Kadomtsev Petviashvili Research Articles

Lie symmetries, exact solution and conservation laws of (2 + 1)-dimensional time fractional Kadomtsev–Petviashvili system

Abstract In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied.

Long wavelength limit of the KP-type equation for the ion dynamics system

In this paper, we consider the modified Kadomtsev–Petviashvili equation (KP-type) limit for ion dynamics system in R 2 + 1 . We derive the two-dimensional homogeneous and inhomogeneous KP-type equation from the non-dimensional ion dynamics system by Gardner–Morikawa transforms. The different Gardner–Morikawa transforms in temporal–spatial directions bring the anisotropic problem for energy estimate. By the energy method, we prove that the solution of Euler–Poisson system converges to KP-type equation globally when ϵ → 0 in R 2 + 1 .

New analytical wave structures for generalized B-type Kadomtsev–Petviashvili equation by improved modified extended tanh function method

Abstract Recently, solving the complicated nonlinear partial differential equations has become very important demand in order to simulate their physical phenomena. This manuscript focuses on extracting the wave solutions of (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili equation (GBKPE), which demonstrates the behavior of nonlinear waves in fluid mechanics. The improved modified extended Tanh function (IMETF) method is the suggested method to do this task as it gives different types of solutions. This method enables us to obtain many solutions, such as Jacobi elliptic, dark soliton, and singular soliton, exponential, and singular periodic wave solutions. Additionally, for more illustrations graphical visual representations of some solutions are provided.

Nonlinear waves for a variable-coefficient modified Kadomtsev–Petviashvili system in plasma physics and electrodynamics

In this paper, a variable-coefficient modified Kadomtsev–Petviashvili (vcmKP) system is investigated by modeling the propagation of electromagnetic waves in an isotropic charge-free infinite ferromagnetic thin film and nonlinear waves in plasma physics and electrodynamics. Painlevé analysis is given out, and an auto-Bäcklund transformation is constructed via the truncated Painlevé expansion. Based on the auto-Bäcklund transformation, analytic solutions are given, including the solitonic, periodic and rational solutions. Using the Lie symmetry approach, infinitesimal generators and symmetry groups are presented. With the Lagrangian, the vcmKP equation is shown to be nonlinearly self-adjoint. Moreover, conservation laws for the vcmKP equation are derived by means of a general conservation theorem. Besides, the physical characteristics of the influences of the coefficient parameters on the solutions are discussed graphically. Those solutions have comprehensive implications for the propagation of solitary waves in nonuniform backgrounds.

Rogue wave patterns in the nonlocal nonlinear Schrödinger equation

This paper investigates rogue wave patterns in the nonlocal nonlinear Schrödinger (NLS) equation. Initially, employing the Kadomtsev–Petviashvili reduction method, rogue wave solutions of the nonlocal NLS equation, whose τ function is a 2×2 block matrix, are simplified. Afterward, utilizing the asymptotic analysis approach, we investigate the rogue wave patterns when two free parameters a2m1+1 and b2m2+1 are considerably large and fulfill the condition |a2m1+1|2/(2m1+1)=O(|b2m2+1|1/(2m2+1)). Our findings reveal that under these conditions, rogue wave solutions of the nonlocal NLS equation exhibit novel patterns, which consist of three regions, which are the outer region, the middle region and the inner region. In the outer and middle regions, only single rogue waves with singularities may occur, and their locations are characterized by roots of two polynomials from the Yablonskii–Vorob'ev polynomial hierarchies. In the inner region, a possible lower order rogue wave may appear, which can be singular or regular, depending on the values of m1,m2, the sizes of τ function, and certain free parameters. Finally, the numerical results indicate that the predicted outcomes are in close alignment with real rogue waves.

Impact of fractional and integer order derivatives on the [formula omitted]-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation

In this study, the closed-form wave solutions of the (4+1)-dimensional fractional Davey–Stewartson–Kadomtsev–Petviashvili equation are investigated using the modified auxiliary equation method and the Jacobi elliptic function method. In the analysis, two fractional derivatives known as M-truncated, beta and integer order derivative are used. The fractional-order partial differential equation is transformed into an integer-order ordinary differential equation by using the wave transformation, fractional derivatives, and integer-order derivatives. As a result, wave function solutions are found, including bell shape, W-shaped, composite dark-bright and periodic wave. The effects of free parameters on the amplitudes and wave behaviors are illustrated. It is demonstrated extensively that changes in the free parameters lead to changes in the wave amplitude. A comparison of solutions using the two types of fractional derivatives and the integer-order derivatives is included. The effects of the beta derivative, the M-truncated derivative and integer order derivative on the considered model are presented using 2D and 3D figures.

A study on analytical solutions of one of the important shallow water wave equations and its stability analysis

Abstract The present research uses the modified w/g-expansion technique to give analytical solutions for the nonlinear time fractional integro-differential Kadomtsev Petviashvili (KP) hierarchy equation with beta fractional derivative. The modified w/g-expansion technique is an important method for finding analytical solutions to nonlinear equations. The suggested approach produces three sorts of solutions trigonometric, hyperbolic, and rational. These answers have been discovered with the aid of a software tool. Additionally, stability analysis of observed governing model solutions is used to validate scientific calculations. Furthermore, the 3-dimensional, contour, and 2-dimensional images are provided for a better understanding of the behavior of the solutions.

Real and complex multi solitons, soliton molecules, asymmetric solitons and diverse wave solutions to the (2+1)-dimensional Sawada–Kotera–Kadomtsev Petviashvili equation

Real and complex multi solitons, soliton molecules, asymmetric solitons and diverse wave solutions to the (2+1)-dimensional Sawada–Kotera–Kadomtsev Petviashvili equation

Integrable Semi-Discretization for a Modified Camassa-Holm Equation with Cubic Nonlinearity

In the present paper, an integrable semi-discretization of the modified Camassa-Holm (mCH) equation with cubic nonlinearity is presented. The key points of the construction are based on the discrete Kadomtsev-Petviashvili (KP) equation and appropriate definition of discrete reciprocal transformations. First, we demonstrate that these bilinear equations and their determinant solutions can be derived from the discrete KP equation through Miwa transformation and some reductions. Then, by scrutinizing the reduction process, we obtain a set of semi-discrete bilinear equations and their general soliton solutions in the Gram-type determinant form. Finally, we obtain an integrable semi-discrete analog of the mCH equation by introducing dependent variables and discrete reciprocal transformation. It is also shown that the semi-discrete mCH equation converges to the continuous one in the continuum limit.

A New (3+1)-Dimensional Extension of the Kadomtsev–Petviashvili–Boussinesq-like Equation: Multiple-Soliton Solutions and Other Particular Solutions

In this study, we focus on investigating a novel extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq-like (KPB-like) equation. Initially, we utilized the Lie symmetry method to determine the symmetry generator by considering the Lie invariance condition. Subsequently, by similar reduction, the equation becomes ordinary differential equations (ODEs). Exact analytical solutions were derived through the power series method, with a comprehensive proof of solution convergence. Employing the (G′/G2)-expansion method enabled the identification of trigonometric, exponential, and rational solutions of the equation. Furthermore, we established the auto-Bäcklund transformation of the equation. Multiple-soliton solutions were identified by utilizing Hirota’s bilinear method. The fundamental properties of these solutions were elucidated through graphical representations. Our results are of certain value to the interpretation of nonlinear problems.

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.

Phase trajectories, chaotic behavior, and solitary wave solutions for (3+1)-dimensional integrable Kadomtsev–Petviashvili equation in fluid dynamics

This paper focuses on finding exact soliton solutions and further examining the qualitative characteristics of these solutions for a nonlinear partial differential equation known as the (3+1)-dimensional integrable Kadomtsev–Petviashvili equation that portrays a unique dispersion effect. The (3+1)-dimensional integrable Kadomtsev–Petviashvili equation models the evolution of weakly nonlinear dispersive waves in three spatial dimensions “x,y,z” and time “t”. It balances nonlinearity and dispersion, leading to solitary wave solutions that are stable and localized. This equation is crucial in understanding complex wave phenomena in fields like fluid dynamics, plasma physics, and optics. Initially, the traveling wave transformation is employed to convert the nonlinear partial differential equations into the non-linear ordinary differential equations. Then, we use the Riccati equation approach to find solitary wave solutions for the resulting ordinary differential equation. To better comprehend the physical significance of these derived solutions, we illustrate them using various visual representations. Diverse novel collections of solitary wave solutions are obtained such as bright solitons, dark solitons, and dark singular solitons. Second, we convert the nonlinear ordinary differential equation into a linear dynamical system using the planar dynamical transformation. Then, we undertake a qualitative analysis of the dynamical system to study its chaotic characteristics and bifurcation phenomena. Phase portraits of bifurcation are observed at the equilibrium points of the planner dynamical system. We discuss various tools to identify the chaos (random and unpredictable behavior) in dynamical systems. Graphical visualization of qualitative analysis of the system is also depicted in 3-D phase portraits, 2-D phase portraits, Poincaré mapping, and time series. Additionally, the Runge–Kutta technique is used to do a thorough sensitivity analysis of the dynamical system. Using this analytical procedure, it is verified that the stability of the solution has negligible impact while there is little change in the beginning circumstances. This paper aims to provide vital views to scientists and researchers seeking to advance their experimental operations. Furthermore, this research might be expanded by incorporating solitons and situations and investigating wave dynamics. The methods employed provide a direct and uncomplicated approximation of all solutions when compared to the previously established techniques. Various combinations and values of the physical parameters are utilized to explore the optical soliton solutions of the resultant system.

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.

Hirota [formula omitted]-operator forms, multiple soliton waves, and other nonlinear patterns of a 2D generalized Kadomtsev–Petviashvili equation

In the present paper, extensive research is conducted on a 2D generalized Kadomtsev–Petviashvili (2D-gKP) equation which models water waves with long wavelengths. The study starts by establishing Hirota D-operator forms of the governing equation using the Bell polynomial method (BPM). Through checking the three-soliton condition for the 2D-gKP equation, its integrability is then examined, and as a consequence, single-, double-, and (special) triple-soliton waves are derived from the modified Hirota method. In the end, after deriving lump and breather waves of the 2D-gKP equation through ansatz methods, a theoretical discussion along with its proof is presented. Attempts have been made to elucidate the physical features of nonlinear waves by displaying such waves in 2D and 3D postures. The paper’s findings contribute certainly to research on nonlinear waves of KP-type equations.

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.

A novel discretization of the Yajima-Oikawa equation: Cauchy matrix approach

The Cauchy matrix approach, rooted in the Sylvester equation, plays a crucial role in defining the τ functions of nonlinear evolution equations. In this paper, the Cauchy matrix approach is employed to introduce a novel integrable semi-discrete counterpart of the one-dimensional Yajima-Oikawa (YO) system. This new system is linked with the differential-difference Kadomtsev-Petviashvili (KP) equation with self-consistent sources (SCS). Based on the Cauchy matrix approach of the KP system, we systematically construct multiple soliton and multiple pole solutions. Furthermore, we analyze and illustrate various examples of soliton solutions for further understanding.

Rossby waves with barotropic–baroclinic coherent structures and dynamics for the (2 + 1)-dimensional coupled cylindrical KP equations with variable coefficients

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.

Extended Direct Method and New Similarity Solutions of Kadomtsev–Petviashvili Equation

Extended Direct Method and New Similarity Solutions of Kadomtsev–Petviashvili Equation

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

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

Bi-Hamiltonian structure of a super KdV equation of Kupershmidt

In this paper, we study a super Korteweg–de Vries (sKdV) equation proposed by Kupershmidt which possesses a Lax operator with three fully nonlocal terms. The Lax operator is reformulated so that it is of the super constrained modified Kadomtsev–Petviashvili (scmKP) type. By calculating the bi-Hamiltonian structure of the scmKP hierarchy and employing Dirac reduction, we obtain the bi-Hamiltonian structure of the sKdV equation. We also present a spectral problem of its modified system.