Abstract
In this article, the generalized (3+1) dimensional Kadomtsev–Petviashvili (KP) and modified Kadomtsev–Petviashvili equations are explored, along with weak non-linearity, dispersion and disturbances which can demonstrate the expansion of surface water and prolonged waves in fluid dynamics. These models explain numerous nonlinear phenomena in the field of fluid dynamics, plasma physics and many more. Modified auxiliary equation method is implemented to derive analytic exact solutions for the governing equations. Some interesting and new travelling wave patterns have been observed. The obtained results include kink soliton, kinky periodic solitary wave, dark-bright soliton and periodic waves. Furthermore, graphical analysis is performed by selecting appropriate values of parameters in these solutions to explain the dynamic behavior of some different types of solitons. The proposed technique is well organized and proficient to discuss various KP-type equations physically.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11082-023-05758-w/MediaObjects/11082_2023_5758_Fig1_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11082-023-05758-w/MediaObjects/11082_2023_5758_Fig2_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11082-023-05758-w/MediaObjects/11082_2023_5758_Fig3_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11082-023-05758-w/MediaObjects/11082_2023_5758_Fig4_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11082-023-05758-w/MediaObjects/11082_2023_5758_Fig5_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11082-023-05758-w/MediaObjects/11082_2023_5758_Fig6_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11082-023-05758-w/MediaObjects/11082_2023_5758_Fig7_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11082-023-05758-w/MediaObjects/11082_2023_5758_Fig8_HTML.png)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs11082-023-05758-w/MediaObjects/11082_2023_5758_Fig9_HTML.png)
Similar content being viewed by others
Availability of data and materials
Not applicable.
References
Akram, G., Sadaf, M., Sarfraz, M., Anum, N.: Dynamics investigation of (1+ 1)-dimensional time-fractional potential Korteweg-de Vries equation. Alex. Eng. J. 61(1), 501–509 (2022)
Akram, G., Sadaf, M., Zainab, I.: The dynamical study of Biswas-Arshed equation via modified auxiliary equation method. Optik 255, 168614 (2022)
Akram, G., Sarfraz, M.: Multiple optical soliton solutions for CGL equation with Kerr law nonlinearity via extended modified auxiliary equation map** method. Optik 242, 167258 (2021)
Bala, P., Kaur, A.: Quantum electron acoustic solitons and double layers with \(\kappa\)-deformed kaniadakis distributed electrons. Indian J. Pure Appl. Phys. 59(8), 577–585 (2021)
Dorranian, D., Sabetkar, A.: Dust acoustic solitary waves in a dusty plasma with two kinds of nonthermal ions at different temperatures. Phys. Plasmas 19, 1 (2012)
Drazin, P.G., Johnson, R.S.: Solitons: An Introduction, vol. 2. Cambridge University Press, Cambridge (1989)
El-Tantawy, S., Moslem, W., Schlickeiser, R.: Ion-acoustic dark solitons collision in an ultracold neutral plasma. Phys. Scr. 90(8), 085606 (2015)
Gurefe, Y., Sonmezoglu, A., Misirli, E.: Application of the trial equation method for solving some nonlinear evolution equations arising in mathematical physics. Pramana 77, 1023–1029 (2011)
Jiang, Y., Tian, B., Wang, P., Li, M.: Bilinear form and soliton interactions for the modified Kadomtsev-Petviashvili equation in fluid dynamics and plasma physics. Nonlinear Dyn. 73, 1343–1352 (2013)
Chawla, J.K.: Effect of non-thermal electrons on ion-acoustic dressed solitons in unmagnetised plasmas. Pramana 95(1), 48 (2021)
Kadomtsev, B., Petviashvili, V.: Soviet Physics Doklady (English translation), pp. 539–541 (1970)
Liu, J.-G., He, Y.: New periodic solitary wave solutions for the (3+ 1)-dimensional generalized shallow water equation. Nonlinear Dyn. 90, 363–369 (2017)
Liu, J.-G., Wazwaz, A.-M., Zhu, W.-H.: Solitary and lump waves interaction in variable-coefficient nonlinear evolution equation by a modified ansätz with variable coefficients. J. Appl. Anal. Comput. 12(2), 517–532 (2022)
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)
Liu, J.-G., Zhu, W.-H.: Multiple rogue wave, breather wave and interaction solutions of a generalized (3+ 1)-dimensional variable-coefficient nonlinear wave equation. Nonlinear Dyn. 103(2), 1841–1850 (2021)
Liu, J.-G., Zhu, W.-H., He, Y.: Variable-coefficient symbolic computation approach for finding multiple rogue wave solutions of nonlinear system with variable coefficients. Z. Angew. Math. Phys. 72(4), 154 (2021)
Liu, J.-G., Zhu, W.-H., Wu, Y.-K., **, G.-H.: Application of multivariate bilinear neural network method to fractional partial differential equations. Res. Phys. 47, 106341 (2023)
Ma, W.X., Qin, Z., Lü, X.: Lump solutions to dimensionally reduced p-gKP and p-gBKP equations. Nonlinear Dyn. 84(2), 923–931 (2016)
Mirzazadeh, M., Eslami, M., Biswas, A.: Soliton solutions of the generalized Klein-Gordon equation by using \(\left(\frac{g^{\prime }}{g}\right)\)-expansion method. Comput. Appl. Math. 33, 831–839 (2014)
Rizvi, S.T., Seadawy, A.R., Ahmed, S., Younis, M., Ali, K.: Study of multiple lump and rogue waves to the generalized unstable space time fractional nonlinear Schrödinger equation. Chaos Solitons Fractals 151, 111251 (2021)
Seadawy, A.R.: Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput. Math. Appl. 67(1), 172–180 (2014)
Seadawy, A.R., Iqbal, M., Lu, D.: Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev-Petviashvili modified equal width dynamical equation. Comput. Math. Appl. 78(11), 3620–3632 (2019)
Seadawy, A.R., Rizvi, S.T.R., Ahmad, S., Younis, M., Baleanu, D.: Lump, lump-one stripe, multiwave and breather solutions for the Hunter–Saxton equation. Open Phys. 19(1), 1–10 (2021)
Sirendaoreji: A new auxiliary equation and exact travelling wave solutions of nonlinear equations. Phys. Lett. A 356(2), 124–130 (2006)
Tala-Tebue, E., Seadawy, A.R., Kamdoum-Tamo, P., Lu, D.: Dispersive optical soliton solutions of the higher-order nonlinear Schrödinger dynamical equation via two different methods and its applications. Eur. Phys. J. Plus 133, 1–10 (2018)
Wang, J., Shehzad, K., Seadawy, A.R., Arshad, M., Asmat, F.: Dynamic study of multi-peak solitons and other wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems with their stability. J. Taibah Univ. Sci. 17(1), 2163872 (2023)
Wazwaz, A.-M., El-Tantawy, S.: A new (3+ 1)-dimensional generalized Kadomtsev-Petviashvili equation. Nonlinear Dyn. 84, 1107–1112 (2016)
Wazwaz, A.-M., El-Tantawy, S.: Solving the (3+ 1)-dimensional KP-Boussinesq and BKP-Boussinesq equations by the simplified Hirota’s method. Nonlinear Dyn. 88, 3017–3021 (2017)
Younas, U., Seadawy, A.R., Younis, M., Rizvi, S.: Optical solitons and closed form solutions to the (3+ 1)-dimensional resonant Schrödinger dynamical wave equation. Int. J. Mod. Phys. B 34, 2050291 (2020)
Acknowledgements
The researchers would like to acknowledge Deanship of Scientific Research, Taif University for funding this work.
Funding
Not available.
Author information
Authors and Affiliations
Contributions
GA: Identification of the research problem, analysis of the outcomes, Funding acquisition, review and editing. MS: Methodology, conceptualization, validation. MS: Supervision, project administration. ASAA, MI: Conceptualization, review and editing, software.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Consent for publication
All the authors have agreed and given their consent for the publication of this research paper.
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.
About this article
Cite this article
Akram, G., Sadaf, M., Perveen, Z. et al. Exact travelling wave solutions for generalized (3+1) dimensional KP and modified KP equations. Opt Quant Electron 56, 325 (2024). https://doi.org/10.1007/s11082-023-05758-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11082-023-05758-w