Dimensional Kadomtsev-Petviashvili Equation Research Articles

Integrated Jacobi elliptic function solutions to the [formula omitted]-dimensional generalized Kadomtsev–Petviashvili equation by utilizing new solutions of the elliptic equation of order six

In this research, the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation ((3+1)-GKPE) that expresses various nonlinear phenomena was studied. An extended Jacobi elliptic function expansion method (JEFEM) was developed by considering new solutions for the Jacobi elliptic equation of order six. Then the extended method was used to (3+1)-GKPE, where new exact Jacobi elliptic function solutions were obtained. This equation is of particular interest as it required a special transformation in order to apply the JEFEM. Moreover, some of the solutions are shown graphically.

Soliton dynamics in (2+1) dimensional Heisenberg spin chain with Dzyaloshinskii–Moriya interaction in nanowire systems

In this study, we delve into the theoretical framework of the (2+1)-dimensional Kadomtsev–Petviashvili equation coupled with Dzyaloshinskii–Moriya (DM) interaction, elucidated from the Landau-Lifshitz equation, a prominent model in the realm of ferromagnetic dynamics. Employing the binary Bell polynomial technique along with the Hirota bilinear form, we construct one- and two-soliton solutions for the considered physical system. Through the manipulation of parameters present in these solutions, we explore the various nonlinear dynamical phenomena characterizing ferromagnetic nano-wires with DM interaction. Our investigations elucidate that solitons can be effectively manipulated in the nanowire system through judicious selection of system parameters. Moreover, our findings illustrate that soliton profiles exhibit compression, amplification, shifting, and changes in orientation during evolution. These insights hold significant implications for experimentalists aiming to analyze the propagation of solitons in ferromagnetic nanowire systems.

Properties of the hybrid solutions for a generalized (3 + 1)-dimensional KP equation

The Kadomtsev-Petviashvili (KP) equation is instrumental in characterizing nonlinear wave phenomena in fluid dynamics. With the incorporation of three-dimensional disturbances, the (3+1)-dimensional KP equation emerges. In this study, we employ the Hirota bilinear method to investigate a (3+1)-dimensional KP equation. We successfully derive both lump and lump-chain solutions for this equation. The characteristics of the resulting lump are thoroughly examined. It is observed that those lumps propagate along a specific ray direction, with the origin of these rays dependent on the imaginary part of σˆjk=σj−σk with 1≤j,k≤3 and j≠k. We have simultaneously obtained the propagation direction of these rays. The periodicity of those lumps is denoted as Tjk dependent on spectrum parameter. Furthermore, we explore two potential lines, including the characteristic line associated with the lumps, to gain a deeper understanding of the nonlinear waves controlled by this system.

Complexiton, complex multiple kink soliton and the rational wave solutions to the generalized (3 + 1)-dimensional kadomtsev-petviashvili equation

Some new exact solutions of the generalized (3 + 1)-dimensional Kadomtsev-Petviashvili equation (KPE) are explored in this study. Firstly, the resonant multiple soltion solutions (RMSs) are discussed via employing the linear superposition principle and weight algorithm. Then, by introducing pairs of the conjugate parameters to the RMSs, the complexiton solutions including the non-singular complexiton and singular complexiton solutions are extracted. In addition, the complex multiple kink soliton solutions are also probed by employing the bilinear approach. Finally, we investigate the rational wave solutions via the test function method and symbolic computation. By choosing the appropriate parameters, the graph descriptions of the derived solutions are presented to show the dynamical properties. The outcomes of this work are desirous to bring some new perspective to the study of the complexiton, complex solutions and rational wave solutions to the other PDEs.

Generalized extended (2+1)-dimensional Kadomtsev-Petviashvili equation in fluid dynamics: analytical solutions, sensitivity and stability analysis

This study discusses a new version of the (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-dimensional Kadomtsev-Petviashvili equation. This equation is used to model the behavior of nonlinear waves in various fields like ferromagnetic media, ion-acoustic waves in plasma physics, and fluid dynamics. It is instrumental in modeling surface and internal waves in straits or channels. The main goal of the research is to determine the exact solutions for this equation and analyze their physical characteristics. We obtain exact solutions using two improved techniques, namely the modified extended tanh-function and the modified generalized Kudryashov methods. These techniques investigate various exact solutions, such as exponential, rational, hyperbolic, and trigonometric. Besides, the sensitivity and the stability analysis of the model are presented. Additionally, three- and two-dimensional contour plots are created to present the physical behavior of the exact solutions.

Decay mode ripple waves within the (3+1)‐dimensional Kadomtsev–Petviashvili equation

AbstractIn this paper, a series of ripple waves with decay modes for the ‐dimensions Kadomtsev–Petviashvili equation, always viewed as nonintegrable KP equation, are investigated. The decay mode wave solutions including ripplon, lump ripples, and lump chain ripples are described by Airy function, and all solutions are constructed from the Gram determinant form. Their propagation dynamics behavior is studied, and a comprehensive analysis of the asymptotic properties of these solutions has been diligently conducted.

Dynamics of heterotypic soliton, high-order breather, M-lump wave, and multi-wave interaction solutions for a ($$3+1$$)-dimensional Kadomtsev–Petviashvili equation

Dynamics of heterotypic soliton, high-order breather, M-lump wave, and multi-wave interaction solutions for a ($$3+1$$)-dimensional Kadomtsev–Petviashvili equation

Bilinear Bäcklund Transformation, Fission/Fusion and Periodic Waves of a (3+1)-dimensional Kadomtsev-Petviashvili Equation for the Shallow Water Waves

Bilinear Bäcklund Transformation, Fission/Fusion and Periodic Waves of a (3+1)-dimensional Kadomtsev-Petviashvili Equation for the Shallow Water Waves

Multiple soliton, soliton molecules and the other diverse wave solutions to the (2+1)-dimensional Kadomtsev–Petviashvili equation

The central purpose of this paper is to explore the nonlinear dynamics of the (2+1)-dimensional Kadomtsev–Petviashvili equation (KPE). The multiple soliton solutions (MSSs) are constructed via applying the Hirota method. Then the soliton molecules on the ([Formula: see text]-, ([Formula: see text]- and ([Formula: see text]-planes are extracted via imposing the velocity resonance conditions to the MSSs. Eventually, two effective techniques, the sub-equation approach (SEA) and the variational approach (VA), are employed to probe some other diverse wave solutions, which are the bright wave, dark wave, singular wave and the singular periodic wave solutions. The dynamics of the extracted solutions are unveiled graphically to exhibit the physical attributes. The attained solutions in this paper can enlarge the exact solutions of the (2+1)-dimensional KPE and enable us to understand the nonlinear dynamic behaviors better.

Breather waves, periodic cross-lump waves and complexiton type solutions for the (2 + 1)-dimensional Kadomtsev-Petviashvili equation in dispersive media

The manuscript under consideration explores the (2+1)-dimensional Kadomtsev-Petviashvili (KP) model using its bilinear representation. The focus is on studying the characteristics of periodic cross-lump waves and breather wave solutions. Breathers, which are localized periodic solutions of continuous media equations or discrete lattice equations, are of particular interest. The research also delves into complexiton solutions using the extended transformed rational function approach. The study graphically analyzes the dynamics of these interaction solutions using appropriate parameters. It's truly fascinating to learn about the application of breathers in continuous media equations or discrete lattice equations.

Exact solutions of the time-fractional extended (3+1)-dimensional Kadomtsev–Petviashvili equation

Exact solutions of the time-fractional extended (3+1)-dimensional Kadomtsev–Petviashvili equation

Interactions of certain localized waves for an extended (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid mechanics

Currently, there is much interest in the study on the localized waves and their interactions in fluid mechanics. An extended (3+1)-dimensional Kadomtsev-Petviashvili equation is considered here. By means of the Hirota method, we obtain the N-soliton solutions, with N as a positive integer. The higher-order breather solutions are obtained from the N-soliton solutions through the complex conjugated transformations. Making use of the long-wave limit method, we determine the higher-order lump solutions via the N-soliton solutions. Besides, some hybrid solutions are presented. Three kinds of the localized waves, namely, the solitons, breathers and lumps, along with their interactions, are investigated via the above solutions. Amplitudes, shapes and velocities of those localized waves remain invariant after the interactions, which indicate that the interactions are elastic. Fluid-mechanically meaningful, all our results rely on the coefficients in that equation.

An Investigation for Soliton Solutions of the Extended (2+1)-Dimensional Kadomtsev–Petviashvili Equation

This article presents an investigation for soliton solutions of the extended (2+1)-dimensional Kadomtsev–Petviashvili equation which describes wave behavior in shallow water. We utilize the unified Riccati equation expansion method. By employing the powerful method, many soliton solutions are successfully derived, and it is verified by Wolfram Mathematica that the solutions satisfy the main equation. Additionally, Matlab is utilized to generate plots and examine the properties of the obtained solitons. The results reveal that the considered equation exhibits a wide range of soliton solutions, including dark, bright, singular, and periodic solutions. This comprehensive investigation of soliton solutions for the Kadomtsev–Petviashvili equation holds significant relevance in various fields such as oceanography and nonlinear optics, contributing to practical applications.

Higher Dimensional Kadomtsev–Petviashvili Equation: New Collision Phenomena

Higher Dimensional Kadomtsev–Petviashvili Equation: New Collision Phenomena

Analytical and numerical simulation of the conformable new extended (2 + 1) dimensional Kadomtsev–Petviashvili equation

Partial differential equations are used to model problems in many scientific fields. To solve these equations, analytical and numerical techniques are created. This study considers the conformable version of the expanded Kadomtsev–Petviashvili equation in -dimensions for the first time. New exact and approximative solutions to the equation that do not exist in the literature are obtained using the analytical techniques of the -expansion and modified Kudryashov methods, as well as the numerical technique residual power series method. To better understand the dynamic nature of these findings, we have given 3D, contour, and 2D plots of some of the solutions. The development of numerous new exact solutions has demonstrated the reliability and effectiveness of the methods. We also provide an approximation table of values for the approximate solutions of the given equation using different values for various parameters.

Traveling Wave Solutions and Bifurcations of a New Generalized (3 + 1)-Dimensional Kadomtsev–Petviashvili Equation

Traveling Wave Solutions and Bifurcations of a New Generalized (3 + 1)-Dimensional Kadomtsev–Petviashvili Equation

Symmetry and complexity: A Lie symmetry approach to bifurcation, chaos, stability and travelling wave solutions of the (3+1)-dimensional Kadomtsev-Petviashvili equation

This work delves into the investigation of the nonlinear dynamics pertaining to the (3+1)-dimensional Kadomtsev-Petviashvili equation, which describes the propagation of long-wave with dissipation and dispersion in nonlinear media. The research entails an exploration of symmetry reductions using Lie group analysis, an analysis of the dynamical system’s characteristics through bifurcation phase portraits, and a study of the perturbed dynamical system’s dynamic behavior through chaos theory. Chaotic behavior is identified using various tools for detecting chaos, including the Lyapunov exponent, 3D phase portrait, Poincare map, time series analysis, and an exploration of the presence of multistability in the autonomous system under different initial conditions. Additionally, the research applies the unified Riccati equation expansion method to solve the considered equation analytically and constructs the general solutions of solitary wave solutions such as trigonometric function solutions, periodic and singular soliton solutions. These solutions come with their associated constraint conditions and are demonstrated through visual representations in the form of 2D, 3D, and density plots with carefully selected parameters. Furthermore, the stability analysis of the considered equation is also discussed and shown graphically. The results of this work are relevant and have applications in describing the propagation of long-wave with dissipation and dispersion in nonlinear media.

A Generalized (3+1)-Dimensional Nonlinear Wave Equation in Liquid with Gas Bubbles: Symmetry Reductions; Exact Solutions; Conservation Laws

The analysis of a generalised (3+1)-dimensional nonlinear wave equation that simulates a variety of nonlinear processes that occur in liquids including gas bubbles will be performed. After some cosmetic adjustments to the underlying equation, this generalised (3+1)-dimensional nonlinear wave equation naturally degenerates into the (3+1)-dimensional Kadomtsev-Petviashvili equation, the (3+1)-dimensional nonlinear wave equation, and the Korteweg-de Vries equation. To completely investigate this fundamental equation, a clear and rigorous technique is used. In order to obtain innovative symmetry reductions, group invariant solutions, conservation laws, and eventually kink wave solutions, the Lie symmetry, multiplier, and simplest equation methods are used. Complex waves and their dealing dynamics in fluids can be well imitated by the verdicts.

Analytical solutions of the extended Kadomtsev–Petviashvili equation in nonlinear media

Abstract This manuscript attempts to construct diverse exact traveling wave solutions for an important model called the (3+1)-dimensional Kadomtsev–Petviashvili equation. In order to achieve that, the Jacobi elliptic function technique and the Kudryashov technique are chosen in favor of their noticeable efficacy in dealing with nonlinear dynamical models. As expected, the used approaches lead to a variety of traveling wave solutions of different types. Finally, we have graphically illustrated some of the obtained wave solutions to further make sense of their representation. Also, we provide an overview of the main results at the end.

(3+1)-dimensional cylindrical dust ion-acoustic solitary waves in dusty plasma

The basic properties of (3+1)-dimensional cylindrical dust ion-acoustic (DIA) solitary waves have been studied in a multi-component plasma medium consisting of inertial ions, Maxwellian electrons, and stationary negative dust particles. The reductive perturbation method has been used to derive a cylindrical (3+1)-dimensional Kadomtsev–Petviashvili equation for small but finite amplitude of dust ion-acoustic waves. The impacts of nonplanar geometry and presence of static negative dust particles on the dust ion-acoustic solitary structures have been investigated. DIA solitary waves with both positive and negative polarity is observed to generate for this existing multi-component and complex plasma medium. The changes in the DIA wave structure in radial, azimuthal, and axial coordinates over time have been presented analytically. The results of this present investigation may contribute to a better understanding of laboratory plasma events as well as nonlinear electrostatic excitations in astrophysical environments.