Advertisement

Nonlinear Dynamics

, Volume 92, Issue 3, pp 1351–1358 | Cite as

Reconstruction of stability for Gaussian spatial solitons in quintic–septimal nonlinear materials under \({{\varvec{\mathcal {P}}}}{\varvec{\mathcal {T}}}\)-symmetric potentials

Original Paper

Abstract

Gaussian spatial soliton solutions of both the constant-coefficient and variable-coefficient (2 + 1)-dimensional nonlinear Schrödinger equations in quintic–septimal nonlinear materials with different diffractions are presented under two kinds of \({\mathcal {P}}{\mathcal {T}}\)-symmetric potentials. The linear stability analysis and direct numerical simulation are jointly utilized to investigate the stability for analytical solutions of the constant-coefficient equation. Results from the linear stability analysis and the direct numerical simulation possess a high degree of consistency, that is, the stable case for Gaussian spatial solitons of the constant-coefficient equation appears only in the defocusing quintic and focusing septimal nonlinear material. Moreover, reconstruction of stable Gaussian spatial solitons of the variable-coefficient equation is studied based on the expression of the effective propagation distance Z(z) by choosing an appropriate form of diffraction \(\beta _1(z)\).

Keywords

Gaussian spatial solitons Reconstruction of stability Quintic–septimal nonlinearities \({\mathcal {P}}{\mathcal {T}}\)-symmetric potential 

Notes

Acknowledgements

This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. LY17F050011) and the National Natural Science Foundation of China (Grant No. 11375007). Dr. Chao-Qing Dai is also sponsored by the Foundation of New Century “151 Talent Engineering” of Zhejiang Province of China, Open Fund of IPOC (BUPT) and Youth Top-notch Talent Development and Training Program of Zhejiang A&F University.

Compliance with ethical standards

Conflict of interest

The authors have declared that no conflict of interest exists.

References

  1. 1.
    Ding, D.J., Jin, D.Q., Dai, C.Q.: Analytical solutions of differential-difference sine-Gordon equation. Therm. Sci. 21, 1701–1705 (2017)CrossRefGoogle Scholar
  2. 2.
    Liu, W.J., Yu, W.T., Liu, M.L., Zhang, Y.J., Lei, M.: Analytic solutions for the generalized complex Ginzburg–Landau equation in fiber lasers. Nonlinear Dyn. 89, 2933–2939 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Zhang, B., Zhang, X.L., Dai, C.Q.: Discussions on localized structures based on equivalent solution with different forms of breaking soliton model. Nonlinear Dyn. 87, 2385–2393 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Wang, Y.Y., Zhang, Y.P., Dai, C.Q.: Re-study on localized structures based on variable separation solutions from the modified tanh-function method. Nonlinear Dyn. 83, 1331–1339 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Zhou, Q., Biswas, A.: Optical solitons in parity-time-symmetric mixed linear and nonlinear lattice with non-Kerr law nonlinearity. Superlattices Microstruct. 109, 588–598 (2017)CrossRefGoogle Scholar
  6. 6.
    Chen, R.P., Dai, C.Q.: Three-dimensional vector solitons and their stabilities in a Kerr medium with spatially inhomogeneous nonlinearity and transverse modulation. Nonlinear Dyn. 88, 2807–2816 (2017)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Zhou, Q.: Analytic study on optical solitons in a Kerr-law medium with an imprinted parity-time-symmetric mixed linear-nonlinear lattice. Proc. Rom. Acad. Ser. A 18, 223–230 (2017)MathSciNetGoogle Scholar
  8. 8.
    Dai, C.Q., Wang, Y.Y., Zhang, J.F.: Analytical spatiotemporal localizations for the generalized (3+1)-dimensional nonlinear Schrodinger equation. Opt. Lett. 35, 1437–1439 (2010)CrossRefGoogle Scholar
  9. 9.
    Agrawal, G.P.: Nonlinear Fiber Optics. Academic Press, New York (1993)MATHGoogle Scholar
  10. 10.
    Dai, C.Q., Zhou, G.Q., Chen, R.P., Lai, X.J., Zheng, J.: Vector multipole and vortex solitons in two-dimensional Kerr media. Nonlinear Dyn. 88, 2629–2635 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Pusharov, D.I., Tanev, S.: Bright and dark solitary wave propagation and bistability in the anomalous dispersion region of optical waveguides with third- and fifth-order nonlinearities. Opt. Commun. 124, 354–364 (1996)CrossRefGoogle Scholar
  12. 12.
    Wang, Y.Y., Chen, L., Dai, C.Q., Zheng, J., Fan, Y.: Exact vector multipole and vortex solitons in the media with spatially modulated cubic–quintic nonlinearity. Nonlinear Dyn. 90, 1269–1275 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Reyna, A.S., Malomed, B.A., de Araújo, C.B.: Stability conditions for one-dimensional optical solitons in cubic–quintic-septimal media. Phys. Rev. A 92, 033810 (2015)CrossRefGoogle Scholar
  14. 14.
    Chen, Y.X., Xu, F.Q., Hu, Y.L.: Two-dimensional Gaussian-type spatial solitons in inhomogeneous cubic–quintic-septimal nonlinear media under PT-symmetric potentials. Nonlinear Dyn. 90, 1115–1122 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Dai, C.Q., Chen, R.P., Wang, Y.Y., Fan, Y.: Dynamics of light bullets in inhomogeneous cubic–quintic-septimal nonlinear media with PT-symmetric potentials. Nonlinear Dyn. 87, 1675–1683 (2017)CrossRefGoogle Scholar
  16. 16.
    Wu, H.Y., Jiang, L.H., Wu, Y.F.: The stability of two-dimensional spatial solitons in cubic–quintic-septimal nonlinear media with different diffractions and PT-symmetric potentials. Nonlinear Dyn. 87, 1667–1674 (2017)CrossRefGoogle Scholar
  17. 17.
    Zhu, H.P., Pan, Z.H.: Stability of Gaussian-type light bullets in the cubic–quintic-septimal nonlinear media with different diffractions under PT-symmetric potentials. Nonlinear Dyn. 89, 1745–1752 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Reyna, A.S., Jorge, K.C., de Araújo, C.B.: Two-dimensional solitons in a quintic-septimal medium. Phys. Rev. A 90, 063835 (2014)CrossRefGoogle Scholar
  19. 19.
    Reyna, A.S., de Araújo, C.B.: Spatial phase modulation due to quintic and septic nonlinearities in metal colloids. Opt. Express 22, 22456–22469 (2014)CrossRefGoogle Scholar
  20. 20.
    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)CrossRefGoogle Scholar
  21. 21.
    Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Optical solitons in PT periodic potentials. Phys. Rev. Lett. 100, 030402 (2008)CrossRefMATHGoogle Scholar
  22. 22.
    Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having PT-symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Bogatyrev, V.A., Bubnov, M.M., Dianov, E.M., et al.: A single-mode fiber with chromatic dispersion varying along the length. J. Lightwave Technol. 9, 561–566 (1991)CrossRefGoogle Scholar
  24. 24.
    Mamyshev, P.V., Cherinkov, S.V., Dianov, M.: Generation of fundamental soliton trains for high-bit-rate optical fiber communication lines. IEEE J. Quantum Electron. 7, 2347–2355 (1991)CrossRefGoogle Scholar
  25. 25.
    Serkin, V.N., Belyaeva, T.L., Alexandrov, I.V., Melchor, G.M.: Novel topological quasi-soliton solutions for the nonlinear cubic–quintic Schrodinger equation model. Proc. SPIE Int. Soc. Opt. Eng. 4271, 292–302 (2001)Google Scholar
  26. 26.
    Belmonte-Beitia, J., Cuevas, J.: Solitons for the cubic–quintic nonlinear Schrodinger equation with time- and space-modulated coefficients. J. Phys. A Math. Theor. 42, 165201 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Dai, C.Q., Wang, Y.Y., Zhang, J.F.: Nonlinear similariton tunneling effect in the birefringent fiber. Opt. Express 18, 17548–17554 (2010)CrossRefGoogle Scholar
  28. 28.
    Dai, C.Q., Zhang, J.F.: Exact spatial similaritons and rogons in 2D graded-index waveguides. Opt. Lett. 35, 2651–2653 (2010)CrossRefGoogle Scholar
  29. 29.
    Chen, Y.X.: Sech-type and Gaussian-type light bullet solutions to the generalized (3+1)-dimensional cubic–quintic Schrödinger equation in \({\cal{P}}{\cal{T}}\)-symmetric potentials. Nonlinear Dyn. 79, 427–436 (2015)CrossRefGoogle Scholar
  30. 30.
    Zhu, Y., Qin, W., Li, J.T., Han, J.Z., Wang, Y.Y., Dai, C.Q.: Recurrence behavior for controllable excitation of rogue waves in a two-dimensional PT-symmetric coupler. Nonlinear Dyn. 88, 1883–1889 (2017)CrossRefGoogle Scholar
  31. 31.
    Li, J.T., Zhu, Y., Liu, Q.T., Han, J.Z., Wang, Y.Y., Dai, C.Q.: Vector combined and crossing Kuznetsov–Ma solitons in PT-symmetric coupled waveguides. Nonlinear Dyn. 85, 973–980 (2016)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Dai, C.Q., Wang, Y.Y.: Controllable combined Peregrine soliton and Kuznetsov–Ma soliton in PT-symmetric nonlinear couplers with gain and loss. Nonlinear Dyn. 80, 715–721 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous solitons in external potentials. Phys. Rev. Lett. 98, 074102 (2007)CrossRefGoogle Scholar
  34. 34.
    Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Solitary waves in nonautonomous nonlinear and dispersive systems: nonautonomous solitons. J. Mod. Opt. 57, 1456–1472 (2010)CrossRefMATHGoogle Scholar
  35. 35.
    Serkin, V.N., Hasegawa, A., Belyaeva, T.L.: Nonautonomous matter-wave solitons near the Feshbach resonance. Phys. Rev. A 81, 023610 (2010)CrossRefGoogle Scholar
  36. 36.
    Zhang, J.F., Tian, Q., Wang, Y.Y., Dai, C.Q., Wu, L.: Self-similar optical pulses in competing cubic–quintic nonlinear media with distributed coefficients. Phys. Rev. A 81, 023832 (2010)CrossRefGoogle Scholar
  37. 37.
    Abramowitz, M., Stegun, I.A.: “Chapter 15”, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, p. 555. Dover, New York (1965)Google Scholar
  38. 38.
    Dai, C.Q., Zhou, G.Q., Zhang, J.F.: Controllable optical rogue waves in the femtosecond regime. Phys. Rev. E 85, 016603 (2012)CrossRefGoogle Scholar
  39. 39.
    Dai, C.Q., Zhu, H.P.: Superposed Kuznetsov–Ma solitons in a two-dimensional graded-index grating waveguide. J. Opt. Soc. Am. B 30, 3291–3297 (2013)CrossRefGoogle Scholar
  40. 40.
    Kruglov, V.I., Peacock, A.C., Harvey, J.D.: Exact self-similar solutions of the generalized nonlinear Schrodinger equation with distributed coefficients. Phys. Rev. Lett. 90, 113902 (2003)CrossRefGoogle Scholar
  41. 41.
    Dai, C.Q., Wang, Y.Y., Zhang, X.F.: Controllable Akhmediev breather and Kuznetsov–Ma soliton trains in PT-symmetric coupled waveguides. Opt. Express 22, 29862–29867 (2014)CrossRefGoogle Scholar
  42. 42.
    Dai, C.Q., Wang, X.G., Zhou, G.Q.: Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials. Phys. Rev. A 89, 013834 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Chao-Qing Dai
    • 1
  • Yue-Yue Wang
    • 1
  • Yan Fan
    • 1
  • Ding-Guo Yu
    • 1
  1. 1.School of SciencesZhejiang A&F UniversityLin’anPeople’s Republic of China

Personalised recommendations