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Fundamental solitons and dynamical analysis in the defocusing Kerr medium and \(\varvec{\mathcal {PT}}\)-symmetric rational potential

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Abstract

We find that a class of parity-time- (\(\mathcal {PT}\)-) symmetric rational potentials can support stable solitons in the defocusing Kerr-nonlinear media, though they may not enjoy entirely real linear spectra. Analytical expressions of spatial solitons are elicited at lots of isolated propagation-constant points, around which several families of numerical fundamental solitons can be found to be stable, which is validated by linear stability analysis and nonlinear wave propagation. Many other intriguing properties of nonlinear localized modes are also discussed in detail, including the interactions, excitations, and transverse power flows. The idea of the \(\mathcal {PT}\)-symmetric rational potentials can also be extended to other types of nonlinear wave models.

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References

  1. Abdullaev, F.K., Kartashov, Y.V., Konotop, V.V., Zezyulin, D.A.: Solitons in PT-symmetric nonlinear lattices. Phys. Rev. A 83(4), 041805 (2011)

    Article  Google Scholar 

  2. Ablowitz, M.J., Musslimani, Z.H.: Spectral renormalization method for computing self-localized solutions to nonlinear systems. Opt. Lett. 30(16), 2140–2142 (2005)

    Article  Google Scholar 

  3. Achilleos, V., Kevrekidis, P., Frantzeskakis, D., Carretero-González, R.: Dark solitons and vortices in PT-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear PT phase transitions. Phys. Rev. A 86(1), 013808 (2012)

    Article  Google Scholar 

  4. Ahmed, Z.: Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential. Phys. Lett. A 282(6), 343–348 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bender, C.M.: Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys. 70(6), 947 (2007)

    Article  MathSciNet  Google Scholar 

  6. Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett. 80(24), 5243 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bender, C.M., Brody, D.C., Jones, H.F.: Must a Hamiltonian be Hermitian? Am. J. Phys. 71(11), 1095–1102 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Biswas, A., Milovic, D., Edwards, M.: Mathematical Theory of Dispersion-Managed Optical Solitons. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  9. Bludov, Y.V., Konotop, V.V., Malomed, B.A.: Stable dark solitons in PT-symmetric dual-core waveguides. Phys. Rev. A 87(1), 013816 (2013)

    Article  Google Scholar 

  10. Burlak, G., Malomed, B.A.: Stability boundary and collisions of two-dimensional solitons in PT-symmetric couplers with the cubic-quintic nonlinearity. Phys. Rev. E 88(6), 062904 (2013)

    Article  Google Scholar 

  11. Cartarius, H., Wunner, G.: Model of a PT-symmetric Bose–Einstein condensate in a \(\delta \)-function double-well potential. Phys. Rev. A 86(1), 013612 (2012)

    Article  Google Scholar 

  12. Castaldi, G., Savoia, S., Galdi, V., Alù, A., Engheta, N.: PT metamaterials via complex-coordinate transformation optics. Phys. Rev. Lett. 110(17), 173901 (2013)

    Article  Google Scholar 

  13. Chen, Y.X.: One-dimensional optical solitons in cubic-quintic-septimal media with PT-symmetric potentials. Nonlinear Dyn. 87(3), 1629–1635 (2017)

    Article  Google Scholar 

  14. Chen, Y., Yan, Z.: Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials. Sci. Rep. 6, 23478 (2016)

    Article  Google Scholar 

  15. Chen, Y., Yan, Z.: Stable parity-time-symmetric nonlinear modes and excitations in a derivative nonlinear Schrödinger equation. Phys. Rev. E 95(1), 012205 (2017)

    Article  Google Scholar 

  16. Chen, Y., Yan, Z.: Stable solitons in the 1D and 2D generalized nonlinear Schrödinger equations with the periodic effective mass and PT-symmetric potentials. Ann. Phys. 386, 44–57 (2017)

    Article  MATH  Google Scholar 

  17. Chen, Y., Yan, Z., Li, X.: One-and two-dimensional gap solitons and dynamics in the PT-symmetric lattice potential and spatially-periodic momentum modulation. Commun. Nonlinear Sci. Numer. Simul. 55, 287–297 (2018)

    Article  Google Scholar 

  18. Chen, Y., Yan, Z., Mihalache, D., Malomed, B.A.: Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses. Sci. Rep. 7(1), 1257 (2017)

    Article  Google Scholar 

  19. Dai, C.Q., Wang, X.G., Zhou, G.Q., et al.: Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials. Phys. Rev. A 89(1), 013834 (2014)

    Article  Google Scholar 

  20. Dai, C.Q., Zhang, X.F., Fan, Y., Chen, L.: Localized modes of the (n+1)-dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials. Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017)

    Article  MathSciNet  Google Scholar 

  21. Dizdarevic, D., Dast, D., Haag, D., Main, J., Cartarius, H., Wunner, G.: Cusp bifurcation in the eigenvalue spectrum of PT- symmetric Bose–Einstein condensates. Phys. Rev. A 91(3), 033636 (2015)

    Article  Google Scholar 

  22. Fortanier, R., Dast, D., Haag, D., Cartarius, H., Main, J., Wunner, G., Gutöhrlein, R.: Dipolar Bose–Einstein condensates in a PT-symmetric double-well potential. Phys. Rev. A 89(6), 063608 (2014)

    Article  Google Scholar 

  23. Guo, A., Salamo, G., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G., Christodoulides, D.: Observation of PT-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103(9), 093902 (2009)

    Article  Google Scholar 

  24. He, Y., Malomed, B.A., Mihalache, D.: Localized modes in dissipative lattice media: an overview. Phil. Trans. R. Soc. A 372(2027), 20140017 (2014)

    Article  Google Scholar 

  25. Hu, S., Ma, X., Lu, D., Yang, Z., Zheng, Y., Hu, W.: Solitons supported by complex PT-symmetric Gaussian potentials. Phys. Rev. A 84(4), 043818 (2011)

    Article  Google Scholar 

  26. Jisha, C.P., Alberucci, A., Brazhnyi, V.A., Assanto, G.: Nonlocal gap solitons in PT-symmetric periodic potentials with defocusing nonlinearity. Phys. Rev. A 89(1), 013812 (2014)

    Article  Google Scholar 

  27. Jisha, C.P., Devassy, L., Alberucci, A., Kuriakose, V.: Influence of the imaginary component of the photonic potential on the properties of solitons in PT-symmetric systems. Phys. Rev. A 90(4), 043855 (2014)

    Article  Google Scholar 

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

    Google Scholar 

  29. Konotop, V.V., Yang, J., Zezyulin, D.A.: Nonlinear waves in PT-symmetric systems. Rev. Mod. Phys. 88(3), 035002 (2016)

    Article  Google Scholar 

  30. Li, P., Mihalache, D., Li, L.: Asymmetric solitons in parity-time-symmetric double-hump scarf-II potentials. Rom. J. Phys. 61, 1028–1039 (2016)

    Google Scholar 

  31. Lumer, Y., Plotnik, Y., Rechtsman, M.C., Segev, M.: Nonlinearly induced PT transition in photonic systems. Phys. Rev. Lett. 111(26), 263901 (2013)

    Article  Google Scholar 

  32. Makris, K., El-Ganainy, R., Christodoulides, D., Musslimani, Z.H.: PT-symmetric periodic optical potentials. Int. J. Theor. Phys. 50(4), 1019–1041 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: Beam dynamics in PT symmetric optical lattices. Phys. Rev. Lett. 100(10), 103904 (2008)

    Article  Google Scholar 

  34. Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: PT-symmetric optical lattices. Phys. Rev. A 81(6), 063807 (2010)

    Article  MATH  Google Scholar 

  35. Mayteevarunyoo, T., Malomed, B.A., Reoksabutr, A.: Solvable model for solitons pinned to a parity-time-symmetric dipole. Phys. Rev. E 88(2), 022919 (2013)

    Article  Google Scholar 

  36. Midya, B., Roychoudhury, R.: Nonlinear localized modes in PT-symmetric Rosen–Morse potential wells. Phys. Rev. A 87(4), 045803 (2013)

    Article  Google Scholar 

  37. Mihalache, D.: Multidimensional localized structures in optical and matter-wave media: a topical survey of recent literature. Rom. Rep. Phys. 69(1), 403 (2017)

    Google Scholar 

  38. Moiseyev, N.: Crossing rule for a PT-symmetric two-level time-periodic system. Phys. Rev. A 83(5), 052125 (2011)

    Article  MathSciNet  Google Scholar 

  39. Musslimani, Z., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100(3), 030402 (2008)

    Article  MATH  Google Scholar 

  40. Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Analytical solutions to a class of nonlinear Schrödinger equations with PT-like potentials. J. Phys. A Math. Theor. 41(24), 244019 (2008)

    Article  MATH  Google Scholar 

  41. Nixon, S., Ge, L., Yang, J.: Stability analysis for solitons in PT-symmetric optical lattices. Phys. Rev. A 85(2), 023822 (2012)

    Article  Google Scholar 

  42. Peng, B., Özdemir, Ş.K., Lei, F., Monifi, F., Gianfreda, M., Long, G.L., Fan, S., Nori, F., Bender, C.M., Yang, L.: Parity-time-symmetric whispering-gallery microcavities. Nat. Phys. 10(5), 394–398 (2014)

    Article  Google Scholar 

  43. Regensburger, A., Bersch, C., Miri, M.A., Onishchukov, G., Christodoulides, D.N., Peschel, U.: Parity-time synthetic photonic lattices. Nature 488(7410), 167–171 (2012)

    Article  Google Scholar 

  44. Regensburger, A., Miri, M.A., Bersch, C., Näger, J., Onishchukov, G., Christodoulides, D.N., Peschel, U.: Observation of defect states in PT-symmetric optical lattices. Phys. Rev. Lett. 110(22), 223902 (2013)

    Article  Google Scholar 

  45. Rüter, C.E., Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Observation of parity-time symmetry in optics. Nat. Phys. 6(3), 192–195 (2010)

    Article  Google Scholar 

  46. Shi, Z., Jiang, X., Zhu, X., Li, H.: Bright spatial solitons in defocusing kerr media with PT-symmetric potentials. Phys. Rev. A 84(5), 053855 (2011)

    Article  Google Scholar 

  47. Single, F., Cartarius, H., Wunner, G., Main, J.: Coupling approach for the realization of a PT-symmetric potential for a Bose–Einstein condensate in a double well. Phys. Rev. A 90(4), 042123 (2014)

    Article  Google Scholar 

  48. Suchkov, S.V., Sukhorukov, A.A., Huang, J., Dmitriev, S.V., Lee, C., Kivshar, Y.S.: Nonlinear switching and solitons in PT-symmetric photonic systems. Laser Photon. Rev. 10(2), 177–213 (2016)

    Article  Google Scholar 

  49. Ultanir, E.A., Stegeman, G.I., Christodoulides, D.N.: Dissipative photonic lattice solitons. Opt. Lett. 29(8), 845–847 (2004)

    Article  Google Scholar 

  50. Wang, H., Christodoulides, D.: Two dimensional gap solitons in self-defocusing media with PT-symmetric superlattice. Commun. Nonlinear Sci. Numer. Simul. 38, 130–139 (2016)

    Article  Google Scholar 

  51. Wen, Z.C., Yan, Z.: Dynamical behaviors of optical solitons in parity-time (PT) symmetric sextic anharmonic double-well potentials. Phys. Lett. A 379(36), 2025–2029 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wen, Z., Yan, Z.: Solitons and their stability in the nonlocal nonlinear Schrödinger equation with PT-symmetric potentials. Chaos 27(5), 053105 (2017)

    Article  MathSciNet  Google Scholar 

  53. Xu, Y.J.: Hollow ring-like soliton and dipole soliton in (2+ 1)-dimensional PT-symmetric nonlinear couplers with gain and loss. Nonlinear Dyn. 83(3), 1497–1501 (2016)

    Article  MathSciNet  Google Scholar 

  54. Yan, Z.: Complex PT-symmetric extensions of the non PT-symmetric Burgers equation. Phys. Scr. 77(2), 025006 (2008)

    Article  MATH  Google Scholar 

  55. Yan, Z.: Complex PT-symmetric extensions of the nonlinear ultra-short light pulse model. J. Phys. A Math. Theor. 45(44), 444035 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  56. Yan, Z.: Complex PT-symmetric nonlinear schrödinger equation and Burgers equation. Philos. Trans. R. Soc. Lond. A 371(1989), 20120059 (2013)

    Article  MathSciNet  Google Scholar 

  57. Yan, Z.: Integrable PT-symmetric local and nonlocal vector nonlinear Schrödinger equations: a unified two-parameter model. Appl. Math. Lett. 47, 61–68 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  58. Yan, Z.: Nonlocal general vector nonlinear Schrödinger equations: Integrability, PT symmetribility, and solutions. Appl. Math. Lett. 62, 101–109 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  59. Yan, Z., Chen, Y.: The nonlinear Schrödinger equation with generalized nonlinearities and PT-symmetric potentials: stable solitons, interactions, and excitations. Chaos 27(7), 073114 (2017)

    Article  MathSciNet  Google Scholar 

  60. Yan, Z., Chen, Y., Wen, Z.: On stable solitons and interactions of the generalized Gross–Pitaevskii equation with PT-and non-PT-symmetric potentials. Chaos 26(8), 083109 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  61. Yan, Z., Wen, Z., Hang, C.: Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarff-II potentials. Phys. Rev. E 92(2), 022913 (2015)

    Article  MathSciNet  Google Scholar 

  62. Yan, Z., Wen, Z., Konotop, V.V.: Solitons in a nonlinear Schrödinger equation with PT-symmetric potentials and inhomogeneous nonlinearity: stability and excitation of nonlinear modes. Phys. Rev. A 92(2), 023821 (2015)

    Article  Google Scholar 

  63. Yang, J.: Nonlinear waves in integrable and nonintegrable systems. SIAM (2010)

  64. Yang, J.: Symmetry breaking of solitons in one-dimensional parity-time-symmetric optical potentials. Opt. Lett. 39(19), 5547–5550 (2014)

    Article  Google Scholar 

  65. Zezyulin, D.A., Konotop, V.V.: Nonlinear modes in the harmonic PT-symmetric potential. Phys. Rev. A 85(4), 043840 (2012)

    Article  Google Scholar 

  66. Zhu, H.P., Pan, Z.H.: Vortex soliton in (2+ 1)-dimensional PT-symmetric nonlinear couplers with gain and loss. Nonlinear Dyn. 83(3), 1325–1330 (2016)

    Article  MathSciNet  Google Scholar 

  67. Zyablovsky, A.A., Vinogradov, A.P., Pukhov, A.A., Dorofeenko, A.V., Lisyansky, A.A.: PT-symmetry in optics. Phys. Usp. 57(11), 1063 (2014)

    Article  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions. This work was partially supported by the NSFC under Grant Nos.11571346 and 11731014, and the Youth Innovation Promotion Association CAS.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Changshu Institute of Technology, Changshu, 215500, China

    Xin Li

  2. Key Lab of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Xin Li, Yong Chen & Zhenya Yan

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Xin Li, Yong Chen & Zhenya Yan

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  1. Xin Li
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Correspondence to Zhenya Yan.

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Li, X., Chen, Y. & Yan, Z. Fundamental solitons and dynamical analysis in the defocusing Kerr medium and \(\varvec{\mathcal {PT}}\)-symmetric rational potential. Nonlinear Dyn 91, 853–861 (2018). https://doi.org/10.1007/s11071-017-3914-2

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