Skip to main content
SpringerLink
Log in

The exact solutions of Schrödinger–Hirota equation based on the auxiliary equation method

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

The Schrödinger–Hirota equation are studied in light transmission, optical fiber communication, and nonlinear effects in optics. The auxiliary equation method is not only suitable for solving specific types of nonlinear partial differential equations, but also has strong applicability to all kinds of different types of equations. It helps us to deduce the exact solution of the equation faster and analyzes the dynamic behavior of the system further. The extended fourth Jacobi elliptic equation is used in this paper to seek different types of exact solutions, which include bright soliton solutions, kink solutions, periodic wave solutions, and singular traveling wave solutions via selecting appropriate parameters.The characteristics of some solutions are graphically presented using two- and three-dimensional graphs such as the real part, the imaginary part, and their modulus via providing suitable values to arbitrary parameters. Compared to other methods, the method is more direct and easier for calculations. Kindly check and confirm the edit made in the title. The edit made in the title is correct. The word "extended“ in the title can be deleted if necessary.

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

Data availibility

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  • Ahmad, A., Seadawy, A.R., Ahmed, S., et al.: Dynamical forms of breathers, rogue waves, lump and their interactions for Schrödinger–Hirota equation. Opt. Quant. Electron. 55(8), 730 (2023)

    Article  Google Scholar 

  • Akram, G., Sadaf, M., Zainab, I.: New graphical observations for KdV equation and KdV-Burgers equation using modified auxiliary equation method. Mod. Phys. Lett. B 36(01), 2150520 (2022a)

    Article  ADS  MathSciNet  Google Scholar 

  • Akram, G., Sadaf, M., Zainab, I.: The dynamical study of Biswas–Arshed equation via modified auxiliary equation method. Optik 255, 168614 (2022b)

    Article  ADS  Google Scholar 

  • Akram, G., Sadaf, M., Zainab, I., et al.: A comparative study of time fractional nonlinear Drinfeld–Sokolov–Wilson system via modified auxiliary equation Method. Fract. Fract. 7(9), 665 (2023)

    Article  Google Scholar 

  • Arnous, A.H., Ullah, M.Z., Asma, M., et al.: Dark and singular dispersive optical solitons of Schrödinger–Hirota equation by modified simple equation method. Optik 136, 445–450 (2017)

    Article  ADS  Google Scholar 

  • Ayati, Z., Badiepour, A.: Solitary solution of Jimbo–Miwa equation by the modified extended and multiple exp-function methods. Int. J. Appl. Comput. Math. 9(1), 1 (2023)

    Article  MathSciNet  Google Scholar 

  • Cakicioglu, H., Ozisik, M., Secer, A., et al.: Optical soliton solutions of Schrödinger–Hirota equation with parabolic law nonlinearity via generalized Kudryashov algorithm. Opt. Quant. Electron. 55(5), 407 (2023)

    Article  Google Scholar 

  • Ekici, M., Mirzazadeh, M., Sonmezoglu, A., et al.: Dispersive optical solitons with Schrödinger–Hirota equation by extended trial equation method. Optik 136, 451–461 (2017)

    Article  ADS  Google Scholar 

  • Eslami, M., Rezazadeh, H., Rezazadeh, M., et al.: Exact solutions to the space-time fractional Schrödinger–Hirota equation and the space-time modified KDV-Zakharov-Kuznetsov equation. Opt. Quant. Electron. 49, 1–15 (2017)

    Article  Google Scholar 

  • Fu, Z., Liu, S., Liu, S., et al.: New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Phys. Lett. A 290(1–2), 72–76 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  • Ghanbari, B., Günerhan, H., Ìlhan, O.A., et al.: Some new families of exact solutions to a new extension of nonlinear Schrödinger equation. Phys. Scr. 95(7), 075208 (2020)

    Article  ADS  Google Scholar 

  • Gonzalez-Gaxiola, O., Biswas, A., Moraru, L., et al.: Dispersive optical solitons with Schrödinger–Hirota equation by Laplace-Adomian decomposition approach. Universe 9(1), 19 (2022)

    Article  ADS  Google Scholar 

  • Guan, X., Yang, H., Meng, X., et al.: Higher-order rogue waves solutions of the modified Gerdjikov–Ivanov equation with dispersion via generalized Darboux transformation. Appl. Math. Lett. 136, 108466 (2023)

    Article  MathSciNet  Google Scholar 

  • Guo, Y.C.: Introduction to Nonlinear Partial Differential Equations. Tsinghua University Press, Beijing (2008)

    Google Scholar 

  • Li, Z.B.: Traveling Wave Solutions of Nonlinear Mathematical Physics Equations. Science Press, Beijing (2007)

    Google Scholar 

  • Liu, S., Fu, Z., Liu, S., et al.: Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 289(1–2), 69–74 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  • Liu, X., Zhang, H., Liu, W.: The dynamic characteristics of pure-quartic solitons and soliton molecules. Appl. Math. Model. 102, 305–312 (2022)

    Article  MathSciNet  Google Scholar 

  • Lu, J.: The exact solutions for the nonlinear variable-coefficient fifth-order Schrödinger equation. Results Phys. 39, 105708 (2022)

    Article  Google Scholar 

  • Ma, H.C., Yu, Y.D., Ge, D.J.: New exact travelling wave solutions for Zakharov–Kuznetsov equation. Commun. Theor. Phys. 51(4), 609 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  • Ma, H.C., Yu, Y.D., Ge, D.J.: The auxiliary equation method for solving the Zakharov–Kuznetsov (ZK) equation. Comput. Math. Appl. 58(11–12), 2523–2527 (2009)

    Article  MathSciNet  Google Scholar 

  • Osman, M.S., Machado, J.A.T., Baleanu, D., et al.: On distinctive solitons type solutions for some important nonlinear Schrödinger equations. Opt. Quant. Electron. 53, 1–24 (2021)

    Article  Google Scholar 

  • Ozisik, M., Onder, I., Esen, H., et al.: On the investigation of optical soliton solutions of cubic-quartic Fokas–Lenells and Schrödinger–Hirota equations. Optik 272, 170389 (2023)

    Article  ADS  Google Scholar 

  • Radha, B., Duraisamy, C.: The homogeneous balance method and its applications for finding the exact solutions for nonlinear equations. J. Ambient. Intell. Humaniz. Comput. 12, 6591–6597 (2021)

    Article  Google Scholar 

  • Shakeel, M., Attaullah, Shah, N.A.: Modified exp-function method to find exact solutions of microtubules nonlinear dynamics models. Symmetry 15(2), 360 (2023)

    Article  ADS  Google Scholar 

  • Sharif, A.: Jacobi elliptic function approach to a conformable fractional nonlinear Schrödinger–Hirota equation. Partial Differ. Equ. Appl. Math. 8, 100541 (2023a)

    Article  Google Scholar 

  • Sharif, A.: Jacobi elliptic function approach to a conformable fractional nonlinear Schrödinger–Hirota equation. Partial Differ. Equ. Appl. Math. 8, 100541 (2023b)

    Article  Google Scholar 

  • Si, R.D.R.J.: Exact travelling wave solutions for four forms of nonlinear Klein–Gordon equations. Phys. Lett. A 363(5–6), 440–447 (2007)

    ADS  MathSciNet  Google Scholar 

  • Si, R.D.R.J.: Traveling wave solutions for nonlinear wave equations: Theory and applications of the auxiliary equation method. Sci. Press 251, 1–184 (2019)

    ADS  Google Scholar 

  • Wang, H., Li, X., Zhou, Q., et al.: Dynamics and spectral analysis of optical rogue waves for a coupled nonlinear Schrödinger equation applicable to pulse propagation in isotropic media. Chaos Solit. Fract. 166, 112924 (2023)

    Article  Google Scholar 

  • Wang, H., Zhou, Q., Liu, W.: Exact analysis and elastic interaction of multi-soliton for a two-dimensional Gross-Pitaevskii equation in the Bose-Einstein condensation. J. Adv. Res. 38, 179–190 (2022)

    Article  Google Scholar 

  • Wang, T.Y., Zhou, Q., Liu, W.J.: Soliton fusion and fission for the high-order coupled nonlinear Schrödinger system in fiber lasers. Chin. Phys. B 31(2), 020501 (2022)

    Article  ADS  Google Scholar 

  • Yan, Y.Y., Liu, W.J.: Soliton rectangular pulses and bound states in a dissipative system modeled by the variable-coefficients complex cubic-quintic Ginzburg-Landau equation. Chin. Phys. Lett. 38(9), 094201 (2021)

    Article  ADS  Google Scholar 

  • Yan, L., Yao, R.X., Lou, S.Y.: An extended Hirota bilinear method and new wave structures of (2+1)-dimensional Sawada–Kotera equation. Appl. Math. Lett. 145, 108760 (2023)

    Article  MathSciNet  Google Scholar 

  • Yin, T., Ji, Y., Pang, J.: Variable coefficient extended cKP equation for Rossby waves and its exact solution with dissipation. Phys. Fluids 35(8), 086605 (2023)

    Article  ADS  Google Scholar 

  • Yin, Z., Jiang, X., Zhang, N., et al.: Stability analysis for linear systems with a differentiable time-varying delay via auxiliary equation-based method. Electronics 11(21), 3492 (2022)

    Article  Google Scholar 

  • Yin, T., Xing, Z., Pang, J.: Modified Hirota bilinear method to (3+ 1)-D variable coefficients generalized shallow water wave equation. Nonlinear Dyn. 111(11), 9741–9752 (2023)

    Article  Google Scholar 

  • Zayed, E.M.E., Shohib, R.M.A., Alngar, M.E.M., et al.: Solitons and conservation laws in magneto-optic waveguides with generalized Kudryashov’s equation by the unified auxiliary equation approach. Optik 245, 167694 (2021)

    Article  ADS  Google Scholar 

  • Zhao, X., Wang, L., Sun, W.: The repeated homogeneous balance method and its applications to nonlinear partial differential equations. Chaos Solit. Fract. 28(2), 448–453 (2006)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Funding

This work was supported by the National Natural Science Foundation of China under Grant No. 10561151, by the Basic Science Research Fund in the Universities Directly under the Inner Mongolia Autonomous Region under Grant No. JY20220003, and by the Basic Research Funds in the Universities directly under the Inner Mongolia Autonomous Region under Grant No. ZTY2023008.

Author information

Authors and Affiliations

  1. College of Sciences, Inner Mongolia University of Technology, Hohhot, 010051, People’s Republic of China

    Yajun Du, Tianle Yin & Jing Pang

  2. Inner Mongolia Key Laboratory of Statistical Analysis Theory for Life Data and Neural Network Modeling, Hohhot, 010051, People’s Republic of China

    Jing Pang

Authors
  1. Yajun Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tianle Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jing Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jing Pang.

Ethics declarations

Conflict of interest

The authors have not disclosed any conflict of interest.

Consent for publication

This article does not include any studies involving humans or animals. Written informed consent for publication of this paper was obtained all authors.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Du, Y., Yin, T. & Pang, J. The exact solutions of Schrödinger–Hirota equation based on the auxiliary equation method. Opt Quant Electron 56, 712 (2024). https://doi.org/10.1007/s11082-024-06283-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-024-06283-0

Keywords

Search

Navigation