On the soliton sensitivity, lump and analytic solutions of the generalized (3+1)-dimensional P-type nonlinear wave equation

Longjun Zhan; Adnan Ahmad Mahmud; Haci Mehmet Baskonus; Longjun Zhan; Adnan Ahmad Mahmud; Haci Mehmet Baskonus

doi:10.3934/math.20261028

AIMS Mathematics

2026, Volume 11, Issue 8: 25652-25671. doi: 10.3934/math.20261028

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Research article

On the soliton sensitivity, lump and analytic solutions of the generalized (3+1)-dimensional P-type nonlinear wave equation

1.
School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, China
2.
Information Technology Department, Tishk International University, Kurdistan Region, Iraq
3.
Harran University, Faculty of Arts and Sciences, Department of Mathematics, Şanlıurfa 63290, Türkiye
4.
Harran University, Faculty of Education, Department of Mathematics and Science Education, Şanlıurfa 63190, Türkiye
5.
Scientific Research Department, Azerbaijan University of Architecture and Construction, Baku AZ1073, Azerbaijan

Received: 20 June 2026 Revised: 03 August 2026 Accepted: 12 August 2026 Published: 19 August 2026
MSC : 35C08, 35C07, 35C05, 35C09, 35Dxx

In this manuscript, we investigated a generalized (3+1)-dimensional P-type nonlinear wave equation. First, for a more detailed analysis of the studied model, we transformed it into a nonlinear ordinary differential equation by utilizing an appropriate wave transformation formula. We utilized a modified Riccati equation method (also called the generalized Riccati equation method), signifying its first application for this model. The extraction of traveling waves and singular solitons used exponential and hyperbolic trigonometric functions, which exhibited features of darkness, brightness, and mixed states. The bell-shaped soliton, or solitary soliton solution, and the rational lump solution were found. To enhance the understanding of our findings, we incorporated numerical simulations in both two- and three-dimensional representations, effectively illustrating the physical implications of our results. The modified Riccati sub-equation function method solutions extended prior results by providing explicit closed-form expressions and a systematic analysis of the role of the dispersion parameter $ \alpha_4 $, which had not been previously quantified for this model. Additionally, we confirmed that all solutions are valid; they were substituted back into their respective partial differential equation and satisfied the required conditions.
Citation: Longjun Zhan, Adnan Ahmad Mahmud, Haci Mehmet Baskonus. On the soliton sensitivity, lump and analytic solutions of the generalized (3+1)-dimensional P-type nonlinear wave equation[J]. AIMS Mathematics, 2026, 11(8): 25652-25671. doi: 10.3934/math.20261028

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Abstract

In this manuscript, we investigated a generalized (3+1)-dimensional P-type nonlinear wave equation. First, for a more detailed analysis of the studied model, we transformed it into a nonlinear ordinary differential equation by utilizing an appropriate wave transformation formula. We utilized a modified Riccati equation method (also called the generalized Riccati equation method), signifying its first application for this model. The extraction of traveling waves and singular solitons used exponential and hyperbolic trigonometric functions, which exhibited features of darkness, brightness, and mixed states. The bell-shaped soliton, or solitary soliton solution, and the rational lump solution were found. To enhance the understanding of our findings, we incorporated numerical simulations in both two- and three-dimensional representations, effectively illustrating the physical implications of our results. The modified Riccati sub-equation function method solutions extended prior results by providing explicit closed-form expressions and a systematic analysis of the role of the dispersion parameter $ \alpha_4 $, which had not been previously quantified for this model. Additionally, we confirmed that all solutions are valid; they were substituted back into their respective partial differential equation and satisfied the required conditions.

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