Skip to main content
SpringerLink
Log in

Painlevé integrability and analytical solutions of variable coefficients negative order KdV–Calogero–Bogoyavlenskii–Schiff equation using auto-Bäcklund transformation

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

Abstract

The generalized form of the time-dependent variable coefficients negative order KdV–CBS equation, which represents the interactions of long wave propagations and have the numerous applications in the field of quantum and fluid mechanics, is being examined in this article. The study of Painlevé analysis effectively produced an integrable version of the considered equation. The several analytical solutions for this equation in the categories of exponential and rational functions have been reported by utilising auto-Bäcklund transformation approach. The periodic, kink-antikink, kink-soliton, and anti-kink soliton wave solutions have all been obtained using this method. To illustrate the physical applicability of the developed solutions, the results are graphically shown.

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

Similar content being viewed by others

Availability of data and materials

All the data and details of the results in this paper can be found within the paper.

References

  • Akinyemi, L., Şenol, M., Iyiola, O.S.: Exact solutions of the generalized multidimensional mathematical physics models via sub-equation method. Math. Comput. Simul. 182, 211–233 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Alharbi, A.R., Almatrafi, M.B., Abdelrahman, M.A.: Analytical and numerical investigation for Kadomtsev-Petviashvili equation arising in plasma physics. Phys. Scr. 95(4), 045215 (2020)

    Article  ADS  Google Scholar 

  • Ali, K.K., Yilmazer, R., Osman, M.S.: Dynamic behavior of the \((3+ 1)\)-dimensional KdV-Calogero-Bogoyavlenskii-Schiff equation. Opt. Quant. Electron. 54(3), 1–15 (2022)

    Article  Google Scholar 

  • Chen, Q., Sun, Z.: The exact solution of the non-linear Schrödinger equation by the exp-function method. Therm. Sci. 25(3), 2057–2062 (2021)

    Article  ADS  Google Scholar 

  • Chen, S.J., Lü, X., Tang, X.F.: Novel evolutionary behaviors of the mixed solutions to a generalized Burgers equation with variable coefficients. Commun. Nonlinear Sci. Numer. Simul. 95, 105628 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Duan, X., Lu, J.: The exact solutions for the \((3+1)\)-dimensional Boiti-Leon-Manna-Pempinelli equation. Res. Phys. 21, 103820 (2021)

    Google Scholar 

  • El-Shiekh, R.M.: Direct similarity reduction and new exact solutions for the variable-coefficient Kadomtsev-Petviashvili equation. Zeitschrift für Naturforschung A 70(6), 445–450 (2015)

    Article  ADS  Google Scholar 

  • El-Shiekh, R.M.: Periodic and solitary wave solutions for a generalized variable-coefficient Boiti-Leon-Pempinlli system. Comput. Math. Appl. 73(7), 1414–1420 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • El-Shiekh, R.M.: Painlevé test, Bäcklund transformation and consistent Riccati expansion solvability for two generalised cylindrical Korteweg-de Vries equations with variable coefficients. Zeitschrift für Naturforschung A 73(3), 207–213 (2018)

    Article  ADS  Google Scholar 

  • El-Shiekh, R.M.: Jacobi elliptic wave solutions for two variable coefficients cylindrical Korteweg-de Vries models arising in dusty plasmas by using direct reduction method. Comput. Math. Appl. 75(5), 1676–1684 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  • El-Shiekh, R.M.: Novel solitary and shock wave solutions for the generalized variable-coefficients \((2+ 1)\)-dimensional KP-Burger equation arising in dusty plasma. Chin. J. Phys. 71, 341–350 (2021)

    Article  MathSciNet  Google Scholar 

  • El-Shiekh, R.M., Gaballah, M.: New analytical solitary and periodic wave solutions for generalized variable-coefficients modified KdV equation with external-force term presenting atmospheric blocking in oceans. J. Ocean Eng. Sci. 7(4), 372–376 (2022)

    Article  Google Scholar 

  • Fetecau, C., Vieru, D.: Exact solutions for unsteady motion between parallel plates of some fluids with power-law dependence of viscosity on the pressure. Appl. Eng. Sci. 1, 100003 (2020)

    Google Scholar 

  • Gao, X.Y., Guo, Y.J., Shan, W.R.: Shallow water in an open sea or a wide channel: auto-and non-auto-Bäcklund transformations with solitons for a generalized \((2+ 1)\)-dimensional dispersive long-wave system. Chaos Solitons Fractals 138, 109950 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Ghanbari Ashrafi, T., Hoseinzadeh, S., Sohani, A., Shahverdian, M.H.: Applying homotopy perturbation method to provide an analytical solution for Newtonian fluid flow on a porous flat plate. Math. Methods Appl. Sci. 44(8), 7017–7030 (2021)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Hyder, A.A., Barakat, M.A.: General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics. Phys. Scr. 95(4), 045212 (2020)

    Article  ADS  Google Scholar 

  • Li, B.Q.: New breather and multiple-wave soliton dynamics for generalized Vakhnenko-Parkes equation with variable coefficients. J. Comput. Nonlinear Dyn. 16(9), 091006 (2021)

    Article  Google Scholar 

  • Lu, H., Zhang, Y., Mei, J.: Some exact solutions and infinite conservation laws of an extended KdV integrable system. Mod. Phys. Lett. B 34(26), 2050285 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  • Ma, Y.L., Wazwaz, A.M., Li, B.Q.: A new \((3+ 1)\)-dimensional Kadomtsev-Petviashvili equation and its integrability, multiple-solitons, breathers and lump waves. Math. Comput. Simul. 187, 505–519 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Montigny, M.D., Hassanabadi, H., Pinfold, J., Zare, S.: Exact solutions of the generalized Klein-Gordon oscillator in a global monopole space-time. Eur. Phys. J. Plus 136(7), 1–14 (2021)

    Article  Google Scholar 

  • Moussa, M.H.M., El-Shiekh, R.M.: Direct reduction and exact solutions for generalized variable coefficients 2D KdV equation under some integrability conditions. Commun. Theor. Phys. 55(4), 551 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Polyanin, A.D., Sorokin, V.G.: New exact solutions of nonlinear wave type PDEs with delay. Appl. Math. Lett. 108, 106512 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Saha Ray, S.: A numerical solution of the coupled sine-Gordon equation using the modified decomposition method. Appl. Math. Comput. 175(2), 1046–1054 (2006)

    MathSciNet  MATH  Google Scholar 

  • Rezazadeh, H., Vahidi, J., Zafar, A., Bekir, A.: The functional variable method to find new exact solutions of the nonlinear evolution equations with dual-power-law nonlinearity. Int. J. Nonlinear Sci. Num. Simul. 21(3–4), 249–257 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Saha Ray, S.: Painlevé analysis, group invariant analysis, similarity reduction, exact solutions, and conservation laws of Mikhailov-Novikov-Wang equation. Int. J. Geometr. Methods Modern Phys. 18(6), 2150094–3985 (2021)

    Article  Google Scholar 

  • Sahoo, S., Saha Ray, S.: Solitary wave solutions for time fractional third order modified KdV equation using two reliable techniques \((G^{\prime }/G)\)-expansion method and improved \((G^{\prime }/G)\)-expansion method. Phys. A 448, 265–282 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Savaissou, N., Gambo, B., Rezazadeh, H., Bekir, A., Doka, S.Y.: Exact optical solitons to the perturbed nonlinear Schrödinger equation with dual-power law of nonlinearity. Opt. Quant. Electron. 52, 1–16 (2020)

    Article  Google Scholar 

  • Sharma, K., Arora, R., Chauhan, A.: Invariance analysis, exact solutions and conservation laws of \((2+ 1)\)-dimensional dispersive long wave equations. Phys. Scr. 95(5), 055207 (2020)

    Article  ADS  Google Scholar 

  • Singh, S., Saha Ray, S.: Painlevé analysis, auto-Bäcklund transformation and analytic solutions for modified KdV equation with variable coefficients describing dust acoustic solitary structures in magnetized dusty plasmas. Mod. Phys. Lett. B 35(30), 2150464 (2021)

    Article  ADS  Google Scholar 

  • Verosky, J.M.: Negative powers of Olver recursion operators. J. Math. Phys. 32(7), 1733–1736 (1991)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Vinita, Saha Ray, S.: Symmetry analysis with similarity reduction, new exact solitary wave solutions and conservation laws of \((3+ 1)\)-dimensional extended quantum Zakharov-Kuznetsov equation in quantum physics. Modern Phys. Lett. B 35(09), 2150163 (2021)

  • Vinita, Saha Ray, S.: Lie symmetry reductions, power series solutions and conservation laws of the coupled Gerdjikov-Ivanov equation using optimal system of Lie subalgebra. Zeitschrift für angewandte Mathematik und Physik 72(4), 1–18 (2021)

  • Wang, K.: Fractal solitary wave solutions for fractal nonlinear dispersive Boussinesq-like models. Fractals 30(04), 1–8 (2022)

    Article  Google Scholar 

  • Wang, C., Fang, H.: Non-auto Bäcklund transformation, nonlocal symmetry and CRE solvability for the Bogoyavlenskii-Kadomtsev-Petviashvili equation. Comput. Math. Appl. 74(12), 3296–3302 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.M.: Optical bright and dark soliton solutions for coupled nonlinear Schrödinger (CNLS) equations by the variational iteration method. Optik 207, 164457 (2020)

    Article  ADS  Google Scholar 

  • Wazwaz, A.M.: Two new Painlevé integrable KdV-Calogero-Bogoyavlenskii-Schiff (KdV-CBS) equation and new negative-order KdV-CBS equation. Nonlinear Dyn. pp. 1-5 (2021)

  • Wazwaz, A.M.: New \((3+ 1)\)-dimensional Date-Jimbo-Kashiwara-Miwa equations with constant and time-dependent coefficients: Painlevé integrability. Phys. Lett. A 384(32), 126787 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Wazwaz, A.M., Xu, G.Q.: Kadomtsev-Petviashvili hierarchy: two integrable equations with time-dependent coefficients. Nonlinear Dyn. 100, 3711–3716 (2020)

    Article  Google Scholar 

  • Weiss, J., Tabor, M., Carnevale, G.: The Painlevé property for partial differential equations. J. Math. Phys. 24(3), 522–526 (1983)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Wei, G.M., Gao, Y.T., Hu, W., Zhang, C.Y.: Painlevé analysis, auto-Bäcklund transformation and new analytic solutions for a generalized variable-coefficient Korteweg-de Vries (KdV) equation. Eur. Phys. J. B Condens. Matter Complex Syst. 53(3), 343–350 (2006)

    Article  ADS  MATH  Google Scholar 

  • Yang, Y., Suzuki, T., Cheng, X.: Darboux transformations and exact solutions for the integrable nonlocal Lakshmanan-Porsezian-Daniel equation. Appl. Math. Lett. 99, 105998 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang, S., Xu, B., Zhang, H.Q.: Exact solutions of a KdV equation hierarchy with variable coefficients. Int. J. Comput. Math. 91(7), 1601–1616 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao, X., Tian, B., Tian, H.Y., Yang, D.Y.: Bilinear Bäcklund transformation, Lax pair and interactions of nonlinear waves for a generalized \((2+ 1)\)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics. Nonlinear Dyn. 103(2), 1785–1794 (2021)

    Article  Google Scholar 

Download references

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology, Rourkela, 769008, India

    Shailendra Singh & S. Saha Ray

Authors
  1. Shailendra Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Saha Ray
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Both the authors contributed equally to the work.

Corresponding author

Correspondence to S. Saha Ray.

Ethics declarations

Conflict of interests

The authors declare that there are no competing interests.

Ethical approval

Not applicable.

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

Singh, S., Ray, S.S. Painlevé integrability and analytical solutions of variable coefficients negative order KdV–Calogero–Bogoyavlenskii–Schiff equation using auto-Bäcklund transformation. Opt Quant Electron 55, 195 (2023). https://doi.org/10.1007/s11082-022-04452-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-022-04452-7

Keywords

Mathematics Subject Classification

Search

Navigation