Home Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation
Article
Licensed
Unlicensed Requires Authentication

Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation

  • Junjie Li , Jalil Manafian ORCID logo EMAIL logo , Nguyen Thi Hang , Dinh Tran Ngoc Huy and Alla Davidyants
Published/Copyright: December 1, 2021
Become an author with De Gruyter Brill
Author Information
International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 1

Abstract

The Hirota bilinear method is prepared for searching the diverse soliton solutions to the (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) equation. Also, the Hirota bilinear method is used to find the lump and interaction with two stripe soliton solutions. Interaction among lumps, periodic waves, and one-kink soliton solutions are investigated. Also, the solitary wave, periodic wave, and cross-kink wave solutions are examined for the KP-BBM equation. The graphs for various parameters are plotted to contain a 3D plot, contour plot, density plot, and 2D plot. We construct the exact lump and interaction among other types of solutions, by solving the underdetermined nonlinear system of algebraic equations with the associated parameters. Finally, analysis and graphical simulation are presented to show the dynamical characteristics of our solutions, and the interaction behaviors are revealed. The existing conditions are employed to discuss the available got solutions.

Keywords: Hirota bilinear method; interaction among lumps, periodic waves, and multi-kink; Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation; lump solitons; solitary wave, periodic wave, and cross-kink wave solutions
Corresponding author: Jalil Manafian, Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran; and Natural Sciences Faculty, Lankaran State University, 50, H. Aslanov str., Lankaran, Azerbaijan, E-mail:

Funding source: The Education and scientific research project for young and middle-aged teachers of Fujian Province

Award Identifier / Grant number: JAT190666

Award Identifier / Grant number: JAT200469

  1. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: This work is supported by the Education and scientific research project for young and middle-aged teachers of Fujian Province (No. JAT190666, No. JAT200469).

  3. Conflict of interest statement: The authors declare that they have no conflict of interest.

References

[1] K. Ali, S. T. R. Rizvi, B. Nawaz, and M. Younis, “Optical solitons for paraxial wave equation in Kerr media,” Mod. Phys. Lett. B, vol. 33, no. 03, pp. 1950020–1950029, 2019. https://doi.org/10.1142/s0217984919500209.Search in Google Scholar

[2] J. Manafian and M. Lakestani, “A new analytical approach to solve some of the fractional-order partial differential equations,” Indian J. Phys., vol. 91, no. 3, pp. 243–258, 2017. https://doi.org/10.1007/s12648-016-0912-z.Search in Google Scholar

[3] M. Dehghan, J. Manafian, and A. Saadatmandi, “Solving nonlinear fractional partial differential equations using the homotopy analysis method,” Numer. Methods Partial Differ. Equ., vol. 26, pp. 448–479, 2010. https://doi.org/10.1002/num.20460.Search in Google Scholar

[4] A. Arif, M. Younis, M. Imran, M. Tantawy, and S. T. R. Rizvi, “Solitons and lump wave solutions to the graphene thermophoretic motion system with a variable heat transmission,” Eur. Phys. J. Plus, vol. 134, no. 6, p. 303, 2019. https://doi.org/10.1140/epjp/i2019-12679-9.Search in Google Scholar

[5] C. Cattani, T. A. Sulaiman, H. M. Baskonus, and H. Bulut, “Solitons in an inhomogeneous Murnaghans rod,” Eur. Phys. J. Plus, vol. 133, p. 228, 2018. https://doi.org/10.1140/epjp/i2018-12085-y.Search in Google Scholar

[6] T. A. Sulaiman, T. A. Sulaiman, H. Bulut, and H. M. Baskonus, “Investigation of the fractional coupled viscous Burgers’ equation involving Mittag-Leffler kernel,” Physica A, vol. 527, p. 121126, 2019. https://doi.org/10.1016/j.physa.2019.121126.Search in Google Scholar

[7] X. Lu, W. X. Ma, J. Yu, and C. M. Khalique, “Solitary waves with the Madelung fluid description: a generalized derivative nonlinear Schrodinger equation,” Commun. Nonlinear Sci. Numer. Simulat., vol. 31, p. 40, 2016. https://doi.org/10.1016/j.cnsns.2015.07.007.Search in Google Scholar

[8] X. Lu, W. X. Ma, Y. Zhou, and C. M. Khalique, “Rational solutions to an ex- tended Kadomtsev-Petviashvili-like equation with symbolic computation,” Comput. Math. Appl., vol. 71, p. 1560, 2016. https://doi.org/10.1016/j.camwa.2016.02.017.Search in Google Scholar

[9] X. Lu and F. Lin, “Soliton excitations and shape-changing collisions in alphahelical proteins with interspine coupling at higher order,” Commun. Nonlinear Sci. Numer. Simulat., vol. 32, p. 241, 2016. https://doi.org/10.1016/j.cnsns.2015.08.008.Search in Google Scholar

[10] M. Singh, “New exact solutions for (3+1)-dimensional Jimbo-Miwa equation,” Nonlinear Dynam., vol. 84, p. 875, 2016. https://doi.org/10.1007/s11071-015-2533-z.Search in Google Scholar

[11] X. Lu, W. X. Ma, S. T. Chen, and C. M. Khalique, “A note on rational solutions to a Hirota-Satsuma-like equation,” Appl. Math. Lett., vol. 58, p. 13, 2016. https://doi.org/10.1016/j.aml.2015.12.019.Search in Google Scholar

[12] Z. H. Xu, H. L. Chen, M. R. Jiang, Z. D. Dai, and W. Chen, “Resonance and deflection of multi-soliton to the (2+1)-dimensional Kadomtsev-Petviashvili equation,” Nonlinear Dynam., vol. 78, p. 461, 2014. https://doi.org/10.1007/s11071-014-1452-8.Search in Google Scholar

[13] W. X. Ma and A. Abdeljabbar, “A bilinear Backlund transformation of a (3 + 1)- dimensional generalized KP equation,” Appl. Math. Lett., vol. 25, p. 1500, 2012. https://doi.org/10.1016/j.aml.2012.01.003.Search in Google Scholar

[14] (a) Y. Zhang and W. X. Ma, “Rational solutions to a KdV-like equation,” Appl. Math. Comput., vol. 256, p. 252, 2015. https://doi.org/10.1016/j.amc.2015.01.027. (b) Y. F. Zhang and W. X. Ma, “A study on rational solutions to a KP-like equation,” Z. Naturforsch., vol. 70a, p. 263, 2015.Search in Google Scholar

[15] W. X. Ma and E. G. Fan, “Linear superposition principle applying to Hirota bilinear equations,” Comput. Math. Appl., vol. 61, p. 950, 2011. https://doi.org/10.1016/j.camwa.2010.12.043.Search in Google Scholar

[16] L. N. Gao, X. Y. Zhao, Y. Y. Zi, J. Yu, and X. Lu, “Resonant behavior of multiple wave solutions to a Hirota bilinear equation,” Comput. Math. Appl., vol. 72, p. 1225, 2016. https://doi.org/10.1016/j.camwa.2016.06.008.Search in Google Scholar

[17] W. X. Ma, Y. Zhang, Y. N. Tang, and J. Y. Tu, “Hirota bilinear equations with linear subspaces of solutions,” Appl. Math. Comput., vol. 218, p. 7174, 2012. https://doi.org/10.1016/j.amc.2011.12.085.Search in Google Scholar

[18] W. X. Ma, Z. Y. Qin, and X. Lu, “Lump solutions to dimensionally reduced p-gKP and p-gBKP equations,” Nonlinear Dynam., vol. 84, p. 923, 2016. https://doi.org/10.1007/s11071-015-2539-6.Search in Google Scholar

[19] W. X. Ma, “Lump solutions to the Kadomtsev-Petviashvili equation,” Phys. Lett. A, vol. 379, p. 1975, 2015. https://doi.org/10.1016/j.physleta.2015.06.061.Search in Google Scholar

[20] X. G. Geng, “Algebraic-geometrical solutions of some multidimensional nonlinear evolution equations,” J. Phys. Math. Gen., vol. 36, p. 2289, 2003. https://doi.org/10.1088/0305-4470/36/9/307.Search in Google Scholar

[21] J. Manafian, “Novel solitary wave solutions for the (3+1)-dimensional extended Jimbo-Miwa equations,” Comput. Math. Appl., vol. 76, no. 5, pp. 1246–1260, 2018. https://doi.org/10.1016/j.camwa.2018.06.018.Search in Google Scholar

[22] W. X. Ma and Y. Zhou, “Lump solutions to nonlinear partial differential equations via Hirota bilinear forms,” J. Differ. Equ., vol. 264, pp. 2633–2659, 2018. https://doi.org/10.1016/j.jde.2017.10.033.Search in Google Scholar

[23] W. X. Ma, “A search for lump solutions to a combined fourthorder nonlinear PDE in (2+1)-dimensions,” J. Appl. Anal. Comput., vol. 9, pp. 1319–1332, 2019. https://doi.org/10.11948/2156-907x.20180227.Search in Google Scholar

[24] W. X. Ma, “Interaction solutions to Hirota-Satsuma-Ito equation in (2+1)-dimensions,” Front. Math. China, vol. 14, pp. 619–629, 2019. https://doi.org/10.1007/s11464-019-0771-y.Search in Google Scholar

[25] W. X. Ma, “Long-time asymptotics of a three-component coupled mKdV system,” Mathematics, vol. 7, no. 7, p. 573, 2019. https://doi.org/10.3390/math7070573.Search in Google Scholar

[26] J. Manafian, B. Mohammadi-Ivatlo, and M. Abapour, “Lump-type solutions and interaction phenomenon to the (2+1)-dimensional Breaking Soliton equation,” Appl. Math. Comput., vol. 13, pp. 13–41, 2019. https://doi.org/10.1016/j.amc.2019.03.016.Search in Google Scholar

[27] O. A. Ilhan, J. Manafian, and M. Shahriari, “Lump wave solutions and the interaction phenomenon for a variable-coefficient Kadomtsev-Petviashvili equation,” Comput. Math. Appl., vol. 78, no. 8, pp. 2429–2448, 2019. https://doi.org/10.1016/j.camwa.2019.03.048.Search in Google Scholar

[28] O. A. Ilhan and J. Manafian, “Periodic type and periodic cross-kink wave solutions to the (2+1)-dimensional breaking soliton equation arising in fluid dynamics,” Mod. Phys. Lett. B, vol. 33, p. 1950277, 2019. https://doi.org/10.1142/S0217984919502774.Search in Google Scholar

[29] W. X. Ma, Y. Zhou, and R. Dougherty, “Lump-type solutions to nonlinear differential equations derived from generalized bilinear equations,” Int. J. Mod. Phys. B, vol. 30, no. 28n29, p. 1640018, 2016. https://doi.org/10.1142/s021797921640018x.Search in Google Scholar

[30] J. Manafian, O. A. Ilhan, and A. Alizadeh, “Periodic wave solutions and stability analysis for the KP-BBM equation with abundant novel interaction solutions,” Phys. Scripta, vol. 95, p. 13, 2020, Art no. 065203. https://doi.org/10.1088/1402-4896/ab68be.Search in Google Scholar

[31] L. Kaur and A. M. Wazwaz, “Lump, breather and solitary wave solutions to new reduced form of the generalized BKP equation,” Int. J. Numer. Methods Heat Fluid Flow, vol. 92, pp. 569–579, 2019. https://doi.org/10.1108/hff-07-2018-0405.Search in Google Scholar

[32] L. Kaur and A. M. Wazwaz, “Bright-dark lump wave solutions for a new form of the (3+1)-dimensional BKP-Boussinesq equation,” Rom. Rep. Phys., vol. 70, no. 3, pp. 1–12, 2018.Search in Google Scholar

[33] L. Kaur and A. M. Wazwaz, “Bright-dark optical solitons for Schrödinger-Hirota equation with variable coefficients,” Optik, vol. 179, pp. 479–484, 2019. https://doi.org/10.1016/j.ijleo.2018.09.035.Search in Google Scholar

[34] L. Kaur and A. M. Wazwaz, “Dynamical analysis of lump solutions for (3+1) dimensional generalized KP-Boussinesq equation and its dimensionally reduced equations,” Phys. Scripta, vol. 93, no. 7, pp. 1402–4896, 2018. https://doi.org/10.1088/1402-4896/aac8b8.Search in Google Scholar

[35] S. Singh, L. Kaur, K. Sakkaravarthi, R. Sakthivel, and K. Murugesan, “Dynamics of higher-order bright and dark rogue waves in a new (2+1)-dimensional integrable Boussinesq model,” Phys. Scripta, vol. 95, no. 7, p. 115213, 2018.10.1088/1402-4896/abbca0Search in Google Scholar

[36] J. H. He, “Exp-function method for fractional differential equations,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 14, no. 6, pp. 363–366, 2013. https://doi.org/10.1155/2013/465723.Search in Google Scholar PubMed PubMed Central

[37] Y. Tian and J. Liu, “A modified exp-function method for fractional partial differential equations,” Int. J. Nonlinear Sci. Numer. Stimul., vol. 25, no. 2, pp. 1237–1241, 2021. https://doi.org/10.2298/tsci200428017t.Search in Google Scholar

[38] F. Y. Ji, C. H. He, J. J. Zhang, and J. H. He, “A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar,” Appl. Math. Model., vol. 82, pp. 437–448, 2020. https://doi.org/10.1016/j.apm.2020.01.027.Search in Google Scholar

[39] J. H. He, N. Qie, and C. H. He, “Solitary waves travelling along an unsmooth boundary,” Results Phys., vol. 24, p. 104104, 2021. https://doi.org/10.1016/j.rinp.2021.104104.Search in Google Scholar

[40] J. H. He, W. F. Hou, C. H. He, T. Saeed, and T. Hayat, “Solitary waves travelling along an unsmooth boundary,” Fractals, vol. 29, 2021. https://doi.org/10.1016/j.rinp.2021.104104.Search in Google Scholar

[41] C. Liu, “Periodic solution of fractal Phi-4 equation,” Therm. Sci., vol. 25, pp. 1345–1350, 2021. https://doi.org/10.2298/tsci200502032l.Search in Google Scholar

[42] J. H. He and Y. O. El-Dib, “Periodic property of the time-fractional Kundu-Mukherjee-Naskar equation,” Results Phys., vol. 19, p. 103345, 2020. https://doi.org/10.1016/j.rinp.2020.103345.Search in Google Scholar

[43] J. H. He, “Variational principle and periodic solution of the Kundu-Mukherjee-Naskar equation,” Results Phys., vol. 17, p. 103031, 2020. https://doi.org/10.1016/j.rinp.2020.103031.Search in Google Scholar
Received: 2020-07-12
Revised: 2021-09-03
Accepted: 2021-11-04
Published Online: 2021-12-01
Published in Print: 2023-02-23

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Modeling and assessment of the flow and air pollutants dispersion during chemical reactions from power plant activities
  4. Stochastic dynamics of dielectric elastomer balloon with viscoelasticity under pressure disturbance
  5. Unsteady MHD natural convection flow of a nanofluid inside an inclined square cavity containing a heated circular obstacle
  6. Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations
  7. Battery discharging model on fractal time sets
  8. Adaptive neural network control of second-order underactuated systems with prescribed performance constraints
  9. Optimal control for dengue eradication program under the media awareness effect
  10. Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations
  11. Modeling and simulations of a Zika virus as a mosquito-borne transmitted disease with environmental fluctuations
  12. Mathematical analysis of the impact of vaccination and poor sanitation on the dynamics of poliomyelitis
  13. Anti-sway method for reducing vibrations on a tower crane structure
  14. Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized G ′ / G -expansion method
  15. Convergence analysis of online learning algorithm with two-stage step size
  16. An estimative (warning) model for recognition of pandemic nature of virus infections
  17. Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation
  18. Global exponential stability of periodic solution of delayed discontinuous Cohen–Grossberg neural networks and its applications
  19. An efficient class of fourth-order derivative-free method for multiple-roots
  20. Numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
  21. Construction of breather solutions and N-soliton for the higher order dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising from wave patterns
  22. Delay-dependent robust stability analysis of uncertain fractional-order neutral systems with distributed delays and nonlinear perturbations subject to input saturation
  23. Construction of complexiton-type solutions using bilinear form of Hirota-type
  24. Inverse estimation of time-varying heat transfer coefficients for a hollow cylinder by using self-learning particle swarm optimization
  25. Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting
  26. Lump solutions to a generalized nonlinear PDE with four fourth-order terms
  27. Quantum motion control for packaging machines
https://doi.org/10.1515/ijnsns-2020-0156
Keywords for this article
Hirota bilinear method; interaction among lumps, periodic waves, and multi-kink; Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation; lump solitons; solitary wave, periodic wave, and cross-kink wave solutions

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Modeling and assessment of the flow and air pollutants dispersion during chemical reactions from power plant activities
  4. Stochastic dynamics of dielectric elastomer balloon with viscoelasticity under pressure disturbance
  5. Unsteady MHD natural convection flow of a nanofluid inside an inclined square cavity containing a heated circular obstacle
  6. Fractional-order generalized Legendre wavelets and their applications to fractional Riccati differential equations
  7. Battery discharging model on fractal time sets
  8. Adaptive neural network control of second-order underactuated systems with prescribed performance constraints
  9. Optimal control for dengue eradication program under the media awareness effect
  10. Shifted Legendre spectral collocation technique for solving stochastic Volterra–Fredholm integral equations
  11. Modeling and simulations of a Zika virus as a mosquito-borne transmitted disease with environmental fluctuations
  12. Mathematical analysis of the impact of vaccination and poor sanitation on the dynamics of poliomyelitis
  13. Anti-sway method for reducing vibrations on a tower crane structure
  14. Stable soliton solutions to the time fractional evolution equations in mathematical physics via the new generalized G ′ / G -expansion method
  15. Convergence analysis of online learning algorithm with two-stage step size
  16. An estimative (warning) model for recognition of pandemic nature of virus infections
  17. Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation
  18. Global exponential stability of periodic solution of delayed discontinuous Cohen–Grossberg neural networks and its applications
  19. An efficient class of fourth-order derivative-free method for multiple-roots
  20. Numerical modeling of thermal influence to pollutant dispersion and dynamics of particles motion with various sizes in idealized street canyon
  21. Construction of breather solutions and N-soliton for the higher order dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising from wave patterns
  22. Delay-dependent robust stability analysis of uncertain fractional-order neutral systems with distributed delays and nonlinear perturbations subject to input saturation
  23. Construction of complexiton-type solutions using bilinear form of Hirota-type
  24. Inverse estimation of time-varying heat transfer coefficients for a hollow cylinder by using self-learning particle swarm optimization
  25. Infinite line of equilibriums in a novel fractional map with coexisting infinitely many attractors and initial offset boosting
  26. Lump solutions to a generalized nonlinear PDE with four fourth-order terms
  27. Quantum motion control for packaging machines
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 28.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2020-0156/html
Scroll to top button