Generalized Kadomtsev Petviashvili Equation Research Articles

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.

The positive multi-complexiton solution to a generalized Kadomtsev–Petviashvili equation

Recently, Lin et al. (Appl. Math. Lett. 78 (2018) 112–117) established the resonant multiple-wave solution of a generalized Kadomtsev–Petviashvili (gKP) equation with applications in applied sciences using the linear superposition technique. In the present paper, according to the results obtained by Lin et al. and adopting a systematic method established by Zhou and Manukure, the positive multi-complexiton solution to the generalized Kadomtsev–Petviashvili equation is constructed for the first time. Several simulations in three dimensions are conducted formally in order to examine the dynamic behavior of positive multi-complexiton solutions, particularly the positive single-, double-, and triple-complexion waves. The current research outcomes definitely will enrich studies related to the gKP equation and its exact solutions.

Bäcklund transformation and interaction solutions of a generalized Kadomtsev–Petviashvili equation with variable coefficients

Kadomtsev–Petviashvili (KP) equation is a classical nonlinear evolution equation applied in many fields such as plasma physics, fluid dynamics and engineering. In general, variable coefficients are of value in reflecting the inhomogeneities of media, nonuniformities of boundaries, and external forces. This paper mainly studies some exact solutions of a generalized KP equation with variable coefficients by using Hirota bilinear method. Based on the bilinear form, one-, two-, N-soliton solution, Bäcklund transformation and Lax pair are obtained. With the use of the test function method, lump solution and interaction solutions of lump-stripe and lump-soliton types are constructed, and the structure of the solutions are shown in three-dimensional figures. The analysis on the interaction solutions reveals new dynamic behaviors and periodic evolutions.

Nonlinearly dispersive KP equations with new compacton solutions

A complete classification of compacton solutions is carried out for a generalization of the Kadomtsev–Petviashvili (KP) equation involving nonlinear dispersion in two and higher spatial dimensions. In particular, precise conditions are given on the nonlinearity powers in this equation under which a travelling wave can be cut off to obtain a compacton. Numerous explicit examples having various wave profiles are derived, including a quadratic function, powers of a cosine, and powers of Jacobi cn functions, all of which are symmetric. The cosine and cn symmetric compactons have an anti-symmetric counterpart. In comparison, explicit solitary waves of the generalized KP equation are found to have profiles given by a power of a sech and a reciprocal quadratic function. Kinematic properties of all of the different types of compactons and solitary waves are discussed, along with conservation laws of the generalized KP equation.

Multiplicity of solutions for a generalized Kadomtsev-Petviashvili equation with potential in R^2

In this article, we study the generalized Kadomtsev-Petviashvili equation witha potential $$ (-u_{xx}+D_{x}^{-2}u_{yy}+V(\varepsilon x,\varepsilon y)u-f(u))_{x}=0\quad \text{in }\mathbb{R}^2, $$ where \(D_{x}^{-2}h(x,y)=\int_{-\infty }^{x}\int_{-\infty }^{t}h(s,y)\,ds\,dt \), \(f\) is a nonlinearity, \(\varepsilon\) is a small positive parameter, and the potential \(V\) satisfies a local condition. We prove the existence of nontrivial solitary waves for the modified problem by applying penalization techniques. The relationship between the number of positive solutions and the topology of the set where \(V\) attains its minimum is obtained by using Ljusternik-Schnirelmann theory. With the help of Moser's iteration method, we verify that the solutions of the modified problem are indeed solutions of the original roblem for \(\varepsilon>0\) small enough.

Existence and concentration of nontrivial solitary waves for a generalized Kadomtsev–Petviashvili equation in [formula omitted

In this paper, we study the existence and concentration of solitary waves for a class of generalized Kadomtsev-Petviashvili equations with potential in R2 via the variational methods.

Existence of nontrivial solitary wave for a generalized Kadomtsev–Petviashvili equation with the potential

In this paper, by using the variational methods, we consider the existence of nontrivial solitary wave for a generalized Kadomtsev–Petviashvili equation with a constant potential and a periodic potential in , respectively. In our problem, the nonlinear term f is only continuous, which allows to consider larger classes of nonlinearities than usually assumed.

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

New conservation laws of the Boussinesq and generalized Kadomtsev–Petviashvili equations via homotopy operator

According to the tools of linear algebra and calculus of variations, the conservation laws of Boussinesq and generalized Kadomtsev–Petviashvili (gKP) equations are investigated using multipliers and scaling methods. Using the Euler–Lagrange operator, the determining equations are calculated to find the multipliers of Boussinesq equation and the actual density of gKP equation. Then, the flux and density pairs of Boussinesq equation and the corresponding flux of gKP equation are obtained through 2-dimensional homotopy operator.

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.

A local Palais-Smale condition and existence of solitary waves for a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations

<abstract><p>This paper is concerned with a class of nonhomogeneous generalized Kadomtsev-Petviashvili equations</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \bigg\{ \begin{array}{rl} & u_t + (|u|^{p-2}u)_x + u_{xxx} +h_x(x-\tau t, y) +\beta \nabla_y v = 0, \\ & v_x = \nabla_y u.\end{array} $\end{document} </tex-math></disp-formula></p> <p>By proving a local Palais-Smale condition, we manage to prove the existence of solitary waves with the help of a variational characterization on the smallest positive constant of an anisotropic Sobolev inequality (Huang and Rocha, J. Inequal. Appl., 2018,163). The novelty is to give an <bold>explicit estimate</bold> on the sufficient condition of $ h $ to get the existence of solitary waves.</p></abstract>

Some Exact Solutions to Generalized Kadomtsev-Petviashvili Equation

Most of the papers have explored the interactions between solitons with a zero background, while reports about exact solutions for nonzero background are rare. Hence, this paper is aimed at exploring the breather, lump, and interaction solutions with a small perturbation to ( 2 + 1 )-dimensional generalized Kadomtsev-Petviashvili (gKP) equation. General high-order periodic breather solutions are obtained using Hirota’s bilinear method with a small perturbation. At the same time, combining the use of long wave limit methods and module resonance constraints, general lump solutions and mixed solutions to gKP equation are generated. Finally, the space-time structures of the breather solutions, lump solutions, and interaction solutions are investigated and discussed.

Dynamics investigation on a Kadomtsev–Petviashvili equation with variable coefficients

Abstract In this work, we investigate a generalized Kadomtsev–Petviashvili equation with variable coefficients and self-consistent sources in plasma and fluid mechanics. The multiple rogue wave solutions, including 1-, 3-, and 6-order rogue waves, are presented by three different functions under a nonlinear transformation. Based on the Hirota bilinear method and a more complex assumption, new lump solutions are constructed, which have not been seen in other literature. The dynamic properties of the obtained results are illustrated graphically.

On the High Energy Solitary Waves Solutions for a Generalized KP Equation in Bounded Domain

In this paper, we are mainly concerned with the existence of infinitely many high energy solitary waves solutions for a class of generalized Kadomtsev Petviashvili equation (KP equation) in bounded domain. The aim of this paper is to fill the gap in the relevant literature stated in a previous paper ( Xu, J., Wei, Z., Ding, Y.: Stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain. Electronic Journal of Qualitative Theory of Differential Equations. (2012)(68), 1-18 (2012)). Under more relaxed assumption on the nonlinearity involved in KP equation, we obtain a new result on the existence of infinitely many high energy solitary waves solutions via a variant fountain theorems.

On a Class of Partial Differential Equations and Their Solution via Local Fractional Integrals and Derivatives

This article investigates the local fractional generalized Kadomtsev–Petviashvili equation and the local fractional Kadomtsev–Petviashvili-modified equal width equation. It presents traveling-wave transformation in a nondifferentiable type for the governing equations, which translate them into local fractional ordinary differential equations. It also investigates nondifferentiable traveling-wave solutions for certain proposed models, using an ansatz method based on some generalized functions defined on fractal sets. Several interesting graphical representations as 2D, 3D, and contour plots at some selected parameters are presented, by considering the integer and fractional derivative orders to illustrate the physical naturality of the inferred solutions. Further results are also introduced in some details.

M-lump waves and their interaction with multi-soliton solutions for a generalized Kadomtsev–Petviashvili equation in (3+1)-dimensions

The aim of this work is a study of a generalized Kadomtsev–Petviashvili equation in (3+1)-dimensions that arise in fluid mechanics and plasma physics. We use a long-wave method to construct some new solutions for a gKP equation such as single-M-lump wave, double-M-lump wave, triple-M-lump wave, interaction phenomena of one-M-lump with one-soliton and two-soliton solutions, as well as an interaction between two-M-lump and one-soliton solutions are constructed. To our knowledge presented solutions are new and have never been described before. Moreover, obtained solutions are shown graphically to more understand its physical phenomena.

Transverse spectral instability in generalized Kadomtsev–Petviashvili equation

We study transverse stability and instability of one-dimensional small-amplitude periodic travelling waves of a generalized Kadomtsev–Petviashvili equation with respect to two-dimensional perturbations, which are either periodic or square-integrable in the direction of the propagation of the underlying one-dimensional wave and periodic in the transverse direction. We obtain transverse instability results in KP-fKdV, KP-ILW and KP-Whitham equations. Moreover, assuming the spectral stability of one-dimensional wave with respect to one-dimensional square-integrable periodic perturbations, we obtain transverse stability results in the aforementioned equations.

Multiple solitary waves for a generalized Kadomtsev–Petviashvili equation with a potential

We prove the existence multiple solitary waves for a generalized Kadomtsev-Petviashvili equation with a potential in dimension two. The number of waves correspond to the number of global minimum points of the potential when a parameter is small enough.