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This work intents to obtain symmetry reductions and exact solutions of Kadomtsev–Petviashvili (KP) equation, which describes propagation of long waves on the surface of shallow water. The Lie symmetry method under one-parameter transformation is used to ensure invariance and derive infinitesimal generators. These generators provide similarity variables, which is directed to symmetry reductions of the test equation. This process of reductions recasts test equation into ordinary differential equations (ODEs). These ODEs have finally been solved under various constraints and as a result, exact solutions consisting of arbitrary functions and several arbitrary constants are produced. The solutions are novel and have not yet been published. Due to the existence of arbitrary functions f 1 (t) , f 2 (t) , f 3 (t) and constants, these solutions present a more generalised form than the existing results and give a wide range of possibilities for the interpretation of various physical phenomena. The physical importance of these solutions is demonstrated by numerical simulation revealing a rich variety of soliton structures including line soliton, doubly soliton, multisoliton, solitons on parabolic surface, soliton fission and annihilation behaviour. [ABSTRACT FROM AUTHOR]

In this study, we deal with the (n + 1)-dimensional generalized Kadomtsev–Petviashvili equation (gKPE). This equation is an extension of the classical KP equation to higher dimensions, allowing for the study of nonlinear wave propagation in (n + 1) dimensions. Like the original KP equation, the (n + 1)-dimensional version can support soliton solutions, which are stable waveforms that retain their shape while traveling. The (n + 1)-dimensional gKPE has several applications in different physical systems, such as optical fibers, Bose–Einstein condensates, fluid physics, and plasma physics. We use two efficient methods to search for analytical solutions to the equation we consider. One of these methods is the generalized unified method and the improved F-expansion method. Using these techniques, we obtain many soliton solutions of rational, hyperbolic, and trigonometric types. We present 3D, contour, and density plots to observe the behavior of some of the solutions we obtained. The acquired results constitute an essential resource for the study of hydrodynamic waves, plasma fluctuations, and optical solitons and offer useful information for understanding the behavior of the KPE under different physical situations. The methodologies used in this study are robust, influential, and practicable for diverse nonlinear partial differential equations; to our knowledge, these methods of investigation have not been explored before for this equation. The accuracy of each solution has been verified using the Maple software program. [ABSTRACT FROM AUTHOR]

We give some of our results over the past few years about rogue waves concerning some partial differential equations, such as the focusing nonlinear Schrödinger equation (NLS), the Kadomtsev–Petviashvili equation (KPI), the Lakshmanan–Porsezian–Daniel equation (LPD) and the Hirota equation (H). For the NLS and KP equations, we give different types of representations of the solutions, in terms of Fredholm determinants, Wronskians and degenerate determinants of order 2 N . These solutions are called solutions of order N. In the case of the NLS equation, the solutions, explicitly constructed, appear as deformations of the Peregrine breathers P N as the last one can be obtained when all parameters are equal to zero. At order N, these solutions are the product of a ratio of two polynomials of degree N (N + 1) in x and t by an exponential depending on time t and depending on 2 N − 2 real parameters: they are called quasi-rational solutions. For the KPI equation, we explicitly obtain solutions at order N depending on 2 N − 2 real parameters. We present different examples of rogue waves for the LPD and Hirota equations. [ABSTRACT FROM AUTHOR]

This work explores the generalized variable-coefficient (3+1)-dimensional Kadomtsev-Petviashvili equation, which surfaces in fluid and multi-component plasma research as an important application. Plasmas and fluids are currently attracting attention because of their capacity to facilitate a wide range of wave phenomena. A diversity of wave structures, including M-lump-like waves and multiple solitons are secured. Moreover, the various hybrid solutions are analyzed and discussed with the assistance of the Hirota bilinear method and the long wave technique. The mechanical features of solutions and collision-related aspects within a diversity of nonlinear systems are demonstrated by the results of this investigation. The extracted solutions are thoroughly analyzed to determine their physical significance. This is done by presenting a range of graphs that illustrate the dynamics of the solutions for specific parametric values. The reliability of the methods we have implemented is confirmed by our findings, which also indicate their potential for future use in identifying unique and varied solutions to nonlinear evolution equations experienced in the fields of engineering and mathematical physics. [ABSTRACT FROM AUTHOR]

The Kadomtsev–Petviashvili (KP) equations are nonlinear partial differential equations which are widely used for the modeling of wave propagation in hydrodynamic and plasma systems. This study aims to make a valuable contribution to the literature by providing new solitary waves to the (2+1)- and (3+1)-dimensional potential Kadomtsev–Petviashvili (pKP)-B-type Kadomtsev–Petviashvili (BKP) equations. For this, the auxiliary equation method associated with Bernoulli equation is used and new solutions for the considered equations are obtained. The stability of obtained solutions is demonstrated using nonlinear analysis. It is shown that this method for the considered pKP–BKP equations is an important step forward in an overall mathematical framework for similar equations. [ABSTRACT FROM AUTHOR]

In this study, we utilized a novel auto-Bäcklund transformation and the extended transformed rational function approach to analyze the extended reduced Jimbo–Miwa equation, a prominent equation within the KP hierarchy. The homogenous balance technique was employed to derive the auto-Bäcklund transformation of the equation, leading to the extraction of new exact solutions exhibiting solitary patterns. Additionally, we applied the extended transformed rational function method, which relies on the Hirota bilinear form of the governing equation, to generate complexiton solutions. Furthermore, we included 3D graphics visualizing the obtained solutions. [ABSTRACT FROM AUTHOR]

In this paper, the extended (3+1)-dimensional potential KP equation in fluid mechanics is studied through Hirota bilinear method. Many types of hybrid solutions, such as the lump–kink solution, lump-two kink solution and periodic lump solution are obtained by assuming different functions in the bilinear equation. The interaction solution between lump and triangular periodic wave is also derived by combining sine and cosine functions with quadratic functions. Dynamical structures of these exact solutions are depicted by presenting the corresponding three-dimensional, two-dimensional structures and density graphs. These diverse interaction solutions could be helpful for understanding physical phenomena in fluid mechanics. [ABSTRACT FROM AUTHOR]

In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy. [ABSTRACT FROM AUTHOR]

A weak formulation is devised for the K(m,n)$K(m,n)$ equation, which is a nonlinearly dispersive generalization of the gKdV equation having compacton solutions. With this formulation, explicit weak compacton solutions are derived, including ones that do not exist as classical (strong) solutions. Similar results are obtained for a nonlinearly dispersive generalization of the gKP equation in two dimensions, which possesses line compacton solutions. [ABSTRACT FROM AUTHOR]

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 (x , y) -, (x , t) - and (y , t) -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. [ABSTRACT FROM AUTHOR]

The Cauchy matrix approach is developed for solving nonisospectral Kadomtsev–Petviashvili equation and the nonisospectral modified Kadomtsev–Petviashvili equation. By means of a Sylvester equation , a set of scalar master functions are defined. We derive the evolution of scalar functions using the nonisospectral dispersion relations. Some explicit solutions are illustrated together with the analysis of their dynamics. [ABSTRACT FROM AUTHOR]

The article is devoted to the finite-dimensional dynamics of the Kadomtsev–Petviashvili equation and their use for constructing exact solutions. Finite-dimensional dynamics of evolutionary differential equations is a natural generalization of the theory of dynamical systems to partial differential equations. The necessary information on the finite-dimensional dynamics theory is given. [ABSTRACT FROM AUTHOR]

This study is focusing on the integrable (3+1)-dimensional equation that combines the potential Kadomtsev-Petviashvili (pKP) equation with B-type Kadomtsev-Petviashvili (BKP) equation, also known as the pKP-BKP equation. The idea of combining integrable equations has the potential to produce a variety of unexpected outcomes such as resonance of solitons. This article provides a wide range of alternative exact solutions for the pKP-BKP equation in three dimensional form, including dark solitons, singular solitons, singular periodic solutions, Jacobi elliptic function (JEF) solutions, rational solutions and exponential solution. The improved modified extended (IME) tanh function method is employed to investigate these solutions. All of the obtained solutions for the investigated model are presented using the Wolfram Mathematica program. To further help in understanding the solutions' physical characteristics and dynamic structure, the article provides visual representations of some derived solutions using 2D representation in addition to the 3D graphs via symbolic computation. This article aims to use a potent strategy using a powerful scheme to derive different solutions with various structures. Additionally, the results greatly improve and enhance the literature's solutions to a combined pKP-BKP equation and allow deep understanding of the nonlinear dynamic system through different exact solutions. [ABSTRACT FROM AUTHOR]

Purpose – This paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results. Design/methodology/approach – This paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results. Findings – This paper is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results. Originality/value – This article is devoted to the generalized Kadomtsev–Petviashvili I equation. This study aims to propose a new approach for investigation for the existence of at least one global classical solution and the existence of at least two nonnegative global classical solutions. The main arguments in this paper are based on some recent theoretical results.

Starting from the classical quasi-geostrophic potential vorticity equation with equal depth two-layer fluid, the coupled cylindrical Kadomtsev–Petviashvili (KP) equations with variable coefficients for Rossby waves are studied. To be more general, the phase velocity is considered an indefinite integral about time and improves the analysis procedure. So the variable coefficients are obtained and some previous studies are reasonably explained. The cylindrical wave theory is therewith utilized to reduce the coupled cylindrical KP equations with variable coefficients, and based on the modified Hirota bilinear method, the lump solutions and interaction solutions are found. Through numerical simulations, the Rossby lump waves on both sides of the y axis move closer to the center, and their amplitude gradually decreases and tends to flatten with the generalized Rossby parameter growth. In the Rossby waves flow field, the dipole structures propagate to the east and lead to the appearance of the compress phenomenon during barotropic–baroclinic interaction. It is possibly useful for further theoretical research on atmospheric phenomena. [ABSTRACT FROM AUTHOR]

In this study, we apply the extended direct method to seek new similarity solutions and reduction equations of the (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation. Through this method, we can reduce the KP equation to a system of ordinary differential equations, and we divide the discussion into four cases via calculation. In some cases, new similarity reductions and exact solutions are obtained, such as elliptic function solutions, polynomial solutions. In other cases, similarity solutions are consistent with the present solutions. Futhermore, a similarity solution we get has more clear and concise form than that obtained by the generalized Clarkson–Kruskal direct method. Therefore, it demonstrates that our study is correct and more efficient than before which can also be applied to the other nonlinear physical models. [ABSTRACT FROM AUTHOR]

This study endeavors to examine the dynamics of the generalized Kadomtsev-Petviashvili (gKP) equation in (n + 1) dimensions. Based on the comprehensive three-wave methodology and the Hirota's bilinear technique, the gKP equation is meticulously examined. By means of symbolic computation, a number of three-wave solutions are derived. Applying the Lie symmetry approach to the governing equation enables the determination of symmetry reduction, which aids in the reduction of the dimensionality of the said equation. Using symmetry reduction, we obtain the second order differential equation. By means of applying symmetry reduction, the second order differential equation is derived. The second order differential equation undergoes Galilean transformation to obtain a system of first order differential equations. The present study presents an analysis of bifurcation and sensitivity for a given dynamical system. Additionally, when an external force impacts the underlying dynamic system, its behavior resembles quasi-periodic phenomena. The presence of quasi-periodic patterns are identified using chaos detecting tools. These findings represent a novel contribution to the studied equation and significantly advance our understanding of dynamics in nonlinear wave models. [ABSTRACT FROM AUTHOR]

In this work, we provide new optical soliton structures of the Kadomtsev-Petviashvili equation in (3 + 1)-dimensional and the Jimbo-Miwa equation in (3 + 1)-dimensional together with some intriguing new analysis, chaotic phenomena, bifurcation properties, and sensitivity analysis. Since soliton structure with three analyses is a very interesting recent topic in nonlinear dynamics, we extract different chaotic structures, bifurcation analysis together with phase portrait and sensitivity of our mentioned nonlinear partial differential equations. Applications of the Kadomtsev-Petviashvili equation are in sonic waves, magneto sonic waves, superfluid, weakly nonlinear quasi-unidirectional waves, shallow water waves with weakly nonlinear restoring forces and frequency dispersion, plasma physics, etc. Advanced intellect could benefit from studying the Jimbo-Miwa equation, which addresses specific fascinating higher-dimensional waves in marine engineering, ocean sciences, various interesting physical structures in the areas of optics, acoustic, mathematical modeling, epidemics, circuit analysis, computational neuroscience, intergalactic modeling, etc. Due to the huge applications of the mentioned equations, there is a high demand to investigate with recently developed three analyses. Making use of the recently developed advanced strategy, the adaptive, compatible, further advanced closed-form solitary wave structures are harvested to the mentioned equations in the present manuscript. All these scientifically accomplished exact soliton structures, which take the forms of rational functions and trigonometric functions could assist in our comprehension of remarkable nonlinear challenging situations. In contrast to the present outcomes, our newly formed discoveries will exhibit unique features. The outcomes that were extracted confirm that the recommended technique is meticulously planned, intuitive, and advantageous for measuring the dynamic behavior of nonlinear evolution equations within contemporary science and technology. [ABSTRACT FROM AUTHOR]

In this paper, a series of ripple waves with decay modes for the (3+1)$$ \left(3+1\right) $$‐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. [ABSTRACT FROM AUTHOR]

In this research, we employ the potent technique of Lie group analysis to derive analytical solutions for the (3+1)-extended Kadomtsev-Petviashvili (3D-EKP) equation. The systematic application of this method enables the identification of Lie point symmetries associated with the equation, leading to the derivation of an optimal system of one-dimensional subalgebras relevant to the equation. This optimal system is utilized to obtain several invariant solutions. The Lie group method is subsequently applied to the reduced governing equations derived from the given equation. We complement our findings with Mathematica simulations illustrating some of the obtained solutions. Furthermore, a direct approach is used to investigate local conservation laws. Importantly, our study addresses a gap in the exploration of the 3D-EXP equation using group theoretic methods, making our findings novel in this context. [ABSTRACT FROM AUTHOR]

In this work, various types of soliton solutions for space–time conformable version of (n+1)-dimensional generalized Kadomtsev–Petviashvili (KP) equation were constructed via the approach of modified extended tanh. The derivatives involved were defined to reflect the sense of space–time conformable derivatives. Furthermore, with the help of a fractional wave transformation, the conformable KP equation was reduced into an ODE of a polynomial nature. Mathematica software was used to obtain a system of algebraic equations and then solved. Finally, a graphical illustration for some of the obtained results were provided to show the effect various values of order of the conformable derivative, α (alpha) as well as k, (the sum of linear terms in the equation). [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce a generalized (3 + 1)- Kadomtsev-Petviashvili equation which is used to describe waves in a ferromagnetic medium. The equation's bilinear form is created and the new homoclinic test approach based on the Hirota bilinear form is used to find numerous novel precise solutions. These accurate solutions, which are depicted in the contour, two-dimensional and three-dimensional graphs, show the evolution of periodic characteristics. The modulation instability is used to investigate the stability of the obtained solutions. Additionally, the development of the fusion soliton is examined, as well as the fusion phenomenon in the traveling wave solution is described in the physical discussion. For this evolution equation, the study indicates new mechanical structures and various characteristics. The derived results back up the model that was proposed. These discoveries open up a new avenue for us to investigate the concept further. [ABSTRACT FROM AUTHOR]

This paper aims to study a generalized extended (2 + 1) -dimensional Kadomstev–Petviashvili (KP) equation. The KP equation models several physical phenomena such as shallow water waves with weakly nonlinear restoring forces. We will use a variety of wave ansatz methods so as to extract bright, singular, shock waves also referred to as dark or topological or kink soliton solutions. In addition to soliton solutions, we will also derive periodic wave solutions and other analytical solutions based on the invariance surface condition. Moreover, we will establish the multiplier method to derive low-order conservation laws. In order to have a better understanding of the results, graphical structures of the derived solutions will be discussed in detail based on some selected appropriate parametric values in 2-dimensions, 3-dimensions and contour plots. The findings can well mimic complex waves and their underlying properties in fluids. [ABSTRACT FROM AUTHOR]

In this work, we use two newly powerful and efficient techniques, the unified method and the unified Riccati equation expansion (UREE) method, to derive solitary wave solutions for the (1 + 1)-dimensional longitudinal wave equation (LWE), presenting a systematic approach to reveal the underlying dynamics. By incorporating mathematical analysis and numerical simulations, we investigate the behavior and properties of these solitary wave solutions in various fields of science. The derived solutions are entirely new, and these findings show the essential complexities of the longitudinal wave equation. These obtained solutions shed light on the fundamental mechanisms governing solitary waves in the longitudinal wave equation, contributing to a deeper understanding of nonlinear wave phenomena. We present effective visualizations of the dynamical wave structures using various graphs, such as 3D, 2D, and contour plots in the obtained solutions to improve our comprehensive understanding. Consequently, numerous types of wave profiles, including periodic wave, interaction between periodic wave and kink wave, W-shape wave, kink wave, interaction between breather wave and kink wave, traveling wave, mixed periodic, singular soliton, and some new types of soliton profiles were found. Consequently, this work presents a comprehensive study of solitary wave solutions using the unified method and UREE method for the (1 + 1)-dimensional LWE equation. The outcomes of the current study manifest that the considered methods are significant and systematic in solving nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

Nonlinear partial integro-differential equations (PIDEs) are applied to present the various practical phenomena in a multitude of sectors of modern science and engineering, especially in optic fiber, quantum electronics, modern physics, and the special field of nonlinear wave motion. Basically, our research demonstrates a way to generate a significant quantity of solutions to these types of two PIDEs. In this research, we have used an efficient mathematical tool namely the generalized G ′ / G -expansion method to acquire the closed form soliton solutions for the (2 + 1)-dimensional first integro-differential Kadomtsev–Petviashivili (KP) hierarchy equation and the (2 + 1)-dimensional second integro-differential KP hierarchy equation utilizing a code likely Mathematica. The explicit closed form soliton solutions of these two PIDEs are found in the pattern of trigonometric, hyperbolic, and rational functions which are compared to all the well-known results that are yielded in the paper. We attain solutions that are graphically displayed in addition to physically described in 3D structure, contour, and 2D, such as a bell-shaped soliton, a singular bell-shaped soliton, and some kink-shaped solitons. As far as the authors' wisdom, the outcomes of these problems gained by the offered expansion method are renewed closed form solitary wave and investigated here for the first time. The analysis of obtained results will be able to provide a constructive explanation of the physical phenomena in optical physics and engineering. [ABSTRACT FROM AUTHOR]

This work studies the first equation of the Kadomtsev-Petviashvili (KP) hierarchy. The sine-Gordon expansion method (SGEM) and the rational SGEM (RSGEM) are applied to the governing model. RSGEM is the developed version of SGEM. New complex travelling wave solutions, logarithmic and complex function properties are obtained. Several simulations such as 2D, 3D and contour surfaces of the obtained results are plotted. Physical meanings of these solutions are also reported. Strain conditions are also extracted. [ABSTRACT FROM AUTHOR]

The goal is to find out the influence of halal labeling, product quality, and price on purchasing decisions on tteokbokki products. This study used literature studies and questionnaires with data collection techniques using a Likert scale. The population of this study were students of Sharia Economics, University of Sultan Ageng Tirtayasa. This type of research uses correlational research, which is a type of research with a problem characteristic in the form of a correlation relationship between two or more variables. As a result of data processing with SPSS Version 26, the r square value at the coefficient of determination of .374 or 37.4% of the variables of halal labeling, product quality, and price affect the variables of purchase good decisions. The regression equation obtained KP = -3.293+.721 LH-.076 KPr+.679+e. The calculated F value is 7.379 and the resulting Ftable is 2.86. Fcount>Ftable (7.379>2.86), which means that Ho4 was rejected and Ha4 was accepted. It can be seen that simultaneously the variables of halal labeling, product quality, and price have a positive effect on purchasing decisions. But partially, product quality does not affect purchasing decisions. [ABSTRACT FROM AUTHOR]

We construct the parity–time symmetric solitons of the complex KP equation using the totally nonnegative Grassmannian. We obtain that every element in the totally nonnegative orthogonal Grassmannian corresponds to a parity–time symmetric soliton solution. [ABSTRACT FROM AUTHOR]

In this paper, our aim is to compute exact solutions for the generalized (3+1)-dimensional KadomtsevPetviashvili Benjamin-Bona–Mahony (gnKP-BBM) equation by invoking an effective method, namely the Lie symmetry technique. Firstly, we derive the infinitesimals and write down the Lie symmetries. Using these symmetries the gnKP-BBM equation is reduced to various nonlinear ordinary differential equations (NLODEs). Thereafter, solutions of the NLODEs are derived by using the Jacobi elliptic cosine method, the (G′ /G)-expansion method, the simplest equation technique and Kudryashov’s method. Conclusively, we derive conservation laws of the gnKP-BBM equation by using the Ibragimov’s method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, we present a method for generating (2 + 1)-dimensional integrable equations, resulting in the generalized Pavlov equation and dispersionless Kadomtsev–Petviashvili (dKP) equation, which can further be reduced to the standard Pavlov equation and dKP equation. Inspired by the inverse spectral transform presented in existing literature, we introduce the Riemann–Hilbert (RH) dressing method to construct the formal solutions of the Cauchy problems for the generalized Pavlov equation and dKP equation, providing a spectral representation of the solutions. Subsequently, we also extensively investigate the longtime behavior of solutions to these two equations in specific space regions. In particular, for the generalized dKP equation, we conduct a dedicated study on its implicit solutions expressed by arbitrary differential function through linearizing their RH problems. In the final section, we elaborate in detail on the analytic aspects of the wave breaking of a localized two-dimensional wave evolving according to the Hopf equation. With the assistance of a transformation, the longtime breaking of solutions to the generalized dKP equation can then be further characterized. [ABSTRACT FROM AUTHOR]

In plasma physics, the Kadomtsev–Petviashvili I (KPI) equation is a fundamental model for investigating the evolution characteristics of nonlinear waves. For the KPI equation, the constraint method is an effective tool for generating solitonic or rational solutions from the solutions of lower-dimensional integrable systems. In this work, various nonsingular, rational lump solutions of the KPI equation are constructed by employing the vector one-constraint method and the generalized Darboux transformation of the (1 + 1)-dimensional vector Ablowitz–Kaup–Newell–Segur system. Furthermore, we investigate the large-time asymptotic behavior of high-order lumps in detail and discover distinct types of patterns. These lump patterns correspond to the high-order rogue wave patterns of the (1 + 1)-dimensional vector integrable equation and are associated with root structures of generalized Wronskian–Hermite polynomials. [ABSTRACT FROM AUTHOR]

Recently, considerable attention has been given to (2+1)-dimensional Kadomtsev-Petviashvili equations due to their extensive applications in solitons that widely exist in nonlinear science. Therefore, developing a reliable numerical algorithm for the Kadomtsev-Petviashvili equations is crucial. The lattice Boltzmann method, which has been an efficient simulation method in the last three decades, is a promising technique for solving Kadomtsev-Petviashvili equations. However, the traditional higher-order moment lattice Boltzmann model for the Kadomtsev-Petviashvili equations suffers from low accuracy because of error accumulation. To overcome this shortcoming, a splitting lattice Boltzmann scheme for (2+1)-dimensional Kadomtsev-Petviashvili-Ⅰ type equations is proposed in this paper. The variable substitution method is applied to transform the Kadomtsev-Petviashvili-Ⅰ type equation into two macroscopic equations. Two sets of distribution functions are employed to construct these two macroscopic equations. Moreover, three types of soliton solutions are numerically simulated by this algorithm. The numerical results imply that the splitting lattice Boltzmann schemes have an advantage over the traditional high-order moment lattice Boltzmann model in simulating the Kadomtsev-Petviashvili-Ⅰ type equations.

Fluid mechanics is a branch of physics that focuses on the study of the behavior and laws of motion of fluids, including gases, liquids, and plasmas. The Yu–Toda–Sasa–Fukuyama equation, a class of Kadomtsev–Petviashvili type equations, is a significant integrable model with applications in fluids and other fields. In this paper, we study breather and soliton solutions of a generalized (3 + 1)-dimensional YTSF equation. By utilizing the Hirota bilinear method and Painlevé analysis, we construct solutions in the form of trigonometric and hyperbolic functions and analyze the interaction between waves graphically. We consider the characteristics of wave distribution along characteristic lines to obtain the distance between each wave and the angle generated, which is beneficial for understanding the ocean wave superposition effect. Additionally, we examine the dynamic characteristics of the wave, such as amplitude, velocity, period, shape, position, width, and phase. Furthermore, we investigate the effects of the system parameters on solitons and breathers. [ABSTRACT FROM AUTHOR]

The main aim of this study is to conduct an in-depth exploration of a recently introduced extended variant of the Kadomtsev–Petviashvili (KP) equation. To achieve this goal, we employ the Galilean transformation to derive the dynamic framework associated with the governing equation. Subsequently, we apply the principles of planar dynamical system theory to perform a bifurcation analysis. By incorporating a perturbed element into the established dynamic framework, we explore the potential emergence of chaotic behaviors within the extended KP equation. This investigation is supported by the presentation of phase portraits in both two and three dimensions. Additionally, to ascertain the stability of solutions, we conduct a sensitivity analysis on the dynamic framework employing the Runge–Kutta method. Our results affirm that minor variations in initial conditions have minimal impact on solution stability. Furthermore, employing the modified tanh method, we construct multiple instances of solitons and kinks for the proposed model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The (4+1)-dimensional Davey-Stewartson-Kadomtsev-Petviashvili equation is explored in the present work, revealing its complex dynamics and solitary wave solutions. Modeling ocean and tidal waves, particularly tsunami and long water waves, depends significantly on this nonlinear equation. Additionally, these models can be used to simulate internal and external waves in rivers and oceans as well as wave packets in water with a finite depth. The Sardar subequation method, new Kudryashov's method, and (1 ϑ (ζ) , ϑ ′ (ζ) ϑ (ζ)) method are investigated to discover novel solitary wave solutions in the terms of hyperbolic, trigonometric and rational functions. A wide variety of solitons, as dark, bright, periodic, singular, combined dark-singular solitons and, combined dark-bright are obtained by these techniques. By taking accurate parameter values, certain three-dimensional and two-dimensional graphs are plotted to improve the physical description of solutions. The intriguing field of nonlinear waves and dynamic systems is signaled to readers by this work, which suggests a major advancement in understanding the intricate and unexpected behavior of this model. [ABSTRACT FROM AUTHOR]

In this research, we used the Riccati equation approach to establish new exact solution sets for a new (3 + 1) -dimensional Korteweg–de Vries (KdV) equation, which is an enlarged version of the time-fractional Kadomtsev–Petviashvili equation. Additionally, using the Mathematica symbolic computation program, approximate solutions to the equation, applying the residual power series method (RPSM), have been obtained. A comparison table and various graphical representations have been presented to demonstrate the validity of the solutions. The numerical findings indicate the effectiveness of both approaches in finding exact and approximate solutions to nonlinear fractional differential equations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 > 4 5 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 ≤ 4 5 [26]. Furthermore, we show that for any α > 4 5 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A variational structure for the potential AKP system is established using the novel formalism of a Lagrangian multiforms. The structure comprises not only the fully discrete equation on the 3D lattice, but also its semi-discrete variants including several differential-difference equations asssociated with, and compatible with, the partial difference equation. To this end, an overview is given of the various (discrete and semi-discrete) variants of the KP system, and their associated Lax representations, including a novel 'generating PDE' for the KP hierarchy. The exterior derivative of the Lagrangian 3-form for the lattice potential KP equation is shown to exhibit a double-zero structure, which implies the corresponding generalised Euler-Lagrange equations. Alongside the 3-form structures, we develop a variational formulation of the corresponding Lax systems via the square eigenfunction representation arising from the relevant direct linearization scheme. [ABSTRACT FROM AUTHOR]

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.