Nonlinear Dynamics

, Volume 98, Issue 2, pp 985–995 | Cite as

One-dimensional gap solitons in quintic and cubic–quintic fractional nonlinear Schrödinger equations with a periodically modulated linear potential

  • Liangwei Zeng
  • Jianhua ZengEmail author
Original paper


Competing nonlinearities, such as the cubic (Kerr) and quintic nonlinear terms whose strengths are of opposite signs (the coefficients in front of the nonlinearities), exist in various physical media (in particular, in optical and matter-wave media). A benign competition between self-focusing cubic and self-defocusing quintic nonlinear nonlinearities (known as cubic–quintic model) plays an important role in creating and stabilizing the self-trapping of D-dimensional localized structures, in the contexts of standard nonlinear Schrödinger equation. We incorporate an external periodic potential (linear lattice) into this model and extend it to the space-fractional scenario that begins to surface in very recent years—the nonlinear fractional Schrödinger equation (NLFSE), therefore obtaining the cubic–quintic or the purely quintic NLFSE, and investigate the propagation and stability properties of self-trapped modes therein. Two types of one-dimensional localized gap modes are found, including the fundamental and dipole-mode gap solitons. Employing the techniques based on the linear-stability analysis and direct numerical simulations, we get the stability regions of all the localized modes; and particularly, the anti-Vakhitov–Kolokolov criterion applies for the stable portions of soliton families generated in the frameworks of quintic-only nonlinearity and competing cubic–quintic nonlinear terms.


Fractional calculus Cubic–quintic nonlinearity Nonlinear Schrödinger equation Gap solitons 



This work was supported, in part, by the Natural Science Foundation of China (Project Nos. 61690222, 61690224), and by the Youth Innovation Promotion Association of the Chinese Academy of Sciences (Project No. 2016357).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having \(\cal{PT}\) symmetry. Phys. Rev. Lett. 80, 5243 (1998)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bender, C.M., Brody, D.C., Jones, H.F.: Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 270401 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bender, C.M., Brody, D.C., Jones, H.F., Meister, B.K.: Faster than Hermitian quantum mechanics. Phys. Rev. Lett. 98, 040403 (2007)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Zeng, J., Lan, Y.: Two-dimensional solitons in \(\cal{PT}\) linear lattice potentials. Phys. Rev. E 85, 047601 (2012)Google Scholar
  5. 5.
    El-Ganainy, R., Makris, K.G., Khajavikhan, M., Musslimani, Z.H., Rotter, S., Christodoulides, D.N.: Non-Hermitian physics and PT symmetry. Nat. Phys. 14, 11 (2018)Google Scholar
  6. 6.
    Konotop, V.V., Yang, J., Zezyulin, D.A.: Nonlinear waves in \(\cal{PT}\)-symmetric systems. Rev. Mod. Phys. 81, 013624 (2016)Google Scholar
  7. 7.
    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 Photonics Rev. 10, 177 (2016)Google Scholar
  8. 8.
    Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A 268, 298–305 (2000)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Laskin, N.: Fractional quantum mechanics. Phys. Rev. E 62, 3135 (2000)zbMATHGoogle Scholar
  10. 10.
    Laskin, N.: Fractional Schrödinger equation. Phys. Rev. E 66, 056108 (2002)MathSciNetGoogle Scholar
  11. 11.
    Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific, Singapore (2011)zbMATHGoogle Scholar
  12. 12.
    Stickler, B.A.: Potential condensed-matter realization of space-fractional quantum mechanics: the one-dimensional Lévy crystal. Phys. Rev. E 88, 012120 (2013)Google Scholar
  13. 13.
    Longhi, S.: Fractional Schrödinger equation in optics. Opt. Lett. 40, 1117 (2015)Google Scholar
  14. 14.
    Zhang, Y., Liu, X., Belić, M.R., Zhong, W., Zhang, Y., Xiao, M.: Propagation dynamics of a light beam in a fractional Schrödinger equation. Phys. Rev. Lett. 115, 180403 (2015)Google Scholar
  15. 15.
    Zhang, Y., Zhong, H., Belić, M.R., Ahmed, N., Zhang, Y., Xiao, M.: Diffraction-free beams in fractional Schrödinger equation. Sci. Rep. 6, 23645 (2016)Google Scholar
  16. 16.
    Zhang, Y., Zhong, H., Belić, M.R., Zhu, Y., Zhong, W., Zhang, Y., Christodoulides, D.N., Xiao, M.: \(\cal{PT}\) symmetry in a fractional Schrödinger equation. Laser Photonics Rev. 10, 526 (2016)Google Scholar
  17. 17.
    Zhang, L., Li, C., Zhong, H., Xu, C., Lei, D., Li, Y., Fan, D.: Propagation dynamics of super-Gaussian beams in fractional Schrödinger equation: from linear to nonlinear regimes. Opt. Express 24, 14406 (2016)Google Scholar
  18. 18.
    Zhong, W.P., Belić, M.R., Malomed, B.A., Zhang, Y., Huang, T.: Spatiotemporal accessible solitons in fractional dimensions. Phys. Rev. E 94, 012216 (2016)Google Scholar
  19. 19.
    Zhong, W.P., Belić, M.R., Zhang, Y.: Accessible solitons of fractional dimension. Ann. Phys. 368, 110 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Zhang, Y., Wang, R., Zhong, H., Zhang, J., Belić, M.R., Zhang, Y.: Optical Bloch oscillation and Zener tunneling in the fractional Schrödinger equation. Sci. Rep. 7, 17872 (2017)Google Scholar
  21. 21.
    Huang, C., Dong, L.: Beam propagation management in a fractional Shrödinger equation. Sci. Rep. 7, 5442 (2017)Google Scholar
  22. 22.
    Zhang, L., He, Z., Conti, C., Wang, Z., Hu, Y., Lei, D., Li, Y., Fan, D.: Modulational instability in fractional nonlinear Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 48, 531 (2017)MathSciNetGoogle Scholar
  23. 23.
    Chen, M., Zeng, S., Lu, D., Hu, W., Guo, Q.: Optical solitons, self-focusing, and wave collapse in a space-fractional Schrödinger equation with a Kerr-type nonlinearity. Phys. Rev. E 98, 022211 (2018)MathSciNetGoogle Scholar
  24. 24.
    Chen, M., Guo, Q., Lu, D., Hu, W.: Variational approach for breathers in a nonlinear fractional Schrödinger equation. Commun. Nonlinear Sci. Numer. Simul. 71, 73 (2019)MathSciNetGoogle Scholar
  25. 25.
    Huang, C., Dong, L.: Gap solitons in the nonlinear fractional Schrödinger equation with an optical lattice. Opt. Lett. 41, 5636 (2016)Google Scholar
  26. 26.
    Dong, L., Huang, C.: Double-hump solitons in fractional dimensions with a \(\cal{PT}\)-symmetric potential. Opt. Express 26, 10509 (2018)Google Scholar
  27. 27.
    Yao, X., Liu, X.: Off-site and on-site vortex solitons in space-fractional photonic lattices. Opt. Lett. 43, 5749 (2018)Google Scholar
  28. 28.
    Zeng, L., Zeng, J.: One-dimensional solitons in fractional Schrödinger equation with a spatially periodical modulated nonlinearity: nonlinear lattice. Opt. Lett. 44, 2661 (2019)Google Scholar
  29. 29.
    Kartashov, Y.V., Malomed, B.A., Torner, L.: Solitons in nonlinear lattices. Rev. Mod. Phys. 83, 247 (2011)Google Scholar
  30. 30.
    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)Google Scholar
  31. 31.
    Triki, H., Porsezian, K., Dinda, P.T., Grelu, P.: Dark spatial solitary waves in a cubic–quintic-septimal nonlinear medium. Phys. Rev. A 95, 023837 (2017)Google Scholar
  32. 32.
    Cisternas, J., Descalzi, O., Albers, T., Radons, G.: Anomalous diffusion of dissipative solitons in the cubic–quintic complex Ginzburg–Landau equation in two spatial dimensions. Phys. Rev. Lett. 116, 203901 (2016)Google Scholar
  33. 33.
    Gao, X., Zeng, J.: Two-dimensional matter-wave solitons and vortices in competing cubic–quintic nonlinear lattices. Front. Phys. 13, 130501 (2018)Google Scholar
  34. 34.
    Zegadlo, K.B., Wasak, T., Malomed, B.A., Karpierz, M.A., Trippenbach, M.: Stabilization of solitons under competing nonlinearities by external potentials. Chaos 24, 043136 (2014)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Burlak, G., Malomed, B.A.: Interactions of three-dimensional solitons in the cubic–quintic model. Chaos 28, 063121 (2018)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Desyatnikov, A., Maimistov, A., Malomed, B.: Three-dimensional spinning solitons in dispersive media with the cubic–quintic nonlinearity. Phys. Rev. E 61, 3107 (2000)Google Scholar
  37. 37.
    Paredes, A., Feijoo, D., Michinel, H.: Coherent cavitation in the liquid of light. Phys. Rev. Lett. 112, 173901 (2014)Google Scholar
  38. 38.
    Falcão-Filho, E.L., de Araújo, C.B., Boudebs, G., Leblond, H., Skarka, V.: Robust two-dimensional spatial solitons in liquid carbon disulfide. Phys. Rev. Lett. 110, 013901 (2013)Google Scholar
  39. 39.
    Reyna, A.S., de Araújo, C.B.: Nonlinearity management of photonic composites and observation of spatial-modulation instability due to quintic nonlinearity. Phys. Rev. A 89, 063803 (2014)Google Scholar
  40. 40.
    Reyna, A.S., de Araújo, C.B.: High-order optical nonlinearities in plasmonic nanocomposites—a review. Adv. Opt. Photonics 9, 720 (2017)Google Scholar
  41. 41.
    Chin, C., Grimm, R., Julienne, P., Tsienga, E.: Feshbach resonances in ultracold gases. Rev. Mod. Phys. 82, 1225 (2010)Google Scholar
  42. 42.
    Zeng, J., Malomed, B.A.: Stabilization of one-dimensional solitons against the critical collapse by quintic nonlinear lattices. Phys. Rev. A 85, 023824 (2012)Google Scholar
  43. 43.
    Shi, J., Zeng, J., Malomed, B.A.: Suppression of the critical collapse for one-dimensional solitons by saturable quintic nonlinear lattices. Chaos 28, 075501 (2018)MathSciNetGoogle Scholar
  44. 44.
    Petrov, D.S.: Quantum mechanical stabilization of a collapsing Bose–Bose mixture. Phys. Rev. Lett. 115, 155302 (2015)Google Scholar
  45. 45.
    Petrov, D.S., Astrakharchik, G.E.: Ultradilute low-dimensional liquids. Phys. Rev. Lett. 117, 100401 (2016)Google Scholar
  46. 46.
    Joannopoulos, J.D., Johnson, S.G., Winn, J.N., Meade, R.D.: Photonic Crystals: Molding the Flow of Light. Princeton University Press, Princeton (2008)zbMATHGoogle Scholar
  47. 47.
    Christodoulides, D.N., Lederer, F., Silberberg, Y.: Discretizing light behaviour in linear and nonlinear waveguide lattices. Nature 424, 817 (2003)Google Scholar
  48. 48.
    Garanovich, I.L., Longhi, S., Sukhorukova, A.A., Kivshar, Y.S.: Light propagation and localization in modulated photonic lattices and waveguides. Phys. Rep. 518, 1 (2012)Google Scholar
  49. 49.
    Chen, Z., Segev, M., Christodoulides, D.N.: Optical spatial solitons: historical overview and recent advances. Rep. Prog. Phys. 75, 086401 (2012)Google Scholar
  50. 50.
    Eggleton, B.J., Slusher, R.E., de Sterke, C.M., Krug, P.A., Sipe, J.E.: Bragg grating solitons. Phys. Rev. Lett. 76, 1627 (1996)Google Scholar
  51. 51.
    Mandelik, D., Morandotti, R., Aitchison, J.S., Silberberg, Y.: Gap solitons in waveguide arrays. Phys. Rev. Lett. 92, 093904 (2004)Google Scholar
  52. 52.
    Kartashov, Y.V., Vysloukh, V.A., Torner, L.: Surface gap solitons. Phys. Rev. Lett. 96, 073901 (2006)Google Scholar
  53. 53.
    Szameit, A., Kartashov, Y.V., Dreisow, F., Pertsch, T., Nolte, S., Tünnermann, A., Torner, L.: Observation of two-dimensional surface solitons in asymmetric waveguide arrays. Phys. Rev. Lett. 98, 173903 (2007)Google Scholar
  54. 54.
    Peleg, O., Bartal, G., Freedman, B., Manela, O., Segev, M., Christodoulides, D.N.: Conical diffraction and gap solitons in honeycomb photonic lattices. Phys. Rev. Lett. 98, 103901 (2007)Google Scholar
  55. 55.
    Fleischer, J.W., Segev, M., Efremidis, N.K., Christodoulides, D.N.: Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature 422, 147 (2003)Google Scholar
  56. 56.
    Baizakov, B.B., Malomed, B.A., Salerno, M.: Multidimensional solitons in periodic potentials. Europhys. Lett. 63, 642 (2003)zbMATHGoogle Scholar
  57. 57.
    Brazhnyi, V.A., Konotop, V.V.: Theory of nonlinear matter waves in optical lattices. Mod. Phys. Lett. B 18, 627 (2004)zbMATHGoogle Scholar
  58. 58.
    Eiermann, B., Anker, Th, Albiez, M., Taglieber, M., Treutlein, P., Marzlin, K.-P., Oberthaler, M.K.: Bright Bose–Einstein gap solitons of atoms with repulsive interaction. Phys. Rev. Lett. 92, 230401 (2004)Google Scholar
  59. 59.
    Morsch, O., Oberthaler, M.: Dynamics of Bose–Einstein condensates in optical lattices. Rev. Mod. Phys. 78, 179 (2006)Google Scholar
  60. 60.
    Zeng, L., Zeng, J.: Gap-type dark localized modes in a Bose–Einstein condensate with optical lattices. Adv. Photonics 1, 046004 (2019)Google Scholar
  61. 61.
    Sakaguchi, H., Malomed, B.A.: Matter-wave solitons in nonlinear optical lattices. Phys. Rev. E 72, 046610 (2005)MathSciNetGoogle Scholar
  62. 62.
    Theocharis, G., Schmelcher, P., Kevrekidis, P.G., Frantzeskakis, D.J.: Matter-wave solitons of collisionally inhomogeneous condensates. Phys. Rev. A 72, 033614 (2005)Google Scholar
  63. 63.
    Sivan, Y., Fibich, G., Weinstein, M.I.: Waves in nonlinear lattices: ultrashort optical pulses and Bose–Einstein condensates. Phys. Rev. Lett. 97, 193902 (2006)Google Scholar
  64. 64.
    Belmonte-Beitia, J., Pérez-García, V.M., Vekslerchik, V., Torres, P.J.: Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities. Phys. Rev. Lett. 98, 064102 (2007)Google Scholar
  65. 65.
    Kartashov, Y.V., Vysloukh, V.A., Torner, L.: Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity. Opt. Lett. 33, 1747 (2008)Google Scholar
  66. 66.
    Kartashov, Y.V., Malomed, B.A., Vysloukh, V.A., Torner, L.: Vector solitons in nonlinear lattices. Opt. Lett. 34, 3625 (2009)Google Scholar
  67. 67.
    Abdullaev, FKh, Gammal, A., Salerno, M., Tomio, L.: Localized modes of binary mixtures of Bose–Einstein condensates in nonlinear optical lattices. Phys. Rev. A 77, 023615 (2008)Google Scholar
  68. 68.
    Lebedev, M.E., Alfimov, G.L., Malomed, B.A.: Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity. Chaos 26, 073110 (2016)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Wen, Z., Yan, Z.: Solitons and their stability in the nonlocal nonlinear Schrödinger equation with \(\cal{PT}\)-symmetric potentials. Chaos 27, 053105 (2017)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Zezyulin, D.A., Konotop, V.V.: Solitons in a Hamiltonian \(\cal{PT}\)-symmetric coupler. J. Phys. A Math. Theor. 51, 015206 (2018)MathSciNetzbMATHGoogle Scholar
  71. 71.
    Kartashov, Y.V., Vysloukh, V.A., Torner, L.: Power-dependent shaping of vortex solitons in optical lattices with spatially modulated nonlinear refractive index. Opt. Lett. 33, 2173 (2008)Google Scholar
  72. 72.
    Sakaguchi, H., Malomed, B.A.: Solitons in combined linear and nonlinear lattice potentials. Phys. Rev. A 81, 013624 (2010)Google Scholar
  73. 73.
    Zeng, J., Malomed, B.A.: Two-dimensional solitons and vortices in media with incommensurate linear and nonlinear lattice potentials. Phys. Scr. T149, 014035 (2012)Google Scholar
  74. 74.
    Shi, J., Zeng, J.: Self-trapped spatially localized states in combined linear-nonlinear periodic potentials. Front. Phys. (submitted) Google Scholar
  75. 75.
    Borovkova, O.V., Kartashov, Y.V., Torner, L., Malomed, B.A.: Bright solitons from defocusing nonlinearities. Phys. Rev. E 84, 035602(R) (2011)Google Scholar
  76. 76.
    Borovkova, O.V., Kartashov, Y.V., Malomed, B.A., Torner, L.: Algebraic bright and vortex solitons in defocusing media. Opt. Lett. 36, 3088 (2011)Google Scholar
  77. 77.
    Zeng, J., Malomed, B.A.: Bright solitons in defocusing media with spatial modulation of the quintic nonlinearity. Phys. Rev. E 86, 036607 (2012)Google Scholar
  78. 78.
    Kartashov, Y.V., Lobanov, V.E., Malomed, B.A., Torner, L.: Asymmetric solitons and domain walls supported by inhomogeneous defocusing nonlinearity. Opt. Lett. 37, 5000 (2012)Google Scholar
  79. 79.
    Young-S, L.E., Salasnich, L., Malomed, B.A.: Self-trapping of Fermi and Bose gases under spatially modulated repulsive nonlinearity and transverse confinement. Phys. Rev. A 87, 043603 (2013)Google Scholar
  80. 80.
    Cardoso, W.B., Zeng, J., Avelar, A.T., Bazeia, D., Malomed, B.A.: Bright solitons from the nonpolynomial Schrödinger equation with inhomogeneous defocusing nonlinearities. Phys. Rev. E 88, 025201 (2013)Google Scholar
  81. 81.
    Driben, R., Kartashov, Y.V., Malomed, B.A., Meier, T., Torner, L.: Soliton gyroscopes in media with spatially growing repulsive nonlinearity. Phys. Rev. Lett. 112, 020404 (2014)Google Scholar
  82. 82.
    Kartashov, Y.V., Malomed, B.A., Shnir, Y., Torner, L.: Twisted toroidal vortex solitons in inhomogeneous media with repulsive nonlinearity. Phys. Rev. Lett. 113, 264101 (2014)Google Scholar
  83. 83.
    Driben, R., Kartashov, Y.V., Malomed, B.A., Meier, T., Torner, L.: Three-dimensional hybrid vortex solitons. New J. Phys. 16, 063035 (2014)MathSciNetGoogle Scholar
  84. 84.
    Kevrekidis, P.G., Malomed, B.A., Saxena, A., Bishop, A.R., Frantzeskakis, D.J.: Solitons and vortices in two-dimensional discrete nonlinear Schrödinger systems with spatially modulated nonlinearity. Phys. Rev. E 91, 043201 (2015)MathSciNetGoogle Scholar
  85. 85.
    Driben, R., Dror, N., Malomed, B.A., Meier, T.: Multipoles and vortex multiplets in multidimensional media with inhomogeneous defocusing nonlinearity. New J. Phys. 17, 083043 (2015)Google Scholar
  86. 86.
    Zeng, J., Malomed, B.A.: Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity. Phys. Rev. E 95, 052214 (2017)MathSciNetGoogle Scholar
  87. 87.
    Huang, C., Ye, Y., Liu, S., He, H., Pang, W., Malomed, B.A., Li, Y.: Excited states of two-dimensional solitons supported by spin–orbit coupling and field-induced dipole–dipole repulsion. Phys. Rev. A 97, 013636 (2018)Google Scholar
  88. 88.
    Zeng, L., Zeng, J., Kartashov, Y.V., Malomed, B.A.: Purely Kerr nonlinear model admitting flat-top solitons. Opt. Lett. 44, 1206 (2019)Google Scholar
  89. 89.
    Zeng, L., Zeng, J.: Gaussian-like and flat-top solitons of atoms with spatially modulated repulsive interactions. J. Opt. Soc. Am. B 36, 002278 (2019)Google Scholar
  90. 90.
    Vakhitov, M., Kolokolov, A.: Stationary solutions of the wave equation in a medium with nonlinearity saturation. Radiophys. Quantum Electron. 16, 783 (1973)Google Scholar
  91. 91.
    Yang, J.: Nonlinear Waves in Integrable and Nonintegrable Systems. SIAM, Philadelphia (2010)zbMATHGoogle Scholar
  92. 92.
    Abdullaev, FKh, Salerno, M.: Gap-Townes solitons and localized excitations in low-dimensional Bose–Einstein condensates in optical lattices. Phys. Rev. A 72, 033617 (2005)Google Scholar

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Authors and Affiliations

  1. 1.State Key Laboratory of Transient Optics and PhotonicsXi’an Institute of Optics and Precision Mechanics of Chinese Academy of SciencesXi’anChina
  2. 2.University of Chinese Academy of SciencesBeijingChina

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