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18 pages, 1717 KiB

Open AccessArticle

Symmetries, Conservation Laws, and Exact Solutions of a Potential Kadomtsev–Petviashvili Equation with Power-Law Nonlinearity

by Dimpho Millicent Mothibi

Symmetry 2025, 17(7), 1053; https://doi.org/10.3390/sym17071053 - 3 Jul 2025

Viewed by 258

Abstract

This study investigates the potential Kadomtsev–Petviashvili equation incorporating a power-type nonlinearity (PKPp), a model that features prominently in various nonlinear phenomena encountered in physics and applied mathematics. A complete Noether symmetry classification of the PKPp equation is conducted, revealing four distinct scenarios based [...] Read more.

p

, namely, the general case where

p \neq 1, - 1, - 2,

and three special cases where

p = 1,

p = - 1,

and

p = - 2 .

Corresponding to each case, conservation laws are derived through a second-order Lagrangian framework. Furthermore, Lie group analysis is employed to reduce the nonlinear partial differential Equation (NLPDE) to ordinary differential Equations (ODEs), thereby enabling the effective application of the Kudryashov method and direct integration techniques to construct exact solutions. In particular, exact solutions of of the considered nonlinear partial differential equation are obtained for the cases

p = 1

and

p = 2,

illustrating the practical implementation of the proposed approach. The solutions obtained include solitary wave, periodic, and rational-type solutions. These results enhance the analytical understanding of the PKPp equation and contribute to the broader theory of nonlinear dispersive equations. Full article

(This article belongs to the Special Issue Symmetries in Differential Equations and Application—2nd Edition)

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18 pages, 750 KiB

Open AccessArticle

Analysis of the(3+1)-Dimensional Fractional Kadomtsev–Petviashvili–Boussinesq Equation: Solitary, Bright, Singular, and Dark Solitons

by Aly R. Seadway, Asghar Ali, Ahmet Bekir and Adem C. Cevikel

Fractal Fract. 2024, 8(9), 515; https://doi.org/10.3390/fractalfract8090515 - 30 Aug 2024

Cited by 3 | Viewed by 1286

Abstract

We looked at the (3+1)-dimensional fractional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which comes up in fluid dynamics, plasma physics, physics, and superfluids, as well as when connecting the optical model and hydrodynamic domains. Furthermore, unlike the Kadomtsev–Petviashvili equation (KPE), which permits the modeling of waves traveling in both directions, the zero-mass assumption, which is required for many scientific applications, is not required by the KP-B equation. In several applications in engineering and physics, taking these features into account allows researchers to acquire more precise conclusions, particularly in studies pertaining to the dynamics of water waves. The foremost purpose of this manuscript is to establish diverse solutions in the form of exponential, trigonometric, hyperbolic, and rational functions of the (3+1)-dimensional fractional (KP-B) via the application of four analytical methods. This KP-B model has fruitful applications in fluid dynamics and plasma physics. Additionally, in order to better explain the potential and physical behavior of the equation, the relevant models of the findings are visually indicated, and 2-dimensional (2D) and 3-dimensional (3D) graphics are drawn. Full article

(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)

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13 pages, 555 KiB

Open AccessArticle

Simulation of a Combined (2+1)-Dimensional Potential Kadomtsev–Petviashvili Equation via Two Different Methods

by Muath Awadalla, Arzu Akbulut and Jihan Alahmadi

Mathematics 2024, 12(3), 427; https://doi.org/10.3390/math12030427 - 29 Jan 2024

Viewed by 1092

Abstract

This paper presents an investigation into original analytical solutions of the (2+1)-dimensional combined potential Kadomtsev–Petviashvili and B-type Kadomtsev–Petviashvili equations. For this purpose, the generalized Kudryashov technique (GKT) and exponential rational function technique (ERFT) have been applied to deal with the equation. These two methods have been applied to the model for the first time, and the the generalized Kudryashov method has an important place in the literature. The characteristics of solitons are unveiled through the use of three-dimensional, two-dimensional, contour, and density plots. Furthermore, we conducted a stability analysis on the acquired results. The results obtained in the article were seen to be different compared to other results in the literature and have not been published anywhere before. Full article

(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations)

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18 pages, 1687 KiB

Open AccessArticle

Lie Symmetries, Closed-Form Solutions, and Various Dynamical Profiles of Solitons for the Variable Coefficient (2+1)-Dimensional KP Equations

by Sachin Kumar, Shubham K. Dhiman, Dumitru Baleanu, Mohamed S. Osman and Abdul-Majid Wazwaz

Symmetry 2022, 14(3), 597; https://doi.org/10.3390/sym14030597 - 17 Mar 2022

Cited by 93 | Viewed by 2785

Abstract

This investigation focuses on two novel Kadomtsev–Petviashvili (KP) equations with time-dependent variable coefficients that describe the nonlinear wave propagation of small-amplitude surface waves in narrow channels or large straits with slowly varying width and depth and non-vanishing vorticity. These two variable coefficients, Kadomtsev–Petviashvili (VCKP) equations in (2+1)-dimensions, are the main extensions of the KP equation. Applying the Lie symmetry technique, we carry out infinitesimal generators, potential vector fields, and various similarity reductions of the considered VCKP equations. These VCKP equations are converted into nonlinear ODEs via two similarity reductions. The closed-form analytic solutions are achieved, including in the shape of distinct complex wave structures of solitons, dark and bright soliton shapes, double W-shaped soliton shapes, multi-peakon shapes, curved-shaped multi-wave solitons, and novel solitary wave solitons. All the obtained solutions are verified and validated by using back substitution to the original equation through Wolfram Mathematica. We analyze the dynamical behaviors of these obtained solutions with some three-dimensional graphics via numerical simulation. The obtained variable coefficient solutions are more relevant and useful for understanding the dynamical structures of nonlinear KP equations and shallow water wave models. Full article

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14 pages, 648 KiB

Open AccessFeature PaperArticle

First Integral Technique for Finding Exact Solutions of Higher Dimensional Mathematical Physics Models

by Shumaila Javeed, Sidra Riaz, Khurram Saleem Alimgeer, M. Atif, Atif Hanif and Dumitru Baleanu

Symmetry 2019, 11(6), 783; https://doi.org/10.3390/sym11060783 - 12 Jun 2019

Cited by 22 | Viewed by 3962

Abstract

In this work, we establish the exact solutions of some mathematical physics models. The first integral method (FIM) is extended to find the explicit exact solutions of high-dimensional nonlinear partial differential equations (PDEs). The considered models are: the space-time modified regularized long wave (mRLW) equation, the (1+2) dimensional space-time potential Kadomtsev Petviashvili (pKP) equation and the (1+2) dimensional space-time coupled dispersive long wave (DLW) system. FIM is a powerful mathematical tool that can be used to obtain the exact solutions of many non-linear PDEs. Full article

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