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Analytical study of solitons for the (2+1)-dimensional Painlevé integrable Burgers equation by using a unified method

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Abstract

In this work, the (2+1)-dimensional Painlevé integrable Burgers equation is investigated. By applying a certain unified method, some analytical solutions, involving rational functions, trigonometric functions and hyperbolic functions, are achieved. In order to predict the wave dynamics, several three-dimensional and two-dimensional graphs and contour profiles are constructed. Bright, dark, periodic, kink, anti-kink, singular, singular periodic, bell-shaped waves are thus obtained. The dynamics of these solutions can be illustrated graphically by choosing appropriate values for the parameters involved. Due to the presence of arbitrary constants in these derived solutions, they can be used to explain a variety of qualitative traits present in wave phenomena. The approach is efficient to algebraic computation and it can be used to categorize a wide range of wave forms, as shown by the demonstrated soliton solutions. Travelling wave solutions are converted into solitary wave solutions when certain values are set for the parameters. Using the Wolfram program Mathematica, we sketch the figures for various values of the associated parameters in order to closely examine the obtained solitons.

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References

  • Abdel-Gawad, H.I.: Towards a unified method for exact solutions of evolution equations: an application to reaction diffusion equations with finite memory transport. J. Stat. Phys. 147, 506–518 (2012)

    ADS  MathSciNet  Google Scholar 

  • Abdel-Gawad, H.I., Osman, M.: Exact solutions of the Korteweg-de Vries equation with space and time dependent coefficients by the extended unified method. Indian J. Pure Appl. Math. 45, 1–12 (2014)

    MathSciNet  Google Scholar 

  • Abdullah, F.A., Islam, M.T., Gómez-Aguilar, J.F., Akbar, M.A.: Impressive and innovative soliton shapes for nonlinear Konno–Oono system relating to electromagnetic field. Opt. Quant. Electron. 55, 69 (2023)

    Google Scholar 

  • Ablowitz, M.J., Ramani, A., Segur, H.: Nonlinear evolution equations and ordinary differential equations of Painlevé type. Lettere al Nuovo Cimento 23, 333–338 (1978)

    MathSciNet  Google Scholar 

  • Akram, G., Sadaf, M., Arshed, S., Sameen, F.: Bright, dark, kink, singular and periodic soliton solutions of Lakshmanan–Porsezian–Daniel model by generalized projective Riccati equations method. Optik 241, 167051 (2021)

    ADS  Google Scholar 

  • Anderson, D., Lisak, M.: Modulational instability of coherent optical-fiber transmission signals. Opt. Lett 9, 468–470 (1984)

    ADS  Google Scholar 

  • Ebaid, A.E.H.: Generalization of He’s Exp-function method and new exact solutions for Burgers equation. Z. Naturforsch. A 64, 604–608 (2009)

    ADS  Google Scholar 

  • Evans, L.C.: Partial Differential Equations, vol. 19. American Mathematical Society, Providence (2022)

    Google Scholar 

  • Faridi, W.A., AlQahtani, S.A.: The explicit power series solution formation and computation of Lie point infinitesimals generators: Lie symmetry approach. Physica Scripta 98, 125249 (2023)

    ADS  Google Scholar 

  • Faridi, W.A., Bakar, M.A., Akgül, A., Abd El-Rahman, M., El Din, S.M.: Exact fractional soliton solutions of thin-film ferroelectric material equation by analytical approaches. Alex. Eng. J. 78, 483–497 (2023)

    Google Scholar 

  • Faridi, W.A., Asjad, M.I., Jhangeer, A., Yusuf, A., Sulaiman, T.A.: The weakly non-linear waves propagation for Kelvin–Helmholtz instability in the magnetohydrodynamics flow impelled by fractional theory. Opt. Quant. Electron. 55(2), 172 (2023)

    Google Scholar 

  • Faridi, W.A., Asghar, U., Asjad, M.I., Zidan, A.M., Eldin, S.M.: Explicit propagating electrostatic potential waves formation and dynamical assessment of generalized Kadomtsev–Petviashvili modified equal width-Burgers model with sensitivity and modulation instability gain spectrum visualization. Results Phys. 44, 106167 (2023)

    Google Scholar 

  • Gao, F., Yang, X.-J., Srivastava, H.M.: Exact travelling-wave solutions for linear and non-linear heat transfer equations. Therm. Sci. 21, 2307–2311 (2017)

    Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  • Ghanbari, B., Akgül, A.: Abundant new analytical and approximate solutions to the generalized Schamel equation. Phys. Scr. 95(7), 075201 (2020)

    ADS  Google Scholar 

  • 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. Plus 134(7), 334 (2019)

    Google Scholar 

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

    ADS  Google Scholar 

  • Griffiths, G.W., Schiesser, W.E.: Linear and nonlinear waves. Scholarpedia 4, 4308 (2009)

    ADS  Google Scholar 

  • Hadhoud, A.R., Srivastava, H.M., Rageh, A.A.M.: Non-polynomial \(B\)-spline and shifted Jacobi spectral collocation techniques to solve time-fractional nonlinear coupled Burgers’ equations numerically. Adv. Differ. Equ. 2021, 439 (2021)

    MathSciNet  Google Scholar 

  • Hirota, R., Satsuma, J.: Soliton solutions of a coupled Korteweg-de Vries equation. Phys. Lett. A 85, 407–408 (1981)

    ADS  MathSciNet  Google Scholar 

  • Hosseini, K., Yazdani Bejarbaneh, E., Bekir, A., Kaplan, M.: New exact solutions of some nonlinear evolution equations of pseudo-parabolic type. Opt. Quant. Electron. 49, 1–10 (2017)

    Google Scholar 

  • Izadi, M., Srivastava, H.M.: An optimized second order numerical scheme applied to the non-linear Fisher’s reaction–diffusion equation. J. Interdiscip. 25, 471–492 (2022)

    Google Scholar 

  • Jhangeer, A., Muddassar, M., Awrejcewicz, J., Naz, Z., Riaz, M.B.: Phase portrait, multi-stability, sensitivity and chaotic analysis of Gardner’s equation with their wave turbulence and solitons solutions. Results Phys. 32, 104981 (2022)

    Google Scholar 

  • Kaplan, M., Ozer, M.N.: Multiple-soliton solutions and analytical solutions to a nonlinear evolution equation. Opt. Quant. Electron. 50, 1–10 (2018)

    Google Scholar 

  • Ke-Zhu, H., Bin, W., Xian-Feng, C.: Painlevé analysis and some solutions of \((2+ 1)\)-dimensional generalized Burgers equations. Commun. Theor. Phys. 39, 393 (2003)

    ADS  MathSciNet  Google Scholar 

  • Khan, K., Akbar, M.A.: Traveling wave solutions of the \((2+ 1)\)-dimensional Zoomeron equation and the Burgers equations via the MSE method and the Exp-function method. Ain Shams Eng. J. 5, 247–256 (2014)

    Google Scholar 

  • Khan, K., Akbar, M.A., Koppelaar, H.: Study of coupled nonlinear partial differential equations for finding exact analytical solutions. R. Soc. Open Sci. 2, 140406 (2015)

    ADS  MathSciNet  Google Scholar 

  • Kumar, R., Kaushal, R.S., Prasad, A.: Solitary wave solutions of selective nonlinear diffusion–reaction equations using homogeneous balance method. Pramana 75, 607–616 (2010)

    ADS  Google Scholar 

  • Kumar, S., Almusawa, H., Hamid, I., Akbar, M.A., Abdou, M.A.: Abundant analytical soliton solutions and evolutionary behaviors of various wave profiles to the Chaffee–Infante equation with gas diffusion in a homogeneous medium. Results Phys. 30, 104866 (2021)

    Google Scholar 

  • Kurkcu, H., Riaz, M.B., Imran, M., Jhangeer, A.: Lie analysis and nonlinear propagating waves of the (3+ 1)-dimensional generalized Boiti–Leon–Manna–Pempinelli equation. Alex. Eng. J. 80, 475–486 (2023)

    Google Scholar 

  • Lin, S., Wang, C., Dai, Z.: New exact traveling and non-traveling wave solutions for \((2+ 1)\)-dimensional Burgers equation. Appl. Math. Comput. 216, 3105–3110 (2010)

    MathSciNet  Google Scholar 

  • Liu, J.G., Yang, X.J.: Symmetry group analysis of several coupled fractional partial differential equations. Chaos Solitons 173, 113603 (2023)

    MathSciNet  Google Scholar 

  • Lonngren, K.E.: Soliton experiments in plasmas. Plasma Phys. 25, 943–982 (1983)

    ADS  Google Scholar 

  • Mahak, N., Akram, G.: Exact solitary wave solutions by extended rational sine-cosine and extended rational sinh–cosh techniques. Phys. Scr. 94, 115212 (2019)

    ADS  Google Scholar 

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

    ADS  MathSciNet  Google Scholar 

  • Peng, Y.-Z., Yomba, E.: New applications of the singular manifold method to the \((2+ 1)\)-dimensional Burgers equations. Appl. Math. Comput. 183, 61–67 (2006)

    MathSciNet  Google Scholar 

  • Rafiq, M.H., Raza, N., Jhangeer, A.: Nonlinear dynamics of the generalized unstable nonlinear Schrödinger equation: a graphical perspective. Opt. Quant. Electron. 55(7), 628 (2023)

    Google Scholar 

  • Serkin, V.N., Hasegawa, A.: Novel soliton solutions of the nonlinear Schrödinger equation model. Phys. Rev. Lett. 85, 4502 (2000)

    ADS  Google Scholar 

  • Shakeel, M., Mohyud-Din, S.T., Iqbal, M.A.: Closed-form solutions for coupled nonlinear Maccari system. Comput. Math. Appl. 76, 799–809 (2018)

    MathSciNet  Google Scholar 

  • Shukla, P.K., Rao, N.N., Yu, M.Y., Tsintsadze, N.L.: Relativistic nonlinear effects in plasmas. Phys. Rep 138, 1–149 (1986)

    ADS  Google Scholar 

  • Srivastava, H.M., Abdel-Gawad, H.I., Saad, K.M.: Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction–diffusion model subjected to the Dirichlet conditions. Discrete Contin. Dyn. Syst. 14, 3785–3801 (2021)

    MathSciNet  Google Scholar 

  • Srivastava, H.M., Ahmad, H., Ahmad, I., Thounthong, P., Khan, M.N.: Numerical simulation of 3-D fractional-order convection–diffusion PDE by a local meshless method. Therm. Sci. 25(1A), 347–358 (2021)

    Google Scholar 

  • Wang, K.J.: Resonant multiple wave, periodic wave and interaction solutions of the new extended (3+ 1)-dimensional Boiti–Leon–Manna–Pempinelli equation. Nonlinear Dyn. 111(17), 16427–16439 (2023)

    Google Scholar 

  • Wang, Q., Chen, Y., Zhang, H.: A new Riccati equation rational expansion method and its application to \((2+ 1)\)-dimensional Burgers equation. Chaos Solitons 25, 1019–1028 (2005)

    ADS  MathSciNet  Google Scholar 

  • Wang, X., Javed, S.A., Majeed, A., Kamran, M., Abbas, M.: Investigation of exact solutions of nonlinear evolution equations using unified method. Mathematics 10, 2996 (2022)

    Google Scholar 

  • Wu, X.-H., He, J.-H.: Solitary solutions, periodic solutions and compacton-like solutions using the exp-function method. Comput. Math. Appl. 54, 966–986 (2007)

    MathSciNet  Google Scholar 

  • Yang, X.-J., Gao, F., Srivastava, H.M.: Exact travelling wave equations for the local fractional two-dimensional Burgers-type equations. Comput. Math. Appl. 73, 203–210 (2017)

    MathSciNet  Google Scholar 

  • Zhou, Q., Zhu, Q.: Optical solitons in medium with parabolic law nonlinearity and higher order dispersion. Waves Random Complex Media 25, 52–59 (2015)

    ADS  Google Scholar 

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Acknowledgements

The researchers would like to acknowledge Deanship of Scientific Research, Taif university for funding this work. The authors are also grateful to anonymous referees for their valuable suggestions, which significantly improved this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan

    Haiqa Ehsan & Muhammad Abbas

  2. School of Mathematical Sciences, Universiti Sains Malaysia, 11800, Penang, Malaysia

    Farah Aini Abdullah

  3. Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, 21944, Taif, Saudi Arabia

    Ahmed S. M. Alzaidi

Authors
  1. Haiqa Ehsan
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  2. Muhammad Abbas
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  3. Farah Aini Abdullah
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  4. Ahmed S. M. Alzaidi
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Contributions

Conceptualization, HE and MA; Formal analysis, MA; Investigation, HE, MA, FAA, and ASMA; Methodology, MA; Project administration, MA and FAA.; Software, HE, MA, and FA.A; Supervision, MA; Validation, MA; Visualization, HE, MA, FAA, and ASMA; Writing—original draft, HE, MA, FAA, and ASMA; Writing—review and editing, HE, MA, FAA, and ASMA. All of the authors read and approved the final manuscript. All authors have read and agreed to the published version of the manuscript.

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Correspondence to Muhammad Abbas.

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Ehsan, H., Abbas, M., Abdullah, F.A. et al. Analytical study of solitons for the (2+1)-dimensional Painlevé integrable Burgers equation by using a unified method. Opt Quant Electron 56, 746 (2024). https://doi.org/10.1007/s11082-023-06212-7

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