Skip to main content
SpringerLink
Log in

Application of the generalized unified method to solve (2+1)-dimensional Kundu–Mukherjee–Naskar equation

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

Abstract

The purpose of this study is to introduce a new extension method named the generalized unified method (GUM) and apply this method to the the (2+1) dimensional Kundu–Mukherjee–Naskar (KMN) equation. The GUM as a powerful method provides more general exact solutions for nonlinear partial differential equations (NPDEs) in a compact form with free parameters. Various exact solutions including also hyperbolic, trigonometric and rational forms can be derived from the obtained exact solutions with tuning these free parameters. The reason of choosing the KMN equation is that this equation as extension of the Schrödinger equation is used to model great numbers of considerable physical phenomena such as ion-acoustic waves in oceanic rogue waves, magnetized plasmas, bending of light beams, propagation pulses in optical fiber. Considering the physical importance of the KMN equation, the obtained wide range solution sets by using the GUM will play a significant role in the applied sciences that use the KMN equation to model their problems.

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

Not applicable.

References

  • Akbulut, A., Islam, R., Arafat, Y., Tascan, F.: A novel scheme for SMCH equation with two different approaches. Comput. Methods Differ. Equ. 11(2), 263–280 (2023)

  • Akbulut, A., Kumar, D.: Conservation laws and optical solutions of the complex modified korteweg-de vries equation. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.04.022

    Article  Google Scholar 

  • Akcagil, S., Aydemir, T.: A new application of the unified method. NTMSCI 6(1), 185–199 (2018)

    Google Scholar 

  • Akinyemi, L., Senol, M., Akpan, U., Oluwasegun, K.: The optical soliton solutions of generalized coupled nonlinear Schrodinger-Korteweg-de vries equations. Opt. Quanum Electron. 53, 1–14 (2021)

    Google Scholar 

  • Ali, I., Seadawy, A.R., Rizvi, S.T.R., Younis, M., Ali, K.: Conserved quantities along with Painleve analysis and optical solitons for the nonlinear dynamics of Heisenberg ferromagnetic spin chains model. Int. J. Mod. Phys. B 34(30), 2050283 (2020)

    ADS  MathSciNet  MATH  Google Scholar 

  • Aliyu, A.I., Li, Y., Baleanu, D.: Single and combined optical solitons, and conservation laws in (2+ 1)-dimensions with Kundu-Mukherjee-naskar equation. Chin. J. Phys. 63, 410–418 (2020)

    MathSciNet  Google Scholar 

  • Althobaiti, A., Althobaiti, S., El-Rashidy, K., Seadawy, A.R.: Exact solutions for the nonlinear extended kdV equation in a stratified shear flow using modified exponential rational method. Results Phys. 29, 104723 (2021)

    Google Scholar 

  • Arafat, S.Y., Islam, S.R., Bashar, M.H.: Influence of the free parameters and obtained wave solutions from CBS equation. Int. J. Appl. Comput. Math. 8(3), 99 (2022)

    MathSciNet  MATH  Google Scholar 

  • Arafat, S.Y., Khan, K., Islam, S.R., Rahman, M.: Parametric effects on paraxial nonlinear Schrodinger equation in Kerr media. Chin. J. Phys. (2022). https://doi.org/10.1016/j.cjph.2022.08.026

    Article  Google Scholar 

  • Asghari, Y., Eslami, M., Rezazadeh, H.: Exact solutions to the conformable time-fractional discretized mKdV lattice system using the fractional transformation method. Opt. Quantum Electron. 55(4), 318 (2023)

    Google Scholar 

  • Asghari, Y., Eslami, M., Rezazadeh, H.: Soliton solutions for the time-fractional nonlinear differential difference equation with conformable derivatives in the ferroelectric materials. Opt. Quantum Electron. 55(4), 289 (2023)

    Google Scholar 

  • Asjad, M.I., Inc, M., Faridi, W.A., Bakar, M.A., Muhammad, T., Rezazadeh, H.: Optical solitonic structures with singular and non-singular kernel for nonlinear fractional model in quantum mechanics. Opt. Quantum Electron. 55(3), 1–20 (2023)

    Google Scholar 

  • Bilal, M., Ahmad, J.: Dynamical nonlinear wave structures of the predator-prey model using conformable derivative and its stability analysis. Pramana 96(3), 149 (2022)

    ADS  Google Scholar 

  • Bilal, M., Ahmad, J.: Investigation of diverse genres exact soliton solutions to the nonlinear dynamical model via three mathematical methods. J. Ocean Eng. Sci. (2022). https://doi.org/10.1016/j.joes.2022.05.031

    Article  Google Scholar 

  • Bilal, M., Ren, J., Younas, U.: Stability analysis and optical soliton solutions to the nonlinear Schrodinger model with efficient computational techniques. Opt. Quantum Electron. 53, 1–19 (2021)

    Google Scholar 

  • Bilal, M., Rehman, S., Ahmad, J.: Analysis in fiber Bragg gratings with Kerr law nonlinearity for diverse optical soliton solutions by reliable analytical techniques. Mod. Phys. Lett. B 36(23), 2250122 (2022)

    ADS  Google Scholar 

  • Biswas, A., Ekici, M., Sonmezoglu, A., Zhou, Q., Moshokoa, S.P., Belic, M.: Optical soliton perturbation with full nonlinearity for Kundu–Eckhaus equation by extended trial function schema. Opt.- Int. J. Light Electron Opt. 160, 17–23 (2018)

    Google Scholar 

  • Biswas, A., Vega-Guzman, J., Bansal, A., Kara, A.H., Alzahrani, A.K., Zhou, Q., Belic, M.R.: Optical dromions, domain walls and conservation laws with Kundu–Mukherjee–naskar equation via traveling waves and lie symmetry. Results Phys. 16, 102850 (2020)

    Google Scholar 

  • Biswas, A., Yildirim, Y., Yasar, E., Triki, H., Zhou, Q., Moshokoa, S.P., Ullah, M.Z., Belic, M., et al.: Optical soliton perturbation with full nonlinearity in polarization preserving fibers using trial equation method. J. Optoelectron. Adv. Mater. 20(7–8), 385–402 (2018)

    Google Scholar 

  • Biswas, A., Zhou, Q., Ullah, M.Z., Triki, H., Moshokoa, S.P., Belic, M.: Optical soliton perturbation with anti-cubic nonlinearity by semi-inverse variational principle. Optik 143, 131–134 (2017)

    ADS  Google Scholar 

  • Bulut, H., Sulaiman, T.A., Demirdag, B.: Dynamics of soliton solutions in the chiral nonlinear Schrodinger equations. Nonlinear Dyn. 91, 1985–1991 (2018)

    Google Scholar 

  • Ekici, M., Sonmezoglu, A., Biswas, A., Belic, M.R.: Optical solitons in (2+1)-dimensions with Kundumukherjee- Naskar equation by extended trial function scheme. Chin. J. Phys. 57, 72–77 (2019)

    Google Scholar 

  • El-Rashidy, K., Seadawy, A.R.: Kinky breathers, multi-peak and multi-wave soliton solutions for the nonlinear propagation of Kundu–Eckhaus dynamical model. Int. J. Mod. Phys. B 34(32), 2050317 (2020)

    ADS  MathSciNet  MATH  Google Scholar 

  • El-Rashidy, K., Seadawy, A.R., Althobaiti, S., Makhlouf, M.: Investigation of interactional phenomena and multi wave solutions of the quantum hydrodynamic Zakharov–Kuznetsov model. Open Phys. 19(1), 91–99 (2021)

    Google Scholar 

  • Eslami, M., Neirameh, A.: New exact solutions for higher order nonlinear Schrodinger equation in optical fibers. Opt. Quantum Electron. 50, 1–8 (2018)

    Google Scholar 

  • Gozukizil, O.F., Akcagil, S., Aydemir, T.: Unification of all hyperbolic tangent function methods. Open Phys. 14(1), 524–541 (2016)

    Google Scholar 

  • Gunerhan, H., Khodadad, F.S., Rezazadeh, H., Khater, M.M.: Exact optical solutions of the (2+ 1) dimensions Kundu–Mukherjee–Naskar model via the new extended direct algebraic method. Mod. Phys. Lett. B 34(22), 2050225 (2020)

    ADS  MathSciNet  Google Scholar 

  • Guo, M., Fu, C., Zhang, Y., Liu, J., Yang, H.: Study of ion-acoustic solitary waves in a magnetized plasma using the three-dimensional time-space fractional schamel-kdv equation. Complexity (2018). https://doi.org/10.1155/2018/6852548

    Article  MATH  Google Scholar 

  • Haas, F., Mahmood, S.: Nonlinear ion-acoustic solitons in a magnetized quantum plasma with arbitrary degeneracy of electrons. Phys. Rev. E 94(3), 033212 (2016)

    ADS  Google Scholar 

  • Helal, M.A.: Introduction to Solitons, p. 289. Solitons, Giza (2022)

    Google Scholar 

  • Hollm, M., Dostal, L., Fischer, H., Seifried, R.: Study on the interaction of nonlinear water waves considering random seas. PAMM 20(1), e202000307 (2021)

    Google Scholar 

  • Islam, S.R., Bashar, M.H., Arafat, S.Y., Wang, H., Roshid, M.M.: Effect of the free parameters on the Biswas-Arshed model with a unified technique. Chin. J. Phys. 77, 2501–2519 (2022)

    MathSciNet  Google Scholar 

  • Islam, S.R., Kumar, D., Fendzi-Donfack, E., et al.: Impacts of nonlinearity and wave dispersion parameters on the soliton pulses of the (2+ 1)-dimensional Kundu–Mukherjee–Naskar equation. Rev. Mexicana de Fisica 68, 061301–06131 (2022)

    Google Scholar 

  • Jhangeer, A., Seadawy, A.R., Ali, F., Ahmed, A.: New complex waves of perturbed Shrodinger equation with Kerr law nonlinearity and Kundu–Mukherjee–Naskar equation. Results Phys. 16, 102816 (2020)

    Google Scholar 

  • Khalil, E.M., Sulaiman, T.A., Yusuf, A., Inc, M.: The m-fractional improved perturbed nonlinear Schrodinger equation: optical solitons and modulation instability analysis. Int. J. Mod. Phys. B 35(08), 2150121 (2021)

    ADS  MathSciNet  MATH  Google Scholar 

  • Kudryashov, N.A.: General solution of traveling wave reduction for the Kundu–Mukherjee–Naskar model. Optik 186, 22–27 (2019)

    ADS  Google Scholar 

  • Kudryashov, N.A.: Optical solitons of the resonant nonlinear Schrodinger equation with arbitrary index. Optik 235, 166626 (2021)

    ADS  Google Scholar 

  • Kumar, D., Hasan, M.M., Paul, G.C., Debnath, D., Mondal, N., Faruk, O.: Revisiting the spatiotemporal dynamics of a diffusive predator-prey system: an analytical approach. Results in Phys. 44, 106122 (2023)

    Google Scholar 

  • Kundu, A.: Novel integrable higher-dimensional nonlinear schroedinger equation: Properties, solutions, applications. arXiv:1305.4023 (2013)

  • Kundu, A., Mukherjee, A., Naskar, T.: Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents. Proc. Royal Soc. A: Math., Phys. Eng. Sci. 470(2164), 20130576 (2014)

    ADS  MathSciNet  MATH  Google Scholar 

  • Lan, Z.-Z., Guo, B.-L.: Nonlinear waves behaviors for a coupled generalized nonlinear Schrodingerboussinesq system in a homogeneous magnetized plasma. Nonlinear Dyn. 100, 3771–3784 (2020)

    Google Scholar 

  • Malik, S., Hashemi, M.S., Kumar, S., Rezazadeh, H., Mahmoud, W., Osman, M.: Application of new Kudryashov method to various nonlinear partial differential equations. Opt. Quantum Electron. 55(1), 8 (2023)

    Google Scholar 

  • Marin, M., Seadawy, A., Vlase, S., Chirila, A.: On mixed problem in Thermoelasticity of type iii for Cosserat media. J. Taibah Univ. Sci. 16(1), 1264–1274 (2022)

    Google Scholar 

  • Mukherjee, A., Janaki, M., Kundu, A.: A new (2+ 1) dimensional integrable evolution equation for an ion acoustic wave in a magnetized plasma. Phys. Plasmas 22(7), 072302 (2015)

    ADS  Google Scholar 

  • Mukherjee, A., Kundu, A.: Novel nonlinear wave equation: regulated rogue waves and accelerated soliton solutions. Phys. Lett. A 383(10), 985–990 (2019)

    ADS  MATH  Google Scholar 

  • Nakatsuji, H., Nakashima, H., Kurokawa, Y.I.: Solving the Schrodinger equation of atoms and molecules: chemical-formula theory, free-complement chemical-formula theory, and intermediate variational theory. J. Chem. Phys. 149(11), 114105 (2018)

    ADS  Google Scholar 

  • Nandi, D.C., Ullah, M.S., Ali, M.Z., et al.: Application of the unified method to solve the ion sound and Langmuir waves model. Heliyon 8(10), e10924 (2022)

    Google Scholar 

  • Rezazadeh, H., Kurt, A., Tozar, A., Tasbozan, O., Mirhosseini-Alizamini, S.M.: Wave behaviors of Kundu–Mukherjee–Naskar model arising in optical fiber communication systems with complex structure. Opt. Quantum Electron. 53, 1–11 (2021)

    Google Scholar 

  • Saha, A., Pradhan, B., Banerjee, S.: Multistability and dynamical properties of ion-acoustic wave for the nonlinear Schrodinger equation in an electron-ion quantum plasma. Phys. Scr. 95(5), 055602 (2020)

    ADS  Google Scholar 

  • San, S., Seadawy, A.R., Yasar, E.: Optical soliton solution analysis for the (2+ 1) dimensional Kundu–Mukherjee–Naskar model with local fractional derivatives. Opt. Quantum Electron. 54(7), 442 (2022)

    Google Scholar 

  • Seadawy, A.R., Ali, A., Althobaiti, S., El-Rashidy, K.: Construction of abundant novel analytical solutions of the space-time fractional nonlinear generalized equal width model via Riemann–Liouville derivative with application of mathematical methods. Open Phys. 19(1), 657–668 (2021)

    Google Scholar 

  • Seadawy, A.R., Cheemaa, N.: Some new families of spiky solitary waves of one-dimensional higherorder k-dV equation with power law nonlinearity in plasma physics. Indian J. Phys. 94(1), 117–126 (2020)

    ADS  Google Scholar 

  • Shahen, N.H.M., Rahman, M., et al.: Dispersive solitary wave structures with mi analysis to the unidirectional DGH equation via the unified method. Partial Differ. Equ. Appl. Math. 6, 100444 (2022)

    Google Scholar 

  • Singh, S., Mukherjee, A., Sakkaravarthi, K., Murugesan, K.: Higher dimensional localized and periodic wave dynamics in an integrable (2+ 1)-dimensional deep water oceanic wave model. Waves Random Complex Med. 33(1), 78–97 (2023)

    ADS  MathSciNet  Google Scholar 

  • Sulaiman, T.A., Akturk, T., Bulut, H., Baskonus, H.M.: Investigation of various soliton solutions to the Heisenberg ferromagnetic spin chain equation. J. Electromagn. Waves Appl. 32(9), 1093–1105 (2018)

    ADS  Google Scholar 

  • Talarposhti, R., Jalili, P., Rezazadeh, H., Jalili, B., Ganji, D., Adel, W., Bekir, A.: Optical soliton solutions to the (2+ 1)-dimensional Kundu–Mukherjee–Naskar equation. Int. J. Mod. Phys. B 34(11), 2050102 (2020)

    ADS  MathSciNet  MATH  Google Scholar 

  • Uddin, S., Karim, S., Alshammari, F., Roshid, H.-O., Noor, N., Hoque, F., Nadeem, M., Akgul, A., et al.: Bifurcation analysis of travelling waves and multi-rogue wave solutions for a nonlinear pseudo-parabolic model of Visco-Elastic Kelvin–Voigt fluid. Math. Probl. Eng. (2022). https://doi.org/10.1155/2022/8227124

    Article  Google Scholar 

  • Ullah, M.S., Abdeljabbar, A., Roshid, H.-O., Ali, M.Z.: Application of the unified method to solve the Biswas–Arshed model. Results Phys. 42, 105946 (2022)

    Google Scholar 

  • Ullah, M.S., Ali, M.Z., Biswas, A., Ekici, M., Khan, S., Moraru, L., Alzahrani, A.K., Belic, M.R., et al.: Optical soliton polarization with Lakshmanan–Porsezian–Daniel model by unified approach. Results Phys. 22, 103958 (2021)

    Google Scholar 

  • Wang, J., Shehzad, K., Seadawy, A.R., Arshad, M., Asmat, F.: Dynamic study of multi-peak solitons and other wave solutions of new coupled kdV and new coupled Zakharov–Kuznetsov systems with their stability. J. Taibah Univ. Sci. 17(1), 2163872 (2023)

    Google Scholar 

  • Wen, X.: Higher-order rational solutions for the (2+1) dimensional kmn equation. In: Proceedings of the Romanian Academy Series A, 18(3), 191–198. Yildirim, Y. (2019a) (2017)

  • Yildirim, Y.: Optical solitons to Kundu–Mukherjee–Naskar model in birefringent fibers with trial equation approach. Optik 183, 1026–1031 (2019)

    ADS  Google Scholar 

  • Yildirim, Y.: Optical solitons to Kundu–Mukherjee–Naskar model with modified simple equation approach. Optik 184, 247–252 (2019)

    ADS  Google Scholar 

  • Yildirim, Y.: Optical solitons to Kundu–Mukherjee–Naskar model with trial equation approach. Optik 183, 1061–1065 (2019)

    ADS  Google Scholar 

  • Yildirim, Y., Mirzazadeh, M.: Optical pulses with Kundu–Mukherjee–Naskar model in fiber communication systems. Chin. J. Phys. 64, 183–193 (2020)

    MathSciNet  Google Scholar 

Download references

Funding

There is no funding for this work.

Author information

Authors and Affiliations

  1. The Faculty of Economics and Administrative Sciences, Yalova University, Yalova, Turkey

    Tuğba Aydemir

Authors
  1. Tuğba Aydemir
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Tuğba Aydemir.

Ethics declarations

Conflict of interest

The author declare that there is no competing interests.

Ethical approval

The author would like to clarify that there is no financial/non-financial interests that are directly or indirectly related to the work submitted for publication.

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

Aydemir, T. Application of the generalized unified method to solve (2+1)-dimensional Kundu–Mukherjee–Naskar equation. Opt Quant Electron 55, 534 (2023). https://doi.org/10.1007/s11082-023-04807-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-023-04807-8

Keywords

Search

Navigation