Dimensional Generalized Kadomtsev Petviashvili 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.

Dynamic analysis and multiple solutions for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid

Dynamic analysis and multiple solutions for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid

Exploring the chaotic structure and soliton solutions for (3 + 1)-dimensional generalized Kadomtsev–Petviashvili model

The study of the Kadomtsev–Petviashvili (KP) model is widely used for simulating several scientific phenomena, including the evolution of water wave surfaces, the processes of soliton diffusion, and the electromagnetic field of transmission. In current study, we explore some multiple soliton solutions of the (3+1)-dimensional generalized KP model via applying modified Sardar sub-equation approach (MSSEA). By extracting the novel soliton solutions, we can effectively obtain singular, dark, combo, periodic and plane wave solutions through a multiple physical regions. We also investigate the chaotic structure of governing model using the chaos theory. The behavior of the collected solutions is visually depicted to demonstrate the physical properties of the proposed model. The solutions obtained in this paper can expand the existing solutions of the (3+1)-dimensional KP model and enhance our understanding of the nonlinear dynamic behaviors. This approach allows for consistent and effective treatment of the computation process for nonlinear KP model.

New exact solutions and related dynamic behaviors of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

New exact solutions and related dynamic behaviors of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

Wronskian solutions, bilinear Bäcklund transformation, quasi-periodic waves and asymptotic behaviors for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

In this paper, we investigate the integrability of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation, which is widely used in fluid mechanics and theoretical physics. The N-soliton solution is obtained via the Hirota’s bilinear method. The Wronskian solution is derived by using the Wronskian technique for the bilinear form. Through the exchange formula, we deduce the bilinear Bäcklund transformation consisting of four equations and six parameters. In order to consider the quasi-periodic wave having complex structure, one-, two- and three-periodic waves are investigated systemically by combining the Hirota’s bilinear method with Riemann theta function. Furthermore, the corresponding graphs of periodic wave are presented by considering the geometric properties between the characteristic lines. The propagation characteristics of periodic waves are investigated by virtue of the characteristic lines. Finally, the asymptotic relationships between quasi-periodic wave solutions and soliton solutions are established theoretically under a condition of the small amplitude limit. The analytical method used in this paper can be applied in other integrable systems.

Nonlinear dynamics of soliton molecules, hybrid interactions and other wave solutions for the (3+1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq equation

This work plumbs the nonlinear dynamics of the ([Formula: see text])-dimensional generalized Kadomtsev–Petviashvili–Boussinesq equation (gKPBe), which is used to describe some interesting physical phenomena in the fields of fluids. The resonance conditions of the soliton molecules on the ([Formula: see text]), ([Formula: see text]) and ([Formula: see text]) planes are investigated and the soliton molecules are obtained on the basis of the N-soliton solutions that are extracted by virtue of the Hirota form. Furthermore, some novel hybrid interactions including the interaction between the soliton and soliton molecule, the interaction between the different soliton molecules are also explored. Finally, the sub-equation approach is exerted to explore the various wave solutions, which include the kinky wave, bright-dark wave and the singular periodic wave solutions. Correspondingly, the graphical descriptions of the attained solutions are drawn to present a better understanding of the physical attributes. The derived solutions can enlarge the exact solutions of the ([Formula: see text])-dimensional gKPBe and lead us to understand the nonlinear dynamic behaviors better.

ON THE SEMI-DOMAIN SOLITON SOLUTIONS FOR THE FRACTAL (3+1)-DIMENSIONAL GENERALIZED KADOMTSEV–PETVIASHVILI– BOUSSINESQ EQUATION

The aim of this study is to explore some semi-domain soliton solutions for the fractal (3+1)-dimensional generalized Kadomtsev–Petviashvili–Boussinesq equation (GKPBe) within He’s fractal derivative. First, the fractal soliton molecules are plumbed by combining the Hirota equation and fractal two-scale transform. Second, the Bernoulli sub-equation function approach together with the fractal two-scale transform is employed to investigate the other soliton solutions, which include the kink soliton and the rough wave soliton solutions. The impact of the different fractal orders on the physical behaviors of the semi-domain soliton solutions is also discussed graphically. The methods mentioned in this research are expected to provide some new viewpoints on the behaviors of the fractal PDEs.

Optical Solitary Waves and Soliton Solutions of the (3+1)-Dimensional Generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony Equation

Optical Solitary Waves and Soliton Solutions of the (3+1)-Dimensional Generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony Equation

OPTICAL SOLITARY WAVES AND SOLITON SOLUTIONS OF THE (3 + 1)-DIMENSIONAL GENERALIZED KADOMTSEV–PETVIASHVILI–BENJAMIN–BONA–MAHONY EQUATION

In this investigation, an appropriate traveling wave transformation has been employed to analyze the fourth-order nonlinear (3+1)-dimensional generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) equation for an offshore structure. In addition to being integrable and having constant coefficients, the examined model represents fluid flow in the context of an offshore structure. The aforesaid nonlinear system is subjected to the initial application of two trustworthy and dependable approaches, namely the improved Bernoulli sub-equation function method and the modified extended-function method. Investigating and obtaining certain explicitly exact traveling waves, periodic waves, and soliton solutions is the major objective. The generated solutions take the form of trigonometric hyperbolic functions, exponential functions, rational functions, and multiple forms of trigonometric functions. The proposed solutions are both novel and important in that they provide light on the relevant aspects of the physical phenomena. The characteristics of the solutions have been displayed in a variety of figures, including two- and three-dimensional ones, for the best visual optical discernment.

Nonautonomous lump-periodic and analytical solutions to the ($$\varvec{3+1}$$)-dimensional generalized Kadomtsev–Petviashvili equation

Nonautonomous lump-periodic and analytical solutions to the ($$\varvec{3+1}$$)-dimensional generalized Kadomtsev–Petviashvili equation

Anomalous scattering of dark lumps to the (2+1)-dimensional generalized Kadomtsev–Petviashvili equation

The anomalous interactions of dark lumps that are completely different from the general lump collisions of the (2+1)-dimensional generalized Kadomtsev–Petviashvili equation were constructed. Different from the normal interaction mechanisms, lumps with same velocity and depth were obtained by extending the expression of rational solution based on Hirota’s Bilinear method. Moreover, interactions between dark lumps and soliton were also investigated. In particular, rogue wave excited from the lumps and stripe soliton in the interaction duration were observed. The dynamic behavior of dark lumps was analyzed in detail, and the results obtained were illustrated graphically.

A new (n+1)-dimensional generalized Kadomtsev–Petviashvili equation: integrability characteristics and localized solutions

Searching for higher-dimensional integrable models is one of the most significant and challenging issues in nonlinear mathematical physics. This paper aims to extend the classic lower-dimensional integrable models to arbitrary spatial dimension. We investigate the celebrated Kadomtsev–Petviashvili (KP) equation and propose its (n+1)-dimensional integrable extension. Based on the singularity manifold analysis and binary Bell polynomial method, it is found that the (n+1)-dimensional generalized KP equation has N-soliton solutions, and it also possesses the Painlevé property, Lax pair, Bäcklund transformation as well as infinite conservation laws, and thus the (n+1)-dimensional generalized KP equation is proven to be completely integrable. Moreover, various types of localized solutions can be constructed starting from the N-soliton solutions. The abundant interactions including overtaking solitons, head-on solitons, one-order lump, two-order lump, breather, breather-soliton mixed solutions are analyzed by some graphs.

Rational and semi‐rational solutions for a (3 + 1)‐dimensional generalized KP–Boussinesq equation in shallow water wave

In this paper, a new extended (3 + 1)‐dimensional generalized Kadomtsev–Petviashvili (KP)–Boussinesq equation, with two additional terms and , is proposed and investigated. This new equation models both left and right going waves like the Boussinesq equation and describes the transmission of tsunami waves at the bottom of the ocean and nonlinear ion‐acoustic waves in the magnetized dusty plasm. We have constructed one set of breathers and first order periodic waves from the two‐soliton solution. Moreover, the four‐soliton solution consists of two sets of breathers, second order periodic waves, and a hybrid of breathers and periodic line waves. Then, we introduce two kinds of “long wave” limits to obtain the rational and semi‐rational solutions for . Under specific parametric constraints, the obtained rational and semi‐rational solutions are nonsingular. The rational solution can be classified as first order line rogue wave, single breather, second order line rogue wave, double breather, a hybrid of breather and line rogue wave, a hybrid of breather, and single soliton. The semi‐rational solution can be classified as the first and second order kink‐shaped rogue wave, a hybrid of breather and one (two) soliton(s), a hybrid of a set of breathers, and a single soliton (breather). In addition, we give Theorem 2.1 for the higher order rational solutions.

Lie group analysis, optimal system and analytic solutions of a (3+1)-dimensional generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation for the fluid flow around an offshore structure

Lie group analysis, optimal system and analytic solutions of a (3+1)-dimensional generalized Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation for the fluid flow around an offshore structure

Bilinear auto-Bäcklund transformation, soliton and periodic-wave solutions for a (2+1)-dimensional generalized Kadomtsev–Petviashvili system in fluid mechanics and plasma physics

Fluid mechanics and plasma physics are currently interesting. In fluid mechanics and plasma physics, a (2+1)-dimensional generalized Kadomtsev–Petviashvili system is hereby investigated via symbolic computation. Based on the Hirota method, a bilinear auto-Bäcklund transformation is determined, along with some soliton solutions. Via the Hirota–Riemann method, periodic-wave solutions are derived. We also disscuss the relation between the periodic-wave and soliton solutions, i.e., how the periodic-wave solutions approach the soliton solutions under a limiting procedure. Our results might be useful in understanding certain nonlinear phenomena in fluid mechanics and plasma physics.

Residual symmetries, consistent-Riccati-expansion integrability, and interaction solutions of a new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation

With the aid of the Painlevé analysis, we obtain residual symmetries for a new (3+1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation. The residual symmetry is localized and the finite transformation is proposed by introducing suitable auxiliary variables. In addition, the interaction solutions of the (3+1)-dimensional gKP equation are constructed via the consistent Riccati expansion method. Particularly, some analytical soliton-cnoidal interaction solutions are discussed in graphical way.

Lump solutions to a generalized Kadomtsev–Petviashvili–Ito equation

A (2+1)-dimensional generalized Kadomtsev–Petviashvili–Ito equation is introduced. Upon adding some second-order derivative terms, its various lump solutions are explicitly constructed by utilizing the Hirota bilinear method and calculated through the symbolic computation system Maple. Furthermore, two specific lump solutions are obtained with particular choices of the parameters and their dynamical behaviors are analyzed through three-dimensional plots and contour plots.

Shape-changed propagations and interactions for the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluids

In this article, we consider the (3+1)-dimensional generalized Kadomtsev–Petviashvili (GKP) equation in fluids. We show that a variety of nonlinear localized waves can be produced by the breath wave of the GKP model, such as the (oscillating-) W- and M-shaped waves, rational W-shaped waves, multi-peak solitary waves, (quasi-) Bell-shaped and W-shaped waves and (quasi-) periodic waves. Based on the characteristic line analysis and nonlinear superposition principle, we give the transition conditions analytically. We find the interesting dynamic behavior of the converted nonlinear waves, which is known as the time-varying feature. We further offer explanations for such phenomenon. We then discuss the classification of the converted solutions. We finally investigate the interactions of the converted waves including the semi-elastic collision, perfectly elastic collision, inelastic collision and one-off collision. And the mechanisms of the collisions are analyzed in detail. The results could enrich the dynamic features of the high-dimensional nonlinear waves in fluids.

Painlevé analysis, Bäcklund transformations and traveling-wave solutions for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation in a fluid

Fluids are seen in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation in a fluid. On the basis of the Painlevé analysis, we find that the equation is Painlevé integrable under a certain constraint. Through the truncated Painlevé expansion, we give an auto-Bäcklund transformation. By virtue of the Hirota method, we derive a bilinear auto-Bäcklund transformation. Via the polynomial-expansion method, traveling-wave solutions are obtained. We observe that the amplitude of a traveling wave remains invariant during the propagation. We graphically demonstrate that the amplitude of the traveling-wave is affected by the coefficients corresponding to the dispersion and nonlinearity effects, while other coefficients have no influence on the traveling-wave amplitude, which represent the perturbed effects and disturbed wave velocity effects.

Bäcklund transformation, Pfaffian, Wronskian and Grammian solutions to the $$(3+1)$$-dimensional generalized Kadomtsev–Petviashvili equation

With the Hirota bilinear method and symbolic computation, we investigate the $$(3+1)$$ -dimensional generalized Kadomtsev–Petviashvili equation. Based on its bilinear form, the bilinear Backlund transformation is constructed, which consists of four equations and five free parameters. The Pfaffian, Wronskian and Grammian form solutions are derived by using the properties of determinant. As an example, the one-, two- and three-soliton solutions are constructed in the context of the Pfaffian, Wronskian and Grammian forms. Moreover, the triangle function solutions are given based on the Pfaffian form solution. A few particular solutions are plotted by choosing the appropriate parameters.