Skip to main content
SpringerLink
Log in

Higher-dimensional soliton structures of a variable-coefficient Gross–Pitaevskii equation with the partially nonlocal nonlinearity under a harmonic potential

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

A reduction correlation between the \((3+1)\)-dimensional variable-coefficient Gross–Pitaevskii equation with the partially nonlocal nonlinearity under a harmonic potential and a \((2+1)\)-dimensional constant-coefficient one is firstly erected. With the aid of solutions via the Hirota method for the \((2+1)\)-dimensional constant-coefficient equation, the \((3+1)\)-dimensional soliton analytical solutions with the Hermite–Gaussian envelope including vortex, diploe soliton and saddle-shaped soliton are firstly unfolded. Expanded and compressed evolutions of these \((3+1)\)-dimensional soliton structures are presented in the periodic amplification and exponential diffraction decreasing systems. In the \(x-z\) plane, the eye-shaped structure appears in all soliton structures, and its number is related to the Hermite parameter.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Availability of data and materials

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

References

  1. Fang, Y., Wu, G.Z., Wang, Y.Y., Dai, C.Q.: Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order NLSE using the PINN. Nonlinear Dyn. 105, 603–616 (2021)

    Article  Google Scholar 

  2. Fei, J., Cao, W.: Explicit soliton-cnoidal wave interaction solutions for the \((2+1)\)-dimensional negative-order breaking soliton equation. Waves Rand. Complex Media 30, 54–64 (2020)

    Article  MathSciNet  Google Scholar 

  3. Cao, Q.H., Dai, C.Q.: Symmetric and anti-symmetric solitons of the fractional second- and third-order nonlinear Schrodinger equation. Chin. Phys. Lett. 38, 090501 (2021)

    Article  Google Scholar 

  4. Wang, B.H., Wang, Y.Y., Dai, C.Q., Chen, Y.X.: Dynamical characteristic of analytical fractional solitons for the space-time fractional Fokas–Lenells equation. Alex. Eng. J. 59, 4699–4707 (2020)

    Article  Google Scholar 

  5. Dai, C.Q., Wang, Y.Y.: Coupled spatial periodic waves and solitons in the photovoltaic photorefractive crystals. Nonlinear Dyn. 102, 1733–1741 (2020)

    Article  Google Scholar 

  6. Zhang, Z., Yang, X., Li, B.: Novel soliton molecules and breather-positon on zero background for the complex modified KdV equation. Nonlinear Dyn. 100, 1551–1557 (2020)

    Article  Google Scholar 

  7. Ain, Q.T., He, J.H., Anjum, N., Ali, M.: The fractional complex transform: a novel approach to the time-fractional Schrodinger equation. Fractals 28, 2050141 (2020)

    Article  Google Scholar 

  8. Chen, J., Luan, Z., Zhou, Q., Alzahrani, A.K., Biswas, A., Liu, W.: Periodic soliton interactions for higher-order nonlinear Schrodinger equation in optical fibers. Nonlinear Dyn. 100, 2817–2821 (2020)

    Article  Google Scholar 

  9. Dai, C.Q., Wu, G.Z., Li, H.J., Wang, Y.Y.: Wick-type stochastic fractional solitons supported by quadratic-cubic nonlinearity. Fractals 29, 2150192 (2021)

    Article  MATH  Google Scholar 

  10. 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)

    Article  MathSciNet  Google Scholar 

  11. Liu, X., Zhou, Q., Biswas, A., Alzahrani, A.K., Liu, W.: The similarities and differences of different plane solitons controlled by \((3+1)\)-dimensional coupled variable coefficient system. J. Adv. Res. 24, 167–173 (2020)

    Article  Google Scholar 

  12. Wang, B.H., Lu, P.H., Dai, C.Q., Chen, Y.X.: Vector optical soliton and periodic solutions of a coupled fractional nonlinear Schrodinger equation. Res. Phys. 17, 103036 (2020)

    Google Scholar 

  13. Khalique, C.M., Simbanefayi, I.: Conserved quantities, optimal system and explicit solutions of a \((1 + 1)\)-dimensional generalised coupled mKdV-type system. J. Adv. Res. 29, 159–166 (2021)

    Article  Google Scholar 

  14. Zhang, R.F., Bilige, S.: Bilinear neural network method to obtain the exact analytical solutions of nonlinear partial differential equations and its application to p-gBKP equation. Nonlinear Dyn. 95, 3041–3048 (2019)

    Article  MATH  Google Scholar 

  15. Zhang, R.F., Li, M.C., Yin, H.M.: Rogue wave solutions and the bright and dark solitons of the \((3+1)\)-dimensional Jimbo–Miwa equation. Nonlinear Dyn. 103, 1071–1079 (2021)

    Article  Google Scholar 

  16. Zhang, R.F., Li, M.C., Albishari, M., Zheng, F.C., Lan, Z.Z.: Generalized lump solutions, classical lump solutions and rogue waves of the \((2+1)\)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada-like equation. Appl. Math. Comput. 403, 126201 (2021)

    MathSciNet  MATH  Google Scholar 

  17. Zhang, R., Bilige, S., Chaolu, T.: Fractal solitons, arbitrary function solutions, exact periodic wave and breathers for a nonlinear partial differential equation by using bilinear neural network method. J. Syst. Sci. Complex 34, 122–139 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  18. Fang, J.J., Mou, D.S., Wang, Y.Y., Zhang, H.C., Dai, C.Q., Chen, Y.X.: Soliton dynamics based on exact solutions of conformable fractional discrete complex cubic Ginzburg–Landau equation. Res. Phys. 20, 103710 (2021)

    Google Scholar 

  19. Dai, C.Q., Wang, Y.Y., Zhang, J.F.: Managements of scalar and vector rogue waves in a partially nonlocal nonlinear medium with linear and harmonic potentials. Nonlinear Dyn. 102, 379–391 (2020)

    Article  Google Scholar 

  20. Dai, C.Q., Wang, Y.Y.: Spatiotemporal localizations in \((3 + 1)\)-dimensional PT-symmetric and strongly nonlocal nonlinear media. Nonlinear Dyn. 83, 2453–2459 (2016)

    Article  MathSciNet  Google Scholar 

  21. Xu, S.L., Belic, M.R.: Three-dimensional Hermite–Bessel solitons in strongly nonlocal media with variable potential coefficients. Opt. Commun. 313, 62–69 (2014)

    Article  Google Scholar 

  22. Zhong, W.P., Xie, R.H., Belic, M., Petrovic, N., Chen, G., Yi, L.: Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrodinger equation with distributed coefficients. Phys. Rev. A 78, 023821 (2008)

    Article  Google Scholar 

  23. Dai, C.Q., Fan, Y., Zhou, G.Q., Zheng, J., Chen, L.: Vector spatiotemporal localized structures in \((3 + 1)\)-dimensional strongly nonlocal nonlinear media. Nonlinear Dyn. 86, 999–1005 (2016)

    Article  MathSciNet  Google Scholar 

  24. Wazwaz, A.M.: Bright and dark optical solitons for \((3+1)\)-dimensional Schrodinger equation with cubic-quintic-septic nonlinearities. Optik 225, 165752 (2021)

    Article  Google Scholar 

  25. Yang, M.: Abundant exact solutions for the \((3+1)\)-dimensional generalized nonlinear Schrodinger equation with variable coefficients. J. Chin. Phys. 65, 491–499 (2020)

    Article  MathSciNet  Google Scholar 

  26. 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)

    Article  Google Scholar 

  27. Hosseini, K., Ansari, R., Zabihi, A., Shafaroody, A., Mirzazadeh, M.: Optical solitons and modulation instability of the resonant nonlinear Schrodinger equations in \((3+1)\)-dimensions. Optik 209, 164584 (2020)

    Article  Google Scholar 

  28. Maruno, K., Ohta, Y.: Localized solitons of a \((2 +1)\)-dimensional nonlocal nonlinear Schrödinger equation. Phys. Lett. A 372, 4446–4450 (2008)

    Article  MATH  Google Scholar 

  29. Yan, Z.Y.: Rogon-like solutions excited in the two-dimensional nonlocal nonlinear Schrödinger equation. J. Math. Anal. Appl. 380, 689–696 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Dai, C.Q., Liu, J., Fan, Y., Yu, D.G.: Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality. Nonlinear Dyn. 88, 1373–1383 (2017)

    Article  Google Scholar 

  31. Liu, Q.: Analytical matter wave solutions of a (2+1)-dimensional partially nonlocal distributed-coefficient Gross–Pitaevskii equation with a linear potential. Optik 200, 163434 (2020)

    Article  Google Scholar 

  32. Chen, Y.X., Ou-Yang, F.Y.: Excitation management of crossed Akhmediev and Ma breather for a nonautonomous partially nonlocal Gross–Pitaevskii equation with an external potential. Nonlinear Dyn. 100, 1543–1550 (2020)

    Article  Google Scholar 

  33. Dai, C.Q., Wang, Y., Liu, J.: Spatiotemporal Hermite-Gaussian solitons of a \((3 + 1)\)-dimensional partially nonlocal nonlinear Schrodinger equation. Nonlinear Dyn. 84, 1157–1161 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  34. Wang, Y.Y., Dai, C.Q., Xu, Y.Q., Zheng, J., Fan, Y.: Dynamics of nonlocal and localized spatiotemporal solitons for a partially nonlocal nonlinear Schrodinger equation. Nonlinear Dyn. 92, 1261–1269 (2018)

  35. Wu, H.Y., Jiang, L.H.: Vortex soliton solutions of a \((3 + 1)\)-dimensional Gross–Pitaevskii equation with partially nonlocal distributed coefficients under a linear potential. Nonlinear Dyn. 101, 2441–2448 (2020)

    Article  Google Scholar 

  36. Luo, Z., Li, Y., Pang, W., Liu, Y.: Dipolar matter-wave soliton in one-dimensional optical lattice with tunable local and nonlocal nonlinearities. J. Phys. Soc. Jpn. 82, 094401 (2013)

    Article  Google Scholar 

  37. Sarkar, S., Bhattacharyay, A.: Non-local interactions in a BEC: an analogue gravity perspective. J. Phys. A: Math. Theor. 47, 092002 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Chen, Y.X., Xu, F.Q., Hu, Y.L.: Excitation control for three-dimensional Peregrine solution and combined breather of a partially nonlocal variable-coefficient nonlinear Schrödinger equation. Nonlinear Dyn. 95, 1957–1964 (2019)

    Article  MATH  Google Scholar 

  39. Zhong, W.P., Belic, M., Assanto, G., Malomed, B.A., Huang, T.W.: Self-trapping of scalar and vector dipole solitary waves in Kerr media. Phys. Rev. A 83, 043833 (2011)

    Article  Google Scholar 

  40. Wang, J., Li, L., Jia, S.: Exact chirped gray soliton solutions of the nonlinear Schrodinger equation with variable coefficients. Opt. Commun. 274, 223–230 (2007)

    Article  Google Scholar 

  41. Yang, R.C., Hao, R.Y., Li, L., Shi, X.J., Li, Z.H., Zhou, G.S.: Exact gray multi-soliton solutions for nonlinear Schrodinger equation with variable coefficients. Opt. Commun. 253, 177–185 (2005)

    Article  Google Scholar 

  42. Hao, R.Y., Li, L., Li, Z.H., Xue, W.R., Zhou, G.S.: A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrodinger equation with variable coefficients. Opt. Commun. 236, 79–86 (2004)

    Article  Google Scholar 

  43. 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 4271, 292–302 (2001)

    Article  Google Scholar 

  44. 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)

    Article  Google Scholar 

  45. 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)

    Article  MathSciNet  Google Scholar 

  46. Dai, C.Q., Wang, Y.Y., Fan, Y., Zhang, J.F.: Interactions between exotic multi-valued solitons of the \((2+1)\)-dimensional Korteweg-de Vries equation describing shallow water wave. Appl. Math. Model. 80, 506–515 (2020)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11905308) and the Science and Technology Development Plan Project of Henan Province (202102210223), the High School Key Scientific Research Project of Henan Province (21A140031) and the Youth Backbone Teacher Training Project of Henan Higher Education Institutions (2020GGJS212).

Author information

Authors and Affiliations

  1. School of Physics and Telecommunications Engineering, Zhoukou Normal University, Zhoukou, 466001, People’s Republic of China

    Jing Yang, Yu Zhu, Wei Qin, Shaohui Wang & Jitao Li

  2. College of Optical, Mechanical and Electrical Engineering, Zhejiang A& F University, Lin’an, 311300, People’s Republic of China

    Chaoqing Dai

Authors
  1. Jing Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wei Qin
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shaohui Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Chaoqing Dai
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Jitao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jitao Li.

Ethics declarations

Conflict of interest

The authors have declared that no conflict of interest exists.

Ethical approval

This article does not contain any studies with human or animal subjects.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, J., Zhu, Y., Qin, W. et al. Higher-dimensional soliton structures of a variable-coefficient Gross–Pitaevskii equation with the partially nonlocal nonlinearity under a harmonic potential. Nonlinear Dyn 108, 2551–2562 (2022). https://doi.org/10.1007/s11071-022-07337-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07337-2

Keywords

Search

Navigation