Kadomstev-Petviashvili Equation Research Articles

Multiple solitons, periodic solutions and other exact solutions of a generalized extended (2 + 1)-dimensional Kadomstev--Petviashvili equation

Abstract This paper aims to study a generalized extended ( 2 + 1 ) {(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.

Solitary wave solution of (3+1)-dimensional cylindrical Kadomstev–Petviashvili equation in warm opposite polarity dusty plasma

A theoretical investigation is done to study the propagation of dust-acoustic (DA) solitary waves in a dusty plasma system containing dynamic positively as well as negatively charged warm dust species and nonthermally distributed ion and electron species. The standard reductive perturbation method is employed to derive the (3+1)-dimensional cylindrical Kadomstev–Petviashvili (KP) equation (also known as cylindrical Korteweg–de Vries equation), which admits solitary wave solution for small but finite amplitudes. The parametric regimes for the existence of DA solitary waves are obtained. Two modes, namely, slow and fast are observed corresponding to different phase velocities. The slow mode has negative solitary pulses, whereas the fast mode contains both positive and negative DA solitary waves. It is shown that the ratio of positively charged dust to negatively charged dust number density significantly modifies the basic features of the DA solitary waves. The noteworthy effects of other parameters (viz. the nonthermal parameter and ion-to-electron temperature ratio) on the basic properties (namely, phase speed, polarity, amplitude, and width) of the DA solitary waves are also mentioned. The fundamental features and the underlying physics of the electrostatic solitary structures that are relevant to cometary tails, upper mesosphere, Jupiter’s magnetosphere, etc., are briefly discussed.

Three-Dimensional Cylindrical Dust-Acoustic Solitary Pulses in Warm Nonthermal Plasma

The nonlinear propagation of 3-D cylindrical dust-acoustic (DA) solitary waves in an unmagnetized dusty plasma system consisting of mobile adiabatic positively charged dust (PCD) grains, nonthermal ions, and electrons is investigated. The basic properties of nonlinear DA solitary structures, which exist in such a warm dusty plasma medium, are studied. The ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3+1$</tex-math> </inline-formula> )-dimensional cylindrical Kadomstev–Petviashvili equation (cKPE) (also known as cylindrical Korteweg-de Vries (KdV) equation) is derived by using the reductive perturbation method (RPM). The possible region of the formation of solitons is shown. The effects of different plasma parameters (viz., the nonthermal parameter, the dust-to-ion temperature ratio, the ion-to-dust number density ratio, etc.) on the characteristics of the DA solitary waves are studied, and also the graphical representation of the cylindrical DA solitary structures is presented. The polarity of DA solitary waves changes with different plasma parameters (viz., the ion-to-electron temperature ratio, the nonthermality of ions and electrons, the ion-to-dust number density, etc.). These results may improve our understanding of DA solitary structures in space and laboratory plasma devices where nonthermal dusty plasma occurs.

Quantum Effects on Cylindrical Dust-ion Acoustic Waves in a Semiclassical Dense Dusty Plasma

The nonlinear cylindrical dust ion acoustic waves are studied in a collisional unmagnetized dense dusty plasma medium containing inertialess degenerated electrons and positrons, fluid ions, negatively charged dust fluid and neutrals in the background, including exchange-correlation effects of both electrons and positrons. Employing the reductive perturbation method, the damped Kadomstev–Petviashvili equation is derived in the framework of two-dimensional cylindrical geometry. An approximate analytical solution of this equation is also discussed. The quantum and geometrical effects on the dust ion acoustic waves have been investigated. It is found that the dust ion acoustic wave is modified by quantum diffraction, Fermi statistics and exchange-correlation potential. Moreover, it has been found that the nebulon structures of quantum dust ions acoustic wave are formed due to the Cartesian geometry and the transverse perturbation. It is also observed that the nebulon structure is significantly modified by the exchange-correlation effects.

Superthermal electron’s effects on lump solitons structures in magnetized auroral plasma

Based on the considerations of the acknowledgements of the numerous space observations by satellites in the auroral plasma, astrophysics plasmas, spacecraft observations, various plasma models have been framed and revealed different interesting features, like electrostatic structures, solitary waves, double layers, supersolitons, etc. Soliton theory is a very efficient and competent way to describe nonlinear features. Using Viking satellite data in the auroral plasma, we have derived lump soliton solutions of the Kadomstev-Petviashvili (KP) equation by employing Hirota bilinear method. Due to its wide range of applications, the study of lump soliton is very attractive and important too. It has been shown that the lump solitons structures as well as in the one-dimensional form of lump soliton are varied with associated parameters in the auroral magnetized plasma. During the analysis of the features of the lump solitons, it is found that the system parameters play a pivotal role on the lump solitons structures.

Analytical behavior of weakly dispersive surface and internal waves in the ocean

The (2+1)-dimensional interaction of a Riemann wave propagating along the y-axis with a long wave along the x-axis is described by the space-time fractional Calogero-Degasperis (CD) and fractional potential Kadomstev-Petviashvili (PKP) equation. It can be modeled according to the Hamiltonian structure, the lax pair with the non-isospectral problem, and the pain level property. The proposed equations are widely used in beachfront ocean and coastal engineering to describe the propagation of shallow-water waves, demonstrate the propagation of waves in dissipative and nonlinear media, and reveal the propagation of waves in dissipative and nonlinear media. In this paper, we have established further exact solutions to the nonlinear fractional partial differential equation (NLFPDEs), namely the space-time fractional CD and fractional PKP equations using the modified Rieman-Liouville fractional derivative of Jumarie through the two variable (G′/G,1/G)-expansion method. As far as trigonometric, hyperbolic, and rational function solutions containing parameters are concerned, solutions are acquired when unique characteristics are assigned to the parameters. Subsequently, the solitary wave solutions are generated from the solutions of the traveling wave. It is important to observe that this method is a realistic, convenient, well-organized, and ground-breaking strategy for solving various types of NLFPDEs.

Solitary wave solutions, lump solutions and interactional solutions to the (2+1)-dimensional potential Kadomstev–Petviashvili equation

In this paper, the [Formula: see text]-solitary wave solution to the (2[Formula: see text]+[Formula: see text]1)-dimensional potential Kadomstev–Petviashvili (PKP) equation is obtained with the Hirota bilinear method. Via the limit technique of long wave, the [Formula: see text]-lump solution can be derived from resulting [Formula: see text]-solitary wave solution. In addition, interactional solutions consisting of lumps and solitary waves for the PKP equation are obtained, which can describe elastic interactions of lumps and solitary waves. These results are illustrated by graphics of several sample examples.

Nonlinear dusty magnetosonic waves in a strongly coupled dusty plasma

The nonlinear propagation of magnetosonic waves in a magnetized strongly coupled dusty plasma consisting of inertialess electrons and ions as well as strongly coupled inertial charged dust particles is presented. A generalized viscoelastic hydrodynamic model for the strongly coupled dust particles and a quantum hydrodynamic model for electrons and ions are considered. In the kinetic regime, we derive a modified Kadomstev-Petviashvili (KP) equation for nonlinear magnetosonic waves of which the amplitude changes slowly with time due to the effect of a small amount of dust viscosity. The approximate analytical solutions of the modified KP equations are obtained with the help of a steady state line-soliton solution of the second type KP equation in a frame with a constant velocity. The dispersion relationship in the kinetic regime shows that the viscosity is no longer a dissipative effect.

The study of lump solution and interaction phenomenon to (2+1)-dimensional potential Kadomstev–Petviashvili equation

In this paper, we obtained a kind of lump solutions of (2+1)-dimensional potential Kadomstev–Petviashvili equation with the assistance of Mathematica by using the Hirota bilinear method. These resulting lump solutions contain a set of five free parameters and some contour plot with different determinant values are sequentially made to show that the corresponding lump solutions tends to zero when the determinant approaches zero. Then, a completely non-elastic interaction between a lump and a stripe of the (2+1)-dimensional potential Kadomstev–Petviashvili equation is obtained, which shows a lump solution is drowned or swallowed by a stripe soliton.

Regarding on The Novel Forms of the (3+1) - Dimensional Kadomstev-Petviashvili Equation

This article, we have applied the Bernoulli Sub-Equation method to the (3+1)-Dimensional Kademstev-Petviashvili equation. We have obtained some new analytical solutions such as exponential function and rational solutions by using this technique. We have observed that two analytical solutions have been verified the (3+1)-Dimentional Kadomstev-Petviashvili equations by using Wolfram Mathematica 9. At the end of this manuscript, we submitted a conclusion in a comprehensive manner.

Exact solution of CKP equation and formation and interaction of two solitons in pair-ion-electron plasma

In the present investigation, cylindrical Kadomstev-Petviashvili (CKP) equation is derived in pair-ion-electron plasmas to study the propagation and interaction of two solitons. Using a novel gauge transformation, two soliton solutions of CKP equation are found analytically by using Hirota's method and to the best of our knowledge have been used in plasma physics for the first time. Interestingly, it is observed that unlike the planar Kadomstev-Petviashvili (KP) equation, the CKP equation admits horseshoe-like solitary structures. Another non-trivial feature of CKP solitary solution is that the interaction parameter gets modified by the plasma parameters contrary to the one obtained for Korteweg–de Vries equation. The importance of the present investigation to understand the formation and interaction of solitons in laboratory produced pair plasmas is also highlighted.

Rogue waves for Kadomstev-Petviashvili solutions in a warm dusty plasma with opposite polarity

Kadomstev-Petviashvili (KP) equation is derived using reductive perturbation method. This equation transformed into a nonlinear Schrodinger equation (NLS) by using appropriate variable transformations. When the carrier wave frequency is much smaller than the dust plasma frequency, the DA waves generating modulated wave packets in the form of rogue waves. The dependence of rogue wave profile on system plasma parameters investigated numerically. The parameters in this model are within the ranges corresponding to upper mesosphere, cometary tails and Jupiter’s magnetosphere.

Shock wave solution of the variable coefficient nonlinear partial differential equations

In this paper, we consider a variable coefficient Calogero–Degasperis equation, a variable coefficient potential Kadomstev–Petviashvili equation and the generalized (3+1)‐dimensional variable coefficient Kadomtsev–Petviashvili equation with time‐dependent coefficients. Shock wave solutions for the considered models are obtained by using ansatz method in the form of tanhp function. The physical parameters in the soliton solutions are obtained as functions of the dependent coefficients. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Phase dynamics of periodic waves leading to the Kadomtsev–Petviashvili equation in 3+1 dimensions

The Kadomstev–Petviashvili (KP) equation is a well-known modulation equation normally derived by starting with the trivial state and an appropriate dispersion relation. In this paper, it is shown that the KP equation is also the relevant modulation equation for bifurcation from periodic travelling waves when the wave action flux has a critical point. Moreover, the emergent KP equation arises in a universal form, with the coefficients determined by the components of the conservation of wave action. The theory is derived for a general class of partial differential equations generated by a Lagrangian using phase modulation. The theory extends to any space dimension and time, but the emphasis in the paper is on the case of 3+1. Motivated by light bullets and quantum vortex dynamics, the theory is illustrated by showing how defocusing NLS in 3+1 bifurcates to KP in 3+1 at criticality. The generalization to N &gt;3 is also discussed.

Rogue waves for Kadomstev-Petviashvili equation in electron-positron-ion plasma

The nonlinear ion-acoustic waves in plasma having excess super-thermal electrons and positrons have been investigated. Reductive perturbation method is used to obtain a Kadomstev-Petviashvili equation describing the system. The dynamics of the modulationally unstable wave packets described by the Kadomstev-Petviashvili equation gives rise to the formation of rogue excitation that is described by a nonlinear Schrodinger equation. The dependence of rogue waves profiles on the system parameters investigated numerically. The result of the present investigation may be applicable to some plasma environments, such as galactic clusters, interstellar medium.

Electron acoustic soliton energy of the Kadomtsev-Petviashvili equation in the Earth’s magnetotail region at critical ion density

The nonlinear properties of small amplitude electron-acoustic solitary waves (EAWs) in a homogeneous system of unmagnetized collisionless plasma consisted of a cold electron fluid and isothermal ions with two different temperatures obeying Boltzmann type distributions have been investigated. A reductive perturbation method was employed to obtain the Kadomstev-Petviashvili (KP) equation. At the critical ion density, the KP equation is not appropriate for describing the system. Hence, a new set of stretched coordinates is considered to derive the modified KP equation. Moreover, the solitary solution, soliton energy and the associated electric field at the critical ion density were computed. The present investigation can be of relevance to the electrostatic solitary structures observed in various space plasma environments, such as Earth’s magnetotail region.

The Improved (G&lt;SUP&gt;’&lt;/SUP&gt;/G)-Expansion Method to the (3+1)-Dimensional Kadomstev-Petviashvili Equation

In this article, the improved (G<SUP>’</SUP>/G)<b>-</b>expansion<b> </b>method has been implemented to generate travelling wave solutions, where G(ξ) satisfies the second order linear ordinary differential equation. To show the advantages of the method, the (3+1)-dimensional Kadomstev-Petviashvili (KP) equation has been investigated. Higher-dimensional nonlinear partial differential equations have many potential applications in mathematical physics and engineering sciences. Some of our solutions are in good agreement with already published results for a special case and others are new. The solutions in this work may express a variety of new features of waves. Furthermore, these solutions can be valuable in the theoretical and numerical studies of the considered equation. Also, in order to understand the behaviour of solutions, the graphical representations of some obtained solutions have been presented.

Nonlinear ion acoustic waveforms for Kadomstev–Petviashvili equation

The reductive perturbation method has been employed to derive the Kadomstev–Petviashvili equation for small but finite amplitude electrostatic ion-acoustic waves. An algebraic method with computerized symbolic computation is applied in obtaining a series of solutions of the Kadomstev–Petviashvili equation. Numerical studies have been made using plasma parameters reveals different waveforms such as bell-shaped solitary pulses, rational pulses and others with singularity at finite points which called blowup solutions in addition to the propagation of explosive pulses. The result of the present investigation may be applicable to some plasma environments, such as ionosphere plasma.

Painlevé analysis, Lie symmetries and invariant solutions of potential Kadomstev–Petviashvili equation with time dependent coefficients

In this paper, the Painlevé analysis of (2+1)-dimensional variable coefficients potential Kadomstev–Petviashvili (VCPKP) equation σxt+α(t)σxσxx+β(t)σxxxx+δ(t)σyy=0, has been performed to study the Painlevé properties. The VCPKP equation is also reduced to a (1+1)-dimensional partial differential equation via classical Lie group method and reduced equations are further studied to obtain certain general solutions.

The quantum dusty magnetosonic solitary wave in magnetized plasma

The effects of quantum statistic and quantum diffraction on the weakly two-dimensional fast quantum dusty magnetosonic wave propagating perpendicular to the external magnetic field are investigated by considering the inertialess electron, inertialess ion, and inertial cold dust in the low frequency limit. A Kadomstev-Petviashvili equation is derived for the magnetosonic solitary wave by using reductive perturbation method. The results show that the amplitude of soliton increases with the increasing of quantum diffraction effects of both electrons and ions, while the amplitude of the soliton decreases with the increasing of the quantum statistic effects. By using the numerical investigations, the interaction law of the nontrivial line-solitons with rich web structure in the interaction area among the line-solitons is studied by the Wronskian determinant method, which shows that there is no exchange of the energy, the momentum, and the angular momentum in the interaction.