Skip to main content
SpringerLink
Log in

Dynamics of Lump, Breather, Two-Waves and Other Interaction Solutions of (2+1)-Dimensional KdV Equation

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this investigation, we address a particular variant of the Korteweg–de Vries (KdV) equation, specifically focusing on the (2+1)-dimensional KdV equation. The equation can model various physical phenomena in different fields, including fluid dynamics, plasma physics, nonlinear optics, and other areas where coupled wave interactions are important. To commence, we establish the Auto-Bäcklund and Cole–Hopf transformations for the given model, resulting in the derivation of numerous soliton-like solutions characterized by hyperbolic, trigonometric, and exponential function waves. Furthermore, we effectively elucidate the behavior of lump, lump–kink, breather, two-wave, and three-wave solutions using the Hirota bilinear technique. Extensive numerical simulations employing 3-D profiles are conducted with meticulous consideration of pertinent parameter values, providing additional insights into the distinctive traits of the obtained solutions. Moreover, employing the extended transformed rational function method grounded in the bilinear form of the underlying equation, we uncover complexiton solutions. These solutions are depicted using 3-D and 2-D visualizations to portray their dynamics. Our findings reveal that the approach adopted to derive analytical solutions for nonlinear partial differential equations proves to be both efficient and potent. The combination of numerical simulations and visual representations enhances our understanding of these solutions, ultimately affirming the effectiveness and robustness of the employed methodology in tackling 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
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Data Availibility

No Data associated in the manuscript.

References

  1. Lou, S.Y.: A novel (2+ 1)-dimensional integrable KdV equation with peculiar solution structures. Chin. Phys. B 29(8), 080502 (2020)

    Article  Google Scholar 

  2. Hosseini, K., Ilie, M., Mirzazadeh, M., Baleanu, D.: A detailed study on a new (2 + 1)-dimensional mKdV equation involving the Caputo–Fabrizio time-fractional derivative. Adv. Differ. Equ. 2020, 331 (2020)

    Article  MathSciNet  Google Scholar 

  3. Boiti, M., Leon, J.P., Manna, M., Pempinelli, F.: On the spectral transform of a Korteweg-de Vries equation in two spatial dimensions. Inverse Probl. 2(3), 271 (1986)

    Article  MathSciNet  Google Scholar 

  4. Wazwaz, A.M.: Single and multiple-soliton solutions for the (2+ 1)-dimensional KdV equation. Appl. Math. Comput. 204(1), 20–26 (2008)

    MathSciNet  Google Scholar 

  5. Raza, N., Seadawy, A.R., Arshed, S., Rafiq, M.H.: A variety of soliton solutions for the Mikhailov–Novikov–Wang dynamical equation via three analytical methods. J. Geom. Phys. 176, 104515 (2022)

    Article  MathSciNet  Google Scholar 

  6. Cao, Y., Hu, P.Y., Cheng, Y., He, J.: Deformed two-dimensional rogue waves in the (2+ 1)-dimensional Korteweg-de Vries equation. Chin. Phys. B 30(3), 030503 (2021)

    Article  Google Scholar 

  7. Özkan, Y.S., Yasar, E., Çelik, N.: On the exact and numerical solutions to a new (2+ 1)-dimensional Korteweg-de Vries equation with conformable derivative. Nonlinear Eng. 10(1), 46–65 (2021)

    Article  Google Scholar 

  8. Peng, Y.Z.: Periodic waves and periodic solitons and their interactions for a (2+ 1)-dimensional KdV equation. Prog. Theor. Phys. 113(5), 927–933 (2005)

    Article  MathSciNet  Google Scholar 

  9. Wang, Z., Zou, L., Zong, Z., Qin, H.: A family of novel exact solutions to 2+ 1-dimensional KdV equation. Abstr. Appl. Anal. 2014, 1–9 (2014)

    MathSciNet  Google Scholar 

  10. Zhang, X., Chen, Y.: Deformation rogue wave to the (2+ 1)-dimensional KdV equation. Nonlinear Dyn. 90, 755–763 (2017)

    Article  MathSciNet  Google Scholar 

  11. Zhai, L., Zhao, J.: The Pfaffian technique: A (2+ 1)-dimensional Korteweg de Vries equation. J. appl. math. phys. 4(10), 1930–1935 (2016)

    Article  Google Scholar 

  12. Lou, S.Y.: Generalized dromion solutions of the (2+ 1)-dimensional KdV equation. J. Phys. A. 28(24), 7227 (1995)

    Article  MathSciNet  Google Scholar 

  13. Foroutan, M.R., Hashemi, M.S., Gholizadeh, L., Akgül, A., Jarad, F.: A new application of the Legendre reproducing kernel method. AIMS Math. 7(6), 10651–10670 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Hashemi, M.S., Akgül, A.: On new exact solutions of the generalized Fitzhugh–Nagumo equation with variable coefficients. Numer. Methods Partial Differ, Equ (2022)

    Google Scholar 

  16. Arnous, A.H., Mirzazadeh, M., Akbulut, A., Akinyemi, L.: Optical solutions and conservation laws of the Chen-Lee-Liu equation with Kudryashov’s refractive index via two integrable techniques. Waves Random Complex Media. 1-17 (2022)

  17. Hosseini, K., Akbulut, A.R.Z.U., Baleanu, D., Salahshour, S.: The Sharma–Tasso–Olver-Burgers equation: its conservation laws and kink solitons. Commun. Theor. Phys. 74(2), 025001 (2022)

    Article  MathSciNet  Google Scholar 

  18. Kaplan, M., Akbulut, A., Raza, N.: Research on sensitivity analysis and traveling wave solutions of the (4+ 1)-dimensional nonlinear Fokas equation via three different techniques. Phys. Scr. 97(1), 015203 (2022)

    Article  Google Scholar 

  19. Ghanbari, B., Akgül, A.: Abundant new analytical and approximate solutions to the generalized Schamel equation. Phys. Scr. 95(7), 075201 (2022)

    Article  Google Scholar 

  20. Akbulut, A., Kaplan, M., Kaabar, M.K.: New conservation laws and exact solutions of the special case of the fifth-order KdV equation. JOES. 7(4), 377–382 (2022)

    Google Scholar 

  21. Khan, K.A., Seadawy, A.R., Raza, N.: The homotopy simulation of MHD time dependent three dimensional shear thinning fluid flow over a stretching plate. Chaos, Solitons Fractals 157, 111888 (2022)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  23. Zayed, E.M., Shohib, R.M.: Optical solitons and other solutions to Biswas–Arshed equation using the extended simplest equation method. Optik 185, 626–635 (2019)

    Article  Google Scholar 

  24. Biswas, A., Sonmezoglu, A., Ekici, M., Kara, A.H., Alzahrani, A.K., Belic, M.R.: Cubic quartic optical solitons and conservation laws with Kudryashov’s law of refractive index by extended trial function. Comput. Math. Math. Phys. 61(12), 1995–2003 (2021)

    Article  MathSciNet  Google Scholar 

  25. Kudryashov, N.A., Loguinova, N.B.: Extended simplest equation method for nonlinear differential equations. Appl. Math. Comput. 205(1), 396–402 (2008)

    MathSciNet  Google Scholar 

  26. Khater, M., Ghanbari, B.: On the solitary wave solutions and physical characterization of gas diffusion in a homogeneous medium via some efficient techniques. Eur. Phys. J. 136(4), 1–28 (2021)

    Google Scholar 

  27. Ghanbari, B., Baleanu, D.: New optical solutions of the fractional Gerdjikov–Ivanov equation with conformable derivative. Front. Phys. 8, 167 (2020)

    Article  Google Scholar 

  28. Ghanbari, B.: Abundant soliton solutions for the Hirota–Maccari equation via the generalized exponential rational function method. Mod. Phys. Lett. B 33(09), 1950106 (2019)

    Article  MathSciNet  Google Scholar 

  29. Ghanbari, B., Baleanu, D., Al Qurashi, M.: New exact solutions of the generalized Benjamin–Bona–Mahony equation. Symmetry. 11(1), 20 (2018)

    Article  Google Scholar 

  30. Ghanbari, B., Kuo, C.K.: New exact wave solutions of the variable-coefficient (1+ 1)-dimensional Benjamin–Bona–Mahony and (2+ 1)-dimensional asymmetric Nizhnik–Novikov–Veselov equations via the generalized exponential rational function method. Eur. Phys. J. 134(7), 334 (2019)

    Google Scholar 

  31. Zhang, H., Ma, W.X.: Extended transformed rational function method and applications to complexiton solutions. Appl. Math. Comput. 230, 509–515 (2014)

    MathSciNet  Google Scholar 

  32. Yasar, E., Yildirim, Y., Rashid Adem, A.: Extended transformed rational function method to nonlinear evolution equations. Int. J. Nonlinear Sci. and Numeri. Simul 20(6), 691–701 (2019)

    Article  MathSciNet  Google Scholar 

  33. Ünsal, Ö., Bekir, A., Tascan, F., Naci Özer, M.: Complexiton solutions for two nonlinear partial differential equations via modification of simplified Hirota method. Waves Random Complex Media. 27(1), 117–128 (2017)

    Article  MathSciNet  Google Scholar 

  34. Mirzazadeh, M.: A couple of solutions to a (3+ 1)-dimensional generalized KP equation with variable coefficients by extended transformed rational function method. J. Math. Anal. Appl. 3(1), 188–194 (2015)

    MathSciNet  Google Scholar 

  35. Ibrahim, I.A., Taha, W.M., Noorani, M.S.M.: Homogenous balance method for solving exact solutions of the nonlinear Benny-Luke equation and Vakhnenko–Parkes equation. Zanco j. pure appl. sci. 31, 52–56 (2019)

    Google Scholar 

  36. Kaplan, M., Ozer, M.N.: Auto-Bäcklund transformations and solitary wave solutions for the nonlinear evolution equation. Opt. Quantum Electron. 50, 1–11 (2018)

    Google Scholar 

  37. Yusuf, A., Sulaiman, T.A.: Dynamics of Lump-periodic, breather and two-wave solutions with the long wave in shallow water under gravity and 2D nonlinear lattice. Nonlinear Sci. Numer. Simul. 99, 105846 (2021)

    Article  MathSciNet  Google Scholar 

  38. Liu, J.G., Du, J.Q., Zeng, Z.F., Nie, B.: New three-wave solutions for the (3+ 1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Nonlinear Dyn. 88, 655–661 (2017)

    Article  MathSciNet  Google Scholar 

  39. Zhang, H., Ma, W.X.: Extended transformed rational function method and applications to complexiton solutions. Appl. Math. Comput. 230, 509–515 (2014)

    MathSciNet  Google Scholar 

  40. Ünsal, Ö.: Application of extended transformed rational function method to some (3+ 1)- dimensional nonlinear evolution equations. Karaelmas Sci. Eng. J. (2018). https://doi.org/10.1080/17455030.2022.2045044

    Article  Google Scholar 

Download references

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore, Pakistan

    Nahal Jannat & Nauman Raza

  2. Department of Computer Engineering, Faculty of Engineering and Architecture, Kastamonu University, Kastamonu, Turkey

    Melike Kaplan

  3. Department of Mathematics, Art-Science Faculty, Uludag University, 16059, Bursa, Turkey

    Arzu Akbulut

Authors
  1. Nahal Jannat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nauman Raza
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Melike Kaplan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Arzu Akbulut
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

NJ methodology, writing—original draft preparation, NR conceptualization, analysis, writing—review and editing, MK data collection, visualization, writing—review and editing, AA supervision, project administration, software development

Corresponding author

Correspondence to Melike Kaplan.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Jannat, N., Raza, N., Kaplan, M. et al. Dynamics of Lump, Breather, Two-Waves and Other Interaction Solutions of (2+1)-Dimensional KdV Equation. Int. J. Appl. Comput. Math 9, 125 (2023). https://doi.org/10.1007/s40819-023-01601-8

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40819-023-01601-8

Keywords

Search

Navigation