Skip to main content
SpringerLink
Log in

Higher-order Sasa–Satsuma equation: Nucci’s reduction and soliton solutions

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

This paper discusses the existence of a diverse range of novel periodic nonlinear waves in the generalized (3+1)-dimensional Sasa–Satsuma equation. This equation models the transmission of femtosecond light pulses through optical fibers, taking into account third-order dispersion, self-frequency shift, and self-steepening effects in all three spatial dimensions of the system. The article presents newly derived periodic wave solutions expressed in terms of Jacobi elliptic functions and also obtains bright and dark soliton solutions in the long-wave limit of the periodic wave solutions. Moreover, a reduction technique employed to obtain another soliton solution and first integral of the considered model. Additionally, the article highlights the necessary fiber parameters required for the existence of these structures and provides graphical illustrations of selected solutions to demonstrate their physical nature.

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

Similar content being viewed by others

Data availibility statement

No Data associated in the manuscript.

Notes

  1. For simplicity we assume \(JSN(f,g):=JacobiSN(f,g),\ JCN(f,g):=JacobiCN(f,g),\ JCD(f,g):=JacobiCD(f,g),\ JSD(f,g):=JacobiSD(f,g),\ \)

References

  1. M.S. Hashemi, E. Darvishi, D. Baleanu, A geometric approach for solving the density-dependent diffusion Nagumo equation. Adv. Differ. Equ. 2016, 1–13 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. M.S. Hashemi, E. Ashpazzadeh, M. Moharrami, M. Lakestani, Fractional order Alpert multiwavelets for discretizing delay fractional differential equation of pantograph type. Appl. Numer. Math. 170, 1–13 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. H.R. Khodabandelo, E. Shivanian, S. Abbasbandy, A novel shifted Jacobi operational matrix method for nonlinear multi-terms delay differential equations of fractional variable-order with periodic and anti-periodic conditions. Math. Methods Appl. Sci. 45(16), 10116–10135 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  4. M.S. Hashemi, D. Baleanu, Lie symmetry analysis of fractional differential equations (CRC Press, 2020)

    Book  MATH  Google Scholar 

  5. M.S. Hashemi, M. Mirzazadeh, Optical solitons of the perturbed nonlinear Schrödinger equation using lie symmetry method. Optik 281, 170816 (2023)

    Article  ADS  Google Scholar 

  6. N. Kadkhoda, E. Lashkarian, M. Inc, M.A. Akinlar, Y.-M. Chu, New exact solutions and conservation laws to the fractional-order Fokker-Planck equations. Symmetry 12(8), 1282 (2020)

    Article  ADS  Google Scholar 

  7. S. Sahoo, S. Saha Ray, M.A. Abdou, M. Inc, Y.M. Chu, New soliton solutions of fractional Jaulent-Miodek system with symmetry analysis. Symmetry 12(6), 1001 (2020)

    Article  ADS  Google Scholar 

  8. Y. Chu, M.A. Shallal, S.M. Mirhosseini-Alizamini, H. Rezazadeh, S. Javeed, D. Baleanu, Application of modified extended tanh technique for solving complex Ginzburg-Landau equation considering Kerr law nonlinearity. Comput. Mater. Contin. 66(2), 1369–1377 (2021)

    Google Scholar 

  9. W. Malfliet, The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations. J. Comput. Appl. Math. 164, 529–541 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. Y. Chu, M.M. Khater, Y. Hamed, Diverse novel analytical and semi-analytical wave solutions of the generalized (2+1)-dimensional shallow water waves model. AIP Adv. 11(1), 015223 (2021)

    Article  ADS  Google Scholar 

  11. H. Srivastava, D. Baleanu, J. Machado, M. Osman, H. Rezazadeh, S. Arshed, H. Günerhan, Traveling wave solutions to nonlinear directional couplers by modified Kudryashov method. Phys. Scr. 95(7), 075217 (2020)

    Article  ADS  Google Scholar 

  12. S. Abbagari, A. Houwe, Y. Saliou, D. Douvagaï, Y.-M. Chu, M. Inc, H. Rezazadeh, S.Y. Doka, Analytical survey of the predator-prey model with fractional derivative order. AIP Adv. 11(3), 035127 (2021)

    Article  ADS  Google Scholar 

  13. P. Marco, C. Conti, G. Assanto, Optical modulational instability in a nonlocal medium. Phys. Rev. E 68(2), 025602 (2003)

    Article  ADS  Google Scholar 

  14. N.A. Kudryashov, Mathematical model of propagation pulse in optical fiber with power nonlinearities. Optik 212, 164750 (2020)

    Article  ADS  Google Scholar 

  15. C. Dai, Y. Wang, C. Yan, Chirped and chirp-free self-similar cnoidal and solitary wave solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients. Opt. Commun. 283(7), 1489–1494 (2010)

    Article  ADS  Google Scholar 

  16. A. Abeya, B. Prinari, G. Biondini, P.G. Kevrekidis, Solitons and soliton interactions in repulsive spinor Bose-Einstein condensates with nonzero background. Eur. Phys. J. Plus 136(11), 1–33 (2021)

    Article  Google Scholar 

  17. M. Scalora, M.S. Syrchin, N. Akozbek, E.Y. Poliakov, G. D’Aguanno, N. Mattiucci, M.J. Bloemer, A.M. Zheltikov, Generalized nonlinear Schrödinger equation for dispersive susceptibility and permeability: application to negative index materials. Phys. Rev. Lett. 95(1), 013902 (2005)

    Article  ADS  Google Scholar 

  18. D. Anderson, M. Lisak, Nonlinear asymmetric self-phase modulation and self-steepening of pulses in long optical waveguides. Phys. Rev. A 27(3), 1393 (1983)

    Article  ADS  Google Scholar 

  19. F.M. Mitschke, L.F. Mollenauer, Discovery of the soliton self-frequency shift. Opt. Lett. 11(10), 659–661 (1986)

    Article  ADS  Google Scholar 

  20. J.P. Gordon, Theory of the soliton self-frequency shift. Opt. Lett. 11(10), 662–664 (1986)

    Article  ADS  Google Scholar 

  21. V. Lashkin, Soliton of modified nonlinear Schrödinger equation with random perturbations. Phys. Rev. E 69(1), 016611 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  22. A.-M. Wazwaz, M. Mehanna, Higher-order Sasa–Satsuma equation: bright and dark optical solitons. Optik 243, 167421 (2021)

    Article  ADS  Google Scholar 

  23. S.-W. Yao, L. Akinyemi, M. Mirzazadeh, M. Inc, K. Hosseini, M. Şenol, Dynamics of optical solitons in higher-order Sasa–Satsuma equation. Results Phys. 30, 104825 (2021)

    Article  Google Scholar 

  24. E. Fadhal, A. Akbulut, M. Kaplan, M. Awadalla, K. Abuasbeh, Extraction of exact solutions of higher order Sasa–Satsuma equation in the sense of beta derivative. Symmetry 14(11), 2390 (2022)

    Article  ADS  Google Scholar 

  25. M. Ozisik, A. Secer, M. Bayram, (3+1)-dimensional Sasa–Satsuma equation under the effect of group velocity dispersion, self-frequency shift and self-steepening. Optik 275, 170609 (2023)

    Article  ADS  Google Scholar 

  26. A. Biswas, 1-soliton solution of (1+2)-dimensional nonlinear Schrödinger’s equation in dual-power law media. Phys. Lett. A 372(38), 5941–5943 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. M. Nucci, P. Leach, The determination of nonlocal symmetries by the technique of reduction of order. J. Math. Anal. Appl. 251(2), 871–884 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. M.S. Hashemi, A novel approach to find exact solutions of fractional evolution equations with non-singular kernel derivative. Chaos, Solitons Fractals 152, 111367 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

No funding received for this paper.

Author information

Authors and Affiliations

  1. Radiation Physics Laboratory, Department of Physics, Faculty of Sciences, Badji Mokhtar University, P.O. Box 12, 23000, Annaba, Algeria

    Houria Triki

  2. Department of Engineering Sciences, Faculty of Technology and Engineering, East of Guilan, University of Guilan, 44891-63157, Rudsar-Vajargah, Iran

    M. Mirzazadeh

  3. Department of Physics and Engineering Mathematics, Higher Institute of Engineering, El Shorouk Academy, Cairo, Egypt

    Hamdy M. Ahmed

  4. Department of Physics and Mathematics Engineering, Faculty of Engineering, Ain Shams University, Cairo, Egypt

    Islam Samir

  5. Department of Computer Engineering, Biruni University, 34010, Istanbul, Turkey

    M. S. Hashemi

Authors
  1. Houria Triki
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Mirzazadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hamdy M. Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Islam Samir
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. M. S. Hashemi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to this work.

Corresponding author

Correspondence to M. Mirzazadeh.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Ethical approval

Not applicable.

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

Triki, H., Mirzazadeh, M., Ahmed, H.M. et al. Higher-order Sasa–Satsuma equation: Nucci’s reduction and soliton solutions. Eur. Phys. J. Plus 138, 472 (2023). https://doi.org/10.1140/epjp/s13360-023-04127-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-023-04127-6

Search

Navigation