Petviashvili Equation Research Articles

Controllable transformed waves of a (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids or plasma

In this paper, we study the modulations of nonlinear transformed waves for a (3 + 1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluids or plasma. By virtue of the phase shift analysis, the shape-changed and unchanged transformed waves are investigated, which shows the inhomogeneity can restrain the time-varying property. The deformation of waves is determined by the phase difference between two wave components. In addition, the evolutions of parabolic transformed waves are illustrated via characteristic lines analysis. The interactions are further explored, which involve the long- and short-lived collisions. In particular, we discuss the dynamics of unidirectional and reciprocating molecular waves based on the velocity resonance condition, including the shape-changed and unchanged atoms. Different from previous results, certain new types of transformed molecular waves with shape-unchanged atoms are discovered. Our results indicate that the inhomogeneity can produce novel transformed waves and further facilitate the modulation of phase transition mechanism.

Some new lump molecules and hybrid molecular states of a (3 + 1)-dimensional generalized variable coefficient Kadomtsev–Petviashvili equation

In this paper, a (3 + 1)-dimensional generalized variable coefficient Kadomtsev–Petviashvili equation is investigated systematically, which can characterize evolution of the long water waves and small amplitude surface waves with the weak nonlinearity, weak dispersion, and weak perturbation in fluid mechanics. We investigate one lump and lump molecules obtained from one breather and breather molecules by a new degenerating breather method, respectively. In addition, the bound state of lump molecules and other localized waves is derived theoretically by velocity resonance. Considering the condition of variable coefficient, the several sets of interesting solutions having a complex structure are obtained, which include the type of parabolic, S-shaped, and periodic. The analysis method can also help us to study lump molecules existing in other integrable systems from a new perspective.

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

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.

Bifurcations, chaotic behavior, sensitivity analysis and soliton solutions of the extended Kadometsev–Petviashvili equation

Bifurcations, chaotic behavior, sensitivity analysis and soliton solutions of the extended Kadometsev–Petviashvili equation

On the Kadomtsev–Petviashvili equation with double-power nonlinearities

In this paper, we study the generalized KP equation with double-power nonlinearities. Our investigation covers various aspects, including the existence of solitary waves, their nonlinear stability, and instability. Notably, we address a broader class of nonlinearities represented by μ1|u|p1−1u+μ2|u|p2−1u, with p1>p2, encompassing cases where μ1>0 and μ1<0<μ2. One of the distinct features of our work is the absence of scaling, which introduces several challenges in establishing the existence of ground states. To overcome these challenges, we employ two different minimization problems, offering novel approaches to address this issue. Furthermore, our study includes a nuanced analysis to ascertain the stability of these ground states. Intriguingly, we extend our stability analysis to encompass cases where the convexity of the Lyapunov function is not guaranteed. This expansion of stability criteria represents a significant contribution to the field. Moving beyond the analysis of solitary waves, we shift our focus to the associated Cauchy problem. Here, we derive criteria that determine whether solutions exhibit finite-time blow-up or remain uniformly bounded within the energy space. Remarkably, our study unveils a notable gap in the existing literature, characterized by the absence of both theoretical evidence of blow-up and uniform boundedness. To explore this intriguing scenario, we employ the integrating factor method, providing a numerical investigation of solution behavior. This method distinguishes itself by offering spectral-order accuracy in space and fourth-order accuracy in time. Lastly, we rigorously establish the strong instability of the ground states, adding another layer of understanding to the complex dynamics inherent in the generalized KP equation.

Two-dimensional reductions of the Whitham modulation system for the Kadomtsev–Petviashvili equation

Two-dimensional reductions of the Kadomtsev–Petviashvili(KP)–Whitham system, namely the overdetermined Whitham modulation system for five dependent variables that describe the periodic solutions of the KP equation, are studied and characterized. Three different reductions are considered corresponding to modulations that are independent of x, independent of y, and of t (i.e. stationary), respectively. Each of these reductions still describes dynamic, two-dimensional spatial configurations since the modulated cnoidal wave, generically, has a nonzero speed and a nonzero slope in the xy plane. In all three of these reductions, the integrability of the resulting systems of equations is proven, and various other properties are elucidated. Compatibility with conservation of waves yields a reduction in the number of dependent variables to two, three and four, respectively. As a byproduct of the stationary case, the Whitham modulation system for the classical Boussinesq equation is explicitly obtained.

Innovative Solutions for the Kadomtsev–Petviashvili Equation via the New Iterative Method

This research paper presents a new iterative method (NIM) for obtaining the solution to the potential Kadomtsev–Petviashvili (PKP) equation. NIM is a promising approach to solving complex mathematical problems, and its effectiveness and efficiency are highlighted through its application to the PKP equation. The results obtained through the use of NIM are compared to the exact solutions of the PKP equation, and it is found that the NIM approach provides results that are in close agreement with the exact solutions. This demonstrates the utility and accuracy of NIM and makes it a valuable tool for solving similar mathematical problems in the future. Furthermore, the lack of discretization in the NIM approach makes it a more convenient method for solving the PKP equation compared to traditional approaches that require discretization. Overall, the findings of this research paper suggest that NIM is a highly effective and convenient method for obtaining approximate analytical solutions to complex mathematical problems, such as the 2+1-dimensional PKP equation.

Soliton Solutions and Other Solutions to the (4+1)-Dimensional Davey–Stewartson–Kadomtsev–Petviashvili Equation using Modified Extended Mapping Method

Soliton Solutions and Other Solutions to the (4+1)-Dimensional Davey–Stewartson–Kadomtsev–Petviashvili Equation using Modified Extended Mapping Method

Lump solutions of the fractional Kadomtsev–Petviashvili equation

Of concern is the fractional Kadomtsev–Petviashvili (fKP) equation and its lump solution. As in the classical Kadomtsev–Petviashvili equation, the fKP equation comes in two versions: fKP-I (strong surface tension case) and fKP-II (weak surface tension case). We prove the existence of nontrivial lump solutions for the fKP-I equation in the energy subcritical case α>45\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >\\frac{4}{5}$$\\end{document} by means of variational methods. It is already known that there exist neither nontrivial lump solutions belonging to the energy space for the fKP-II equation [9] nor for the fKP-I when α≤45\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\le \\frac{4}{5}$$\\end{document} [26]. Furthermore, we show that for any α>45\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha >\\frac{4}{5}$$\\end{document} lump solutions for the fKP-I equation are smooth and decay quadratically at infinity. Numerical experiments are performed for the existence of lump solutions and their decay. Moreover, numerically, we observe cross-sectional symmetry of lump solutions for the fKP-I equation.

New dynamical analysis of the exact traveling wave solutions to a (3+1)-dimensional Gardner-KP equation by three efficient architecture

In this research article, the (3 + 1) dimensional nonlinear Gardner- Kadomtsov–Petviashvili (Gardner-KP) equation, demonstrating the nonlinear modulation of periodic waves, has been analyzed using G′G, G′G2, and the generalized improved G′G methods and a large number of traveling wave solutions have been generated. These solutions exhibit structures with physical properties of hyperbolic, trigonometric, and rational solution types. Numerical simulations were performed for the obtained solutions at different values of the parameters. The stability property of the obtained solutions was tested to demonstrate their reliability. Our novelty in this study lies in obtaining distinct and enriched solutions compared to existing literature, conducting a comparative analysis of three methods, and investigating the stability property of solutions for the Gardner-KP equation, which has not been explored in the existing literature.

Solitons and lump waves to the elliptic cylindrical Kadomtsev–Petviashvili equation

We construct various kinds of solutions to the elliptic cylindrical Kadomtsev–Petviashvili equation, which can be derived in the context of surface gravity waves. This investigation yields a diverse set of analytical outcomes, encompassing the dynamics of horseshoe-like solitons, lump chains, and lump waves. Based on the Hirota bilinear method, the lump chain is derived under specified conjugate constraint conditions. Additionally, we uncover three distinct types of novel interactions among lump waves through the utilization of the long wave limit method. A detailed analysis is conducted to elucidate the features and patterns inherent in these lump waves. The analytic findings presented may providing a descriptive framework for generic weakly nonlinear and weakly dispersive waves.

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.

Decompositions of the Kadomtsev–Petviashvili equation and their symmetry reductions

Starting with a decomposition conjecture, we carefully explain the basic decompositions for the Kadomtsev–Petviashvili (KP) equation as well as the necessary calculation procedures, and it is shown that the KP equation allows the Burgers–STO (BSTO) decomposition, two types of reducible coupled BSTO decompositions and the BSTO–KdV decomposition. Furthermore, we concentrate ourselves on pointing out the main idea and result of Bäcklund transformation of the KP equation based on a special superposition principle in the particular context of the BSTO decompositions. Using the framework of standard Lie point symmetry theory, these decompositions are studied and the problem of computing the corresponding symmetry constraints is treated.

EXACT SOLUTIONS AND BIFURCATION OF A MODIFIED GENERALIZED MULTIDIMENSIONAL FRACTIONAL KADOMTSEV–PETVIASHVILI EQUATION

In this paper, we investigate the exact solutions of a modified generalized multidimensional fractional Kadomtsev–Petviashvili (KP) equation by the bifurcation method. First, the equation is converted into a planar dynamical system through fractional complex wave transformation. The phase portraits of the equation and qualitative analysis are presented under different bifurcation conditions. Then, the bounded and unbounded traveling wave solutions, including periodic, kink, anti-kink, dark-solitary, bright-solitary and breaking wave solutions, are acquired by integrating along different orbits. Finally, numerical simulations of the dynamic behaviors of the solutions obtained are graphically illustrated by choosing appropriate parameters.

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.

New type solutions to the (2+1)-dimensional extended Bogoyavlenskii–Kadomtsev–Petviashvili equation calculated via generalized Kudryashov technique

New type solutions to the (2+1)-dimensional extended Bogoyavlenskii–Kadomtsev–Petviashvili equation calculated via generalized Kudryashov technique

Analyzing Soliton Solutions of the [formula omitted]-dimensional generalized Kadomtsev–Petviashvili equation: Comprehensive study of dark, bright, and periodic dynamics

This paper investigates the (n+1) dimensional integrable extension of the Kadomtsev–Petviashvili (KP) equation. The study focuses on extracting Auto-Bäcklund transformations for the given model using the extended homogeneous balance (HB) method in conjunction with Maple. These transformations are then employed to obtain explicit analytic solutions for the equation. Additionally, a Bilinear Bäcklund transformation is constructed based on the Hirota bilinear form, leading to the acquisition of exponential function solutions. Furthermore, complexiton solutions for the KP equation are obtained utilizing the bilinear form and the extended transformed rational function technique. The physical properties of the obtained solutions are explored through visualization techniques, including 3D, 2D, and contour plots. The analysis reveals the presence of dark, bright, and periodic solitons. The obtained solutions of the KP equation have practical applications in nonlinear optics, plasma physics, and wave propagation analysis.

The partial-rogue ripple solutions of nonlocal Kadomtsev–Petviashvili equation

The derivation of new exact solutions for integrable systems has always been an open problem in mathematical physics and engineering. In this paper, a large class of partial-rogue ripple solutions of a nonlocal Kadomtsev–Petviashvili [KP] equation are derived, which means that these ripple waves are semi-localized in time. Firstly, the Wronskian and Grammian determinant expression of the nonlocal KP equation are constructed. Then, three kinds of partial-rogue ripplon solutions of nonlocal KP equations, including (i) partial-rogue ripple soliton solutions, (ii) partial-rogue ripple lump chain solutions, and (iii) partial-rogue ripple lump solutions, are proposed. These solutions approach to the zero background plane as t→∞, which is the characteristic of partial-rogue ripple waves. These solutions are rarely reported, and their dynamics have been analytically and numerically discussed. The results of this paper are more practical and closely match the attenuated ripple waves stirred up in water waves. Additionally, the research method in this paper provides new references for existing results.

Nonlinear localized waves and their interactions for a (2+1)-dimensional extended Bogoyavlenskii-Kadomtsev–Petviashvili equation in a fluid

In fluid dynamics, a (2+1)-dimensional extended Bogoyavlenskii-Kadomtsev-Petvi-ashvili equation is hereby investigated. Bilinear form and N-soliton solutions are determined with the Hirota method, where N is a positive integer. N-soliton solutions are used to build the higher-order breather and lump solutions with the complex parameter relation and long-wave-limit method. Elastic and inelastic interactions between the two breathers and elastic interaction between the two lumps are constructed and graphically depicted. We find that the coefficients of the weak dispersion terms have an effect on the velocities and amplitudes of the breathers and lumps. Based on the hybrid solutions containing the breathers, lumps and solitons, several interactions of those nonlinear localized waves are graphically illustrated, including the fission of the breather and soliton, elastic or inelastic interaction between the one breather and one soliton, elastic interaction among one breather and two solitons, as well as elastic interaction between the one lump and one breather.