Skip to main content
SpringerLink
Log in

Resonant multiple wave, periodic wave and interaction solutions of the new extended (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This paper is concerned with the new extended (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation (BLMPE), which is regarded as an extension of the well-known (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation and acts a key role in the incompressible fluid. By means of the Cole-Hopf transform, the Hirota bilinear equation of the studied equation is developed. Then based on the Hirota bilinear equation, we give a detailed investigation into the new exact solutions of the BLMPE. Firstly, the linear superposition principle combined with the weight algorithm is utilized to construct the resonant multiple wave solutions by introducing the parameterization for the wave numbers and frequencies. Secondly, the new homoclinic approach is applied to seek for the periodic wave solutions. Finally, the symbolic computation with the ansatz function scheme is adopted to construct the interaction solutions. The 3-D plots and 2-D contours are figured out to reveal the nonlinear dynamic behaviors of the different solutions. To the best of our knowledge, the results obtained in this work are all new and have not been reported elsewhere yet. The attained solutions can help us understand the nonlinear dynamic behaviors of the new extended (3 + 1)-dimensional BLMPE better. What’s more, the proposed methods in this work can be also adopted to study the resonant multiple wave, periodic wave and interaction solutions of the other PDEs arising in physics.

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

Similar content being viewed by others

Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

Abbreviations

BLMPE:

Boiti-Leon-Manna-Pempinelli equation

\(\Pi\) :

The expression of BLMPE

\(D_{x}^{m} D_{t}^{n}\) :

The Hirota bilinear operator

\(\ell\) :

A multivariate polynomial of M variables

\(\left[ {w\left( x \right),w\left( y \right),w\left( z \right),w\left( t \right)} \right]\) :

Computing the weights

References

  1. Ankiewicz, A., Bokaeeyan, M.: Integral relations for rogue wave formations of Gardner equation. Nonlinear Dyn. 99(4), 2939–2944 (2020)

    Google Scholar 

  2. Lü, X., Chen, S. J.: New general interaction solutions to the KPI equation via an optional decoupling condition approach. Commun. Nonlinear Sci. Numer. Simulat. 103, 105939 (2021)

  3. Wang, K.J.: Diverse wave structures to the modified Benjamin-Bona-Mahony equation in the optical illusions field. Mod. Phys. Lett. B 37(11), 2350012 (2023). https://doi.org/10.1142/S0217984923500124

    Article  MathSciNet  Google Scholar 

  4. Wang, K.J., Shi, F., Si, J., Liu, J.H., Wang, G.D.: Non-differentiable exact solutions of the local fractional Zakharov-Kuznetsov equation on the Cantor sets. Fractals 31(3), 2350028 (2023). https://doi.org/10.1142/S0218348X23500287

    Article  MATH  Google Scholar 

  5. Sohail, M., Nazir, U., Bazighifan, O., El-Nabulsi, R.A., Selim, M.M., Alrabaiah, H., Thounthong, P.: Significant involvement of double diffusion theories on viscoelastic fluid comprising variable thermophysical properties. Micromachines. 12(8), 951 (2021)

    Google Scholar 

  6. Alghamdi, M., Memon, A.A., Muhammad, T., et al.: A numerical investigation of a photovoltaic thermal system contained a trapezoidal channel with transport of silver and titanium oxide using the water as base fluids. Case Stud. Thermal Eng. 103056 (2023).

  7. Munguía-Medina, S. J., García-Sandoval, J. P., González-Álvarez, A.: Stability analysis of a class of electronic circuits based on thermodynamic principles part I: analysis of limit cycles: Nonlinear Dyn. 105(3), 2453–2471 (2021)

  8. Lü, X., Hui, H.W., Liu, F.F., Bai, Y.L.: Stability and optimal control strategies for a novel epidemic model of COVID-19. Nonlinear Dyn. 106, 1491 (2021)

    Google Scholar 

  9. Yin, M.Z., Zhu, Q.W., Lü, X.: Parameter estimation of the incubation period of COVID-19 based on the doubly interval-censored data model. Nonlinear Dyn. 106(2), 1347–1358 (2021)

    Google Scholar 

  10. He, J.H., Yang, Q., He, C.H.: A simple frequency formulation for the tangent oscillator. Axioms. 10(4), 320 (2021)

    Google Scholar 

  11. He, J.H.: The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators. J. Low Frequency Noise Vibrat. Active Control. 38(3–4), 1252–1260 (2019)

    Google Scholar 

  12. Wang, K.J., Si, J.: Dynamic properties of the attachment oscillator arising in the nanophysics. Open Physics. 21(1), 20220214 (2023)

    Google Scholar 

  13. Zeng, X., Liang, C., Yuan, C.: Solitary wave and singular wave solutions for Ivancevic Option Pricing model. Math. Probl. Eng. 2022, 4599194 (2022)

    Google Scholar 

  14. Gao, L.N., Zi, Y.Y., Yin, Y.H.: Bäcklund transformation, multiple wave solutions and lump solutions to a (3+1)-dimensional nonlinear evolution equation. Nonlinear Dyn. 89(3), 2233–2240 (2017)

    Google Scholar 

  15. Yin, Y.H., Lü, X., Ma, W.X.: Bäcklund transformation, exact solutions and diverse interaction phenomena to a (3+1)-dimensional nonlinear evolution equation. Nonlinear Dyn. 108(4), 4181–4194 (2022)

    Google Scholar 

  16. Du, Z., Tian, B., Xie, X.Y.: Bäcklund transformation and soliton solutions in terms of the Wronskian for the Kadomtsev–Petviashvili-based system in fluid dynamics. Pramana 90(4), 1–6 (2018)

    Google Scholar 

  17. Wang, K.J., Liu, J.H.: Diverse optical solitons to the nonlinear Schrödinger equation via two novel techniques. Euro. Phys. J. Plus. 138(1), 74 (2023). https://doi.org/10.1140/epjp/s13360-023-03710-1

    Article  Google Scholar 

  18. Bhrawy, A.H., Alhuthali, M.S., Abdelkawy, M.A.: New solutions for (1+ 1)-dimensional and (2+1)-dimensional Ito equations. Math. Probl. Eng. 2012, 537930 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Abdou, M.A.: The extended F-expansion method and its application for a class of nonlinear evolution equations. Chaos, Solitons Fractals 31(1), 95–104 (2007)

    MathSciNet  MATH  Google Scholar 

  20. Rabie, W.B., Ahmed, H.M.: Cubic-quartic optical solitons and other solutions for twin-core couplers with polynomial law of nonlinearity using the extended F-expansion method. Optik 253, 168575 (2022)

    Google Scholar 

  21. Shang, D.: Exact solutions of coupled nonlinear Klein-Gordon equation. Appl. Math. Comput. 217(4), 1577–1583 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Zayed, E.M.E., Gepreel, K.A., El-Horbaty, M.: Optical solitons in birefringent fibers with Kaup-Newell equation using two integration schemes. Optik 251, 167992 (2022)

    Google Scholar 

  23. Wang, K.J., Shi, F., Wang, G.D.: Abundant soliton structures to the (2+1)-dimensional Heisenberg ferromagnetic spin chain dynamical model. Adv. Math. Phys. 2023, 4348758 (2023)

    MathSciNet  MATH  Google Scholar 

  24. Wang, K. L., New perspective to the fractal Konopelchenko-Dubrovsky equations with M-truncated fractional derivative. Int. J. Geometric Methods Modern Phys. 2023, 2350072 (2023)

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

    MathSciNet  MATH  Google Scholar 

  26. Mohyud-Din, S.T., Khan, Y., Faraz, N.: Exp-function method for solitary and periodic solutions of Fitzhugh-Nagumo equation. Int. J. Numer. Meth. Heat Fluid Flow 22(3), 335–341 (2012)

    MathSciNet  MATH  Google Scholar 

  27. Rezazadeh, H., Inc, M., Baleanu, D.: New solitary wave solutions for variants of (3+1)-dimensional Wazwaz-Benjamin-Bona-Mahony equations. Front. Phys.. 8, 332 (2020)

    Google Scholar 

  28. Asjad, M.I., Munawar, N., Muhammad, T.: Traveling wave solutions to the Boussinesq equation via Sardar sub-equation technique. AIMS Math. 7(6), 11134–11149 (2022)

    MathSciNet  Google Scholar 

  29. Wang, K.J., Si, J.: Diverse optical solitons to the complex Ginzburg-Landau equation with Kerr law nonlinearity in the nonlinear optical fiber. Euro. Phys. J. Plus.. 138(3), 187 (2023). https://doi.org/10.1140/epjp/s13360-023-03804-w

    Article  Google Scholar 

  30. Seadawy, A.R., Kumar, D., Chakrabarty, A.K.: Dispersive optical soliton solutions for the hyperbolic and cubic-quintic nonlinear Schrödinger equations via the extended sinh-Gordon equation expansion method. Euro. Phys. J. Plus. 133(5), 182 (2018)

    Google Scholar 

  31. Raza, N., Arshed, S., Sial, S.: Optical solitons for coupled Fokas-Lenells equation in birefringence fibers. Mod. Phys. Lett. B 33(26), 1950317 (2019)

    MathSciNet  Google Scholar 

  32. Afzal, U., Raza, N., Murtaza, I.G.: On soliton solutions of time fractional form of Sawada-Kotera equation. Nonlinear Dyn. 95(1), 391–405 (2019)

    MATH  Google Scholar 

  33. Khater, M.M.A., Alfalqi, S.H., Alzaidi, J.F., et al.: Analytically and numerically, dispersive, weakly nonlinear wave packets are presented in a quasi-monochromatic medium. Results Phys. 46, 106312 (2023)

    Google Scholar 

  34. Khater, M.M.A.: Multi-vector with nonlocal and non-singular kernel ultrashort optical solitons pulses waves in birefringent fibers. Chaos Solitons Fractals 167, 113098 (2023)

    MathSciNet  Google Scholar 

  35. Attia, R.A.M., Zhang, X., Khater, M.M.A.: Analytical and hybrid numerical simulations for the (2+1)-dimensional Heisenberg ferromagnetic spin chain. Results Phys. 43, 106045 (2022)

    Google Scholar 

  36. Sağlam Özkan, Y., Seadawy, A.R., Yaşar, E.: Multi-wave, breather and interaction solutions to (3+1) dimensional Vakhnenko-Parkes equation arising at propagation of high-frequency waves in a relaxing medium. J. Taibah Univ. Sci. 15(1), 666–678 (2021)

    Google Scholar 

  37. Seadawy, A. R., Bilal, M., Younis, M.: Analytical mathematical approaches for the double-chain model of DNA by a novel computational technique. Chaos, Solitons & Fractals. 144, 110669 (2021)

  38. Khater, M. M. A.: Prorogation of waves in shallow water through unidirectional Dullin-Gottwald-Holm model; computational simulations. Int. J. Modern Phys. B, 2350071 (2022)

  39. Khater, M.M.A.: In solid physics equations, accurate and novel soliton wave structures for heating a single crystal of sodium fluoride. Int. J. Mod. Phys. B 37(7), 2350068 (2022)

    Google Scholar 

  40. Khater, M.M.A.: Nonlinear elastic circular rod with lateral inertia and finite radius: Dynamical attributive of longitudinal oscillation. Int. J. Mod. Phys. B 37(6), 2350052 (2023)

    Google Scholar 

  41. Khater, M.M.A., Zhang, X., Attia, R.A.M.: Accurate computational simulations of perturbed Chen-Lee-Liu equation. Results Phys. 45, 106227 (2023)

    Google Scholar 

  42. Rizvi, S.T.R., Seadawy, A.R., Ali, I.: Chirp-free optical dromions for the presence of higher order spatio-temporal dispersions and absence of self-phase modulation in birefringent fibers. Mod. Phys. Lett. B 34(35), 2050399 (2020)

    MathSciNet  Google Scholar 

  43. Khater, M.M.A.: Novel computational simulation of the propagation of pulses in optical fibers regarding the dispersion effect. Int. J. Mod. Phys. B 37(9), 2350083 (2023)

    MathSciNet  Google Scholar 

  44. Wazwaz, A.M.: Painleve analysis for new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equations with constant and time-dependent coefficients. Int. J. Numer. Method H. 30(9), 4259–4266 (2019)

    Google Scholar 

  45. Liu, J. G.: Wazwaz A M. Breather wave and lump-type solutions of new (3+ 1)-dimensional Boiti-Leon-Manna-Pempinelli equation in incompressible fluid. Math. Methods Appl. Sci. 44(2), 2200–2208 (2021)

  46. Yuan, N.: Rich analytical solutions of a new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Results Phys. 22, 103927 (2021)

    Google Scholar 

  47. Wang, K.J., Liu, J.H., Si, J., Shi, F., Wang, G.D.: N-soliton, breather, lump solutions and diverse travelling wave solutions of the fractional (2+1)-dimensional Boussinesq equation. Fractals 31(3), 2350023 (2023). https://doi.org/10.1142/S0218348X23500238

    Article  MATH  Google Scholar 

  48. Ma, W.X.: N-soliton solution and the Hirota condition of a (2+1)-dimensional combined equation. Math. Comput. Simul. 190, 270–279 (2021)

    MathSciNet  MATH  Google Scholar 

  49. Ma, W.X., Zhou, Y.: Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J. Differential Equations 264(4), 2633–2659 (2018)

    MathSciNet  MATH  Google Scholar 

  50. Lü, X., Chen, S.J.: Interaction solutions to nonlinear partial differential equations via Hirota bilinear forms: One-lump-multi-stripe and one-lump-multi-soliton types. Nonlinear Dyn. 103, 947–977 (2021)

    MATH  Google Scholar 

  51. Liu, J.G., Ye, Q.: Stripe solitons and lump solutions for a generalized Kadomtsev-Petviashvili equation with variable coefficients in fluid mechanics. Nonlinear Dyn. 96, 23–29 (2019)

    MATH  Google Scholar 

  52. Ma, W.X.: Bilinear equations, Bell polynomials and linear superposition principle. J. Phys: Conf. Ser. 411(1), 012021 (2013)

    Google Scholar 

  53. Zhang, H.Q., Ma, W.X.: Resonant multiple wave solutions for a (3+ 1)-dimensional nonlinear evolution equation by linear superposition principle. Comput. Math. Appl. 73(10), 2339–2343 (2017)

    MathSciNet  MATH  Google Scholar 

  54. Liu, J.G., Yang, X.J., Feng, Y.Y.: Resonant multiple wave solutions to some integrable soliton equations. Chin. Phys. B 28(11), 110202 (2019)

    Google Scholar 

  55. Ma, W.X., Fan, E.G.: Linear superposition principle applying to Hirota bilinear equations. Comput. Math. Appl. 61(4), 950–959 (2011)

    MathSciNet  MATH  Google Scholar 

  56. Liu, J.G., Yang, X.J., Feng, Y.: Y: Nonlinear dynamic behaviors of the generalized (3+1)-dimensional KP equation. ZAMM-J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik. 102(10), e202000168 (2022)

    MathSciNet  Google Scholar 

  57. Yokus, A., Isah, M.A.: Stability analysis and solutions of (2+ 1)-Kadomtsev–Petviashvili equation by homoclinic technique based on Hirota bilinear form. Nonlinear Dyn. 109, 3029–3040 (2022)

    Google Scholar 

  58. Yin, Y.H., Chen, S.J., Lü, X.: Localized characteristics of lump and interaction solutions to two extended Jimbo-Miwa equations. Chin. Phys. B 29(12), 120502 (2020)

    Google Scholar 

Download references

Funding

This work was supported by the Key Programs of Universities in Henan Province of China (22A140006), Program of Henan Polytechnic University (B2018-40).

Author information

Authors and Affiliations

  1. School of Physics and Electronic Information Engineering, Henan Polytechnic University, Jiaozuo, 454003, China

    Kang-Jia Wang

Authors
  1. Kang-Jia Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Material preparation, data collection and analysis were performed by Kang-Jia Wang. The draft of the manuscript was written by Kang-Jia Wang.

Corresponding author

Correspondence to Kang-Jia Wang.

Ethics declarations

Coonflict of interest

The authors have no relevant financial or non-financial interests to disclose.

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

Wang, KJ. Resonant multiple wave, periodic wave and interaction solutions of the new extended (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation. Nonlinear Dyn 111, 16427–16439 (2023). https://doi.org/10.1007/s11071-023-08699-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-08699-x

Keywords

Search

Navigation