett. A 381, 1791 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  60. Yang, B., Chen, Y.: Dynamics of rogue waves in the partially \(PT\)-symmetric nonlocal Davey–Stewartson systems. Stud. Appl. Math. 141, 131 (2018)

    Google Scholar 

  61. 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 

  62. Wu, Z.W., He, J.S.: New hierarchies of derivative nonlinear Schrödinger-type equation. Rom. Rep. Phys. 68, 79 (2016)

    Google Scholar 

  63. Sato, M.: Soliton equations as dynamical systems on a infinite dimensional Grassmann manifolds. RIMS Kokyuroku 439, 30 (1981)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  65. Date, E., Kashiwara, M., Jimbo, M., Miwa, T.: Transformation groups for soliton equations. In: Jimbo, M., Miwa, T. (eds) Nonlinear Integrable Systems–Classical Theory and Quantum Theory, pp. 39–119. World Scientific, Singapore (1983)

  66. 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 

  67. Ohta, Y., Wang, D., Yang, J.: General N-Dark-Dark solitons in the coupled nonlinear Schrödinger equations. Stud. Appl. Math. 127, 345 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  68. Ohta, Y., Yang, J.: Dynamics of rogue waves in the Davey–Stewartson II equation. J. Phys. A Math. Theor. 46, 105202 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  69. Hirota, R.: The Direct Method in Soliton Theory. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank Prof. D. Zuo of USTC for his fruitful suggestions. The work of J. He was supported by the National Natural Science Foundation of China (Grant 12071304) and Shenzhen Science and Technology Program (Grant No. RCBS20200714114922203).The work of J. Rao was supported by the Guangdong Basic and Applied Basic Research Foundation (Grant 2019A1515110208). The work of Y. Cheng was supported by the National Natural Science Foundation of China (Grant 11871446).

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Authors and Affiliations

  1. Institute for Advanced Study, Shenzhen University, Shenzhen, 518060, Guangdong, People’s Republic of China

    Jiguang Rao & Jingsong He

  2. Institute of Microscale Optoelectronics, Shenzhen University, Shenzhen, 518060, Guangdong, People’s Republic of China

    Jiguang Rao

  3. Horia Hulubei National Institute of Physics and Nuclear Engineering, P.O. Box MG–6, 077125, Magurele, Romania

    Dumitru Mihalache

  4. School of Mathematical Sciences, USTC, Hefei, 230026, Anhui, People’s Republic of China

    Yi Cheng

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  1. Jiguang Rao
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Rao, J., He, J., Mihalache, D. et al. On general solitons in the parity-time-symmetric defocusing nonlinear Schrödinger equation. Z. Angew. Math. Phys. 72, 65 (2021). https://doi.org/10.1007/s00033-021-01487-w

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