Skip to main content
SpringerLink
Log in

Solitons of the \((1 + 1)\)- and \((2 + 1)\)-Dimensional Chiral Nonlinear Schrodinger Equations with the Jacobi Elliptical Function Method

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

Our objective is to find new analytical solutions of the \((1+1)\)- and \((2+1)\)-dimensional Chiral nonlinear Schrodinger (CNLS) equations using the Jacobi elliptical function method. The CNLS equations play a significant role in the development of quantum mechanics, particularly in the field of quantum Hall effect. Soliton solutions of the considered models are obtained such as, cnoidal solutions, the hyperbolic solutions and the trigonometric solutions. The obtained analytical solutions are new in the literature. The stability conditions of these solutions are also given. The obtained stable solutions are presented graphically for some specific parameters. Moreover, the conditions of modulational instability for both models are provided. The proposed method can be useful to obtain the analytical solutions of nonlinear partial differential equations.

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

Similar content being viewed by others

Data Availability

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

References

  1. Achab, A.E., Rezazadeh, H., Baleanu, D., Leta, T.D., Javeed, S., Alimgeer, K.S.: Ginzburg Landau equation’s Innovative Solution (GLEIS). Phys. Scr. 96(3), 035204 (2020)

    Google Scholar 

  2. Gao, X.-T., Tian, B.: Water-wave studies on a (2+1)-dimensional generalized variable-coefficient Boiti-Leon-Pempinelli system. Appl. Math. Lett. 128, 107858 (2022)

    MathSciNet  MATH  Google Scholar 

  3. Chu, Y.M., Shallal, M.A., Alizamini, S.M.M., Rezazadeh, H., Javeed, S., Baleanu, D.: Application of modified extended tanh technique for solving complex Ginzburg-Landau equation considering kerr law nonlinearity, Comput. Mater. Continua (2020)

  4. Ma, W.-X.: Nonlocal PT-symmetric integrable equations and related Riemann-Hilbert problems. Partial Differ. Equ. Appl. Math. 4, 100190 (2021)

    Google Scholar 

  5. Ma, W.: Riemann-Hilbert problems and soliton solutions of nonlocal reverse-time NLS hierarchies. Acta Math. Sci. 42(1), 127–140 (2022)

    MathSciNet  MATH  Google Scholar 

  6. Akinyemi, L., Inc, M., Khater, M.A., Rezazadeh, H.: Dynamical behaviour of Chiral nonlinear Schrödinger equation. Opt. Quant. Electron. 54, 191 (2021)

    Google Scholar 

  7. Tala-Tebue, E., Djoufack, Z.I., Kamdoum-Tamo, P.H., Kenfack-Jiotsa, A.: Cnoidal and solitary waves of a nonlinear Schrodinger equation in an optical fiber. Optik 174, 508–512 (2018)

    Google Scholar 

  8. Rezazadeh, H., Younis, M., Eslami, M., Bilal, M., Younas, U.: New exact traveling wave solutions to the (2+1) -dimensional Chiral nonlinear Schrödinger equation. Math. Model. Nat. Phenom. 16, 1–15 (2021)

    MathSciNet  MATH  Google Scholar 

  9. Zayed, E.M., Alurrfi, K.A.: Solitons and other solutions for two nonlinear Schrodinger equations using the new mapping method. Optik 144, 132–148 (2017)

    Google Scholar 

  10. Javeed, S., Saleem Alimgeer, K., Nawaz, S., Waheed, A., Suleman, M., Baleanu, D., Atif, M.: Soliton solutions of mathematical physics models using the exponential function technique. Symmetry 12(1), 176 (2020)

    Google Scholar 

  11. Javeed, S., Baleanu, D., Nawaz, S., Rezazadeh, H.: Soliton solutions of nonlinear Boussinesq models using the exponential function technique. Phys. Scr. 96, 105209 (2021)

    Google Scholar 

  12. Javeed, S., Riaz, S., Saleem Alimgeer, K., Atif, M., Hanif, A., Baleanu, D.: First integral technique for finding exact solutions of higher dimensional mathematical physics models. Symmetry 11(6), 783 (2019)

    Google Scholar 

  13. Zayed, E.M., Al-Nowehy, A.G., Elshater, M.E.: New-model expansion method and its applications to the resonant nonlinear Schrodinger equation with parabolic law nonlinearity. Europ. Phys. J. Plus 133(10), 417 (2018)

    Google Scholar 

  14. Hong, Z., Ji-Guang, H., Wei-Tao, W., Hong-Yong, A.: Applications of extended hyperbolic function method for quintic discrete nonlinear Schrodinger equation. Commun. Theor. Phys. 47(3), 474 (2007)

    MATH  Google Scholar 

  15. Korkmaz, A.: Complex wave solutions to mathematical biology models I: Newell-Whitehead-Segal and Zeldovich equations. Journal of Computational and Nonlinear Dynamics 13(8), 081004 (2018)

    Google Scholar 

  16. Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60(7), 650–654 (1992)

    MathSciNet  MATH  Google Scholar 

  17. Yamgoué, S.B., Deffo, G.R., Tala-Tebue, E., Pelap, F.B.: Exact transverse solitary and periodic wave solutions in a coupled nonlinear inductor-capacitor network. Chin. Phys. B 27(9), 096301 (2018)

    Google Scholar 

  18. Zaman, D.M.S., Amina, M., Dip, P.R., Mamun, A.A.: Nucleus-acoustic solitary waves in self-gravitating degenerate quantum plasmas. Chin. Phys. B 27(4), 040402 (2018)

    Google Scholar 

  19. Rezazadeh, H., Korkmaz, A., Eslami, M., Vahidi, J., Asghari, R.: Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method. Opt. Quant. Electron. 50(3), 150 (2018)

    Google Scholar 

  20. Manafian, J.: Optical soliton solutions for Schrodinger type nonlinear evolution equations by the tan(\(\phi \)/2)-expansion method. Optik 127(10), 4222–4245 (2016)

    Google Scholar 

  21. Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27(18), 1192 (1971)

    MATH  Google Scholar 

  22. Babalic, C.N., Constantinescu, R., Gerdjikov, V.S.: On the soliton solutions of a family of Tzitzeica equations. J. Geometr. Symmetry Phys. 37, 1–24 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Ablowitz, M.A., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149. Cambridge University Press, Cambridge (1991)

    MATH  Google Scholar 

  24. Cole, J.D.: On a quasi-linear parabolic equation occurring in aerodynamics. Q. Appl. Math. 9(3), 225–236 (1951)

    MathSciNet  MATH  Google Scholar 

  25. Kumar, S., Mann, N.: Abundant closed-form solutions of the (3+1)-Dimensional Vakhnenko-Parkes equation describing the dynamics of various solitary waves in ocean engineering. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.04.007

    Article  Google Scholar 

  26. Kumar, S., Niwas, M., Mann, N.: Abundant analytical closed-form solutions and various solitonic wave forms to the ZK-BBM and GZK-BBM equations in fluids and plasma physics. Partial Differ. Equ. Appl. Math. 4(December), 100200 (2021)

    Google Scholar 

  27. Kumar, S., Setu, R.: Lie symmetry reductions and dynamics of soliton solutions of (\(2+1\))-Dimensional Pavlov Equation. Pramana 94(1), 116 (2020)

    Google Scholar 

  28. Ma, W.-X.: Reduced nonlocal integrable mKdV equations of type \((-\lambda , \lambda )\) and their exact soliton solutions. Commun. Theor. Phys. 74(6), 065002 (2022)

    MathSciNet  MATH  Google Scholar 

  29. Ma, W.-X.: Soliton solutions by means of Hirota bilinear forms. Partial Differ. Equ. Appl. Math. 5, 100220 (2022)

    Google Scholar 

  30. Ma, W.-X.: Nonlocal integrable mKdV equations by two nonlocal reductions and their soliton solutions. J. Geom. Phys. 177, 104522 (2022)

    MathSciNet  MATH  Google Scholar 

  31. Gao, X., Guo, Y., Shan, W., Zhou, T., Wang, M., Yang, D.: In the atmosphere and oceanic fluids: scaling transformations, bilinear forms, Bäcklund Transformations and solitons for a generalized variable-coefficient Korteweg-de Vries-Modified Korteweg-de Vries Equation. China Ocean Eng. 35(4), 518–530 (2021)

    Google Scholar 

  32. Gao, X.-Y., Guo, Y.-J., Shan, W.-R.: Bilinear forms through the binary Bell polynomials, N solitons and Bäcklund transformations of the Boussinesq-Burgers system for the shallow water waves in a lake or near an ocean beach. Commun. Theor. Phys. 72(9), 095002 (2020)

    MATH  Google Scholar 

  33. Gao, X.-Y., Guo, Y.-J., Shan, W.-R.: Regarding the shallow water in an ocean via a Whitham-Broer-Kaup-like system: Hetero-Bäcklund transformations, bilinear forms and M solitons. Chaos Solitons Fract. 162, 112486 (2022)

    MATH  Google Scholar 

  34. Gao, X.-Y., Guo, Y.-J., Shan, W.-R.: Thinking about the oceanic shallow water via a generalized Whitham-Broer-Kaup-Boussinesq-Kupershmidt system. Chaos Solitons Fract. 164, 112672 (2022)

    MathSciNet  MATH  Google Scholar 

  35. Kumar, S., Setu, R.: Invariance analysis, optimal system, closed-form solutions and dynamical wave structures of a (2+1)-Dimensional dissipative long wave system. Phys. Scr. 96(12), 125202 (2021)

    Google Scholar 

  36. Tariq, H., Ahmed, H., Rezazadeh, H., Javeed, S., Alimgeer, K.S., Nonlaopon, K., Baili, J., Khedher, K.M.: New travelling wave analytic and residual power series solutions of conformable Caudrey-Dodd-Gibbon-Sawada-Kotera equation. Res. Phys. 29, 104591 (2021)

    Google Scholar 

  37. Chu, Y., Shallal, M.A., Mirhosseini-Alizamini, S.M., Rezazadeh, H., Javeed, S., Baleanu, D.: Application of modified extended Tanh technique for solving complex Ginzburg-Landau equation considering Kerr law nonlinearity. Comp. Mater. Continua 66(2), 1369–1378 (2021)

    Google Scholar 

  38. Nishino, A., Umeno, Y., Wadati, M.: Chiral nonlinear Schrodinger equation. Chaos Solitons Fract. 9(7), 1063–1069 (1998)

    MathSciNet  MATH  Google Scholar 

  39. Eslami, M.: Trial solution technique to chiral nonlinear Schrodinger’s equation in (1+2)-dimensions. Nonlinear Dyn. 85(2), 813–816 (2016)

    MathSciNet  Google Scholar 

  40. Biswas, A., Mirzazadeh, M., Eslami, M.: Soliton solution of generalized chiral nonlinear schrodinger’s equation with time-dependent coefficients. Acta Phys. Pol. B 45(4), 849–866 (2014)

    MathSciNet  MATH  Google Scholar 

  41. Raza, N., Javid, A.: Optical dark and dark-singular soliton solutions of (1+ 2)-dimensional chiral nonlinear Schrodinger’s equation. Waves Rand. Comp. Med. 29(3), 496–508 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Osman, M.S., Baleanu, D., Tariq, K.U.H., Kaplan, M., Younis, M., Rizvi, S.T.R.: Different types of progressive wave solutions via the 2D-chiral nonlinear Schrodinger equation. Front. Phys. 8, 215 (2020)

    Google Scholar 

  43. Giannini, J.A., Joseph, R.I.: The propagation of bright and dark solitons in lossy optical fibers. IEEE J. Quant. Electron. 26(12), 2109–2114 (1990)

    Google Scholar 

  44. Królikowski, W., Bang, O.: Solitons in nonlocal nonlinear media: exact solutions. Phys. Rev. E 63(1), 016610 (2000)

    Google Scholar 

  45. Arshad, M., Seadawy, A.R., Lu, D., Jun, W.: Modulation instability analysis of modify unstable nonlinear schrodinger dynamical equation and its optical soliton solutions. Res. Phys. 7, 4153–4161 (2017)

    Google Scholar 

  46. Tala-Tebue, E., Kenfack-Jiotsa, A., Tatchou-Ntemfack, M.H., Kofané, T.C.: Modulational instability in a pair of non-identical coupled nonlinear electrical transmission lines. Commun. Theor. Phys. 60(1), 93 (2013)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire d’Automatique et d’Informatique Appliquee (LAIA), IUT-FV of Bandjoun, The University Of Dschang, BP 134, Bandjoun, Cameroon

    Eric Tala-Tebue

  2. Faculty of Engineering Technology, Amol University of Special Modern Technologies, Amol, Iran

    Hadi Rezazadeh

  3. Department of Mathematics, COMSATS University Islamabad, Park Road Chak Shahzad, Islamabad, Pakistan

    Shumaila Javeed

  4. Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

    Shumaila Javeed

  5. Mathematics Research Center, Department of Mathematics, Near East University, Near East Boulevard, 99138, Mersin 10, Turkey

    Shumaila Javeed

  6. Department of Mathematics, Cankaya University, Ankara, Turkey

    Dumitru Baleanu

  7. Institute of Space Sciences, Magurele-Bucharest, Romania

    Dumitru Baleanu

  8. Department of Medical Research, China Medical University, Taichung, Taiwan

    Dumitru Baleanu

  9. Nord Staße 9, Weimar, Germany

    Alper Korkmaz

Authors
  1. Eric Tala-Tebue
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hadi Rezazadeh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shumaila Javeed
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dumitru Baleanu
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Alper Korkmaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shumaila Javeed.

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

Tala-Tebue, E., Rezazadeh, H., Javeed, S. et al. Solitons of the \((1 + 1)\)- and \((2 + 1)\)-Dimensional Chiral Nonlinear Schrodinger Equations with the Jacobi Elliptical Function Method. Qual. Theory Dyn. Syst. 22, 106 (2023). https://doi.org/10.1007/s12346-023-00801-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12346-023-00801-3

Keywords

Search

Navigation