Petviashvili Equation Research Articles

The KP limit of a reduced quantum Euler–Poisson equation

AbstractIn this paper, we consider the derivation of the Kadomtsev–Petviashvili (KP) equation for cold ion‐acoustic wave in the long wavelength limit of a two‐dimensional reduced quantum Euler–Poisson system under different scalings for varying directions in the Gardner–Morikawa transform. It is shown that the types of the KP equation depend on the scaled quantum parameter . The KP‐I is derived for , KP‐II for , and the dispersiveless KP (dKP) equation for the critical case . The rigorous proof for these limits is given in the well‐prepared initial data case, and the norm that is chosen to close the proof is anisotropic in the two directions, in accordance with the anisotropic structure of the KP equation as well as the Gardner–Morikawa transform. The results can be generalized in several directions.

Dynamics of nonlinear dark waves and multi-dark wave interactions for a new extended (3+1)-dimensional Kadomtsev–Petviashvili equation

Dynamics of nonlinear dark waves and multi-dark wave interactions for a new extended (3+1)-dimensional Kadomtsev–Petviashvili equation

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.

Wronskian solutions and linear superposition of rational solutions to B-type Kadomtsev–Petviashvili equation

The Wronskian solutions to the B-type Kadomtsev–Petviashvili (BKP) equation are discussed based on the Plücker relation. Rational solutions, positon solutions, negaton solutions, and complexiton solutions to the BKP equation are directly constructed. The Wronskian formulation is employed to generate rational solutions in the form of determinants. A polynomial identity is demonstrated that an arbitrary linear combination of two Wronskian polynomial solutions of different orders is again a solution to the bilinear BKP equation. The validity of the linear superposition principle can be inferred for two Wronskian rational solutions to certain equations under specific conditions.

Variable coefficient extended cKP equation for Rossby waves and its exact solution with dissipation

In this article, the variable coefficient (2 + 1)-dimensional extended cylindrical Kadomtsev–Petviashvili (cKP) equation describing Rossby waves was derived from the quasi-geostrophic potential vorticity equation. It is difficult for the variable coefficient cKP equation with dissipation to calculate the exact solution. For obtaining the exact solution, a new transformation was constructed for the first time to reduce the extended cKP equation to the extended KP equation. We emphasize that the exact solution, and not just approximate solution, in Rossby waves flow field can be obtained when dissipation is included. The exact lump and interaction solutions with dissipative effect are given according to the modified Hirota bilinear method, and physics for the evolution of Rossby waves is analyzed based on the obtained solutions. When the dissipative parameter μ0 increases, the structure of the amplitude A changes in the spatial scale y. And when the dissipative parameter increases to a certain value, the structure of Rossby waves tends to be stable. It is pointed out that the dissipative parameter μ0 determines not only the amplitude A of Rossby waves but also structures of Rossby waves flow field, with μ0 acting on the spatial scale y and the timescale t.

Effects of the Wiener Process and Beta Derivative on the Exact Solutions of the Kadomtsev–Petviashvili Equation

We take into account the (2 + 1)-dimensional stochastic Kadomtsev–Petviashvili equation with beta-derivative (SKPE-BD) in this paper. To develop new hyperbolic, trigonometric, elliptic, and rational solutions, the Riccati equation and Jacobi elliptic function methods are employed. Because the KP equation is required for explaining the development of quasi-one-dimensional shallow-water waves, the solutions obtained can be used to interpret various attractive physical phenomena. To display how the multiplicative white noise and beta-derivative impact the exact solutions of the SKPE-BD, we plot a few graphs in MATLAB and display different 3D and 2D figures. We deduce how multiplicative noise stabilizes the solutions of SKPE-BD at zero.

Breather, lump, and interaction solutions to a nonlocal KP system

A new high-dimensional two-place Alice–Bob-Kadomtsev–Petviashvili (AB-KP) equation is proposed by applying the Alice–Bob-Bob–Alice principle and shifted-parity, delayed time reversal, charge conjugation () principle to the usual KP equation. Based on the dependent variable transformation, the bilinear form of the AB-KP system is constructed. Explicit trigonometric-hyperbolic, rational and rational-hyperbolic solutions are presented by taking advantage of the Hirota bilinear method. The obtained breather, lump, and interaction solutions enrich the solution structure of nonlocal nonlinear systems. The three-dimensional graphs of these nonlinear wave solutions are demonstrated by choosing the specific parameters.

Accurate Coiflet wavelet solution of extended [formula omitted]-dimensional Kadomtsev–Petviashvili equation using the novel wavelet-homotopy analysis approach

We established a novel wavelet-homotopy analysis technique to give accurate solutions to nonlinear wave problems with unsteady state, which is the originality of the paper. The extended (2+1)-dimensional Kadomtsev–Petviashvili equation is taken as the research object owing to its complexity and importance. The convergence of the proposed technique is proved. The modulation instability is employed to determine the range of the normalized wave numbers of the transformed traveling wave equation. The validity and reliability of our results are checked via a rigid comparison with analytical solutions. Results and the comparison with the numerical solutions reveal that the proposed technique maintains capabilities of the homotopy analysis method for strong nonlinear processing and the wavelet technique for excellent local detail representation. It is expected that the proposed method can be used as an alternative tool to analyze more complex wave problems such as rogue wave or shock wave problems.

Application of the polynomial function method to the variable-coefficient Kadomtsev–Petviashvili equation

In this paper, we research a (2+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics. The lump, lump-soliton and lump-periodic solutions are derived based on the variable-coefficient polynomial function method. The effect of variable coefficients on the amplitude and velocity of solitons is analyzed and shown by some 3D graphs, contour plots and density graphs.

Resonance solutions and hybrid solutions of an extended (2+1)-dimensional Kadomtsev–Petviashvili equation in fluid mechanics

Resonance solutions and hybrid solutions of an extended (2+1)-dimensional Kadomtsev–Petviashvili equation in fluid mechanics

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.

(2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, cnoidal and superposition solutions

We study the problem of gravity surface waves for an ideal fluid model in the (2+1)-dimensional case. We apply a systematic procedure to derive the Boussinesq equations for a given relation between the orders of four expansion parameters, the amplitude parameter α, the long-wavelength parameter β, the transverse wavelength parameter γ, and the bottom variation parameter δ. Such an approach, known in (1+1)-dimensional theory, is extended for the first time for (2+1)-dimensional shallow water waves. Using this procedure, we derived the only possible (2+1)-dimensional extensions of the Korteweg–de Vries equation, the fifth-order KdV equation and the Gardner equation in three cases of the relationship between these parameters. All these equations are non-local. When the bottom is flat, the (2+1)-dimensional KdV equation can be transformed to the Kadomtsev–Petviashvili equation in a fixed reference frame and next to the classical KP equation in a moving frame. Thus, the Kadomtsev–Petviashvili equation gained a derivation from the fundamental laws of hydrodynamics. We have found soliton, cnoidal, and superposition solutions (essentially one-dimensional) to the (2+1)-dimensional Korteweg–de Vries equation and the Kadomtsev–Petviashvili equation.

Solitary wave analysis of the Kadomtsev–Petviashvili model in mathematical physics

The (3 + 1)-dimensional Kadomtsev–Petviashvili equation has significant applications that emerge in the study of fluids and multi-component plasmas. The main object of our study is to investigate stable and efficient solitary solutions to the stated wave equations through two different schemes that provide generic solutions like exponential functions, trigonometric functions, hyperbolic functions, etc. The graphical illustrations for different values of the corresponding parameters of the attained solutions are explained in order to expose the intimate structure of the substantial phenomena. In particular, changes in wave position and category with respect to time are deliberated extensively through 2D figures. It has been found that wave velocity depends on some free parameters associated with the methods used to solve the equation. It has been demonstrated that for different values of the free parameters, the obtained solutions of the KP model exhibit dark soliton, peakon shape soliton, dark soliton-type wave, bright soliton, bell-shaped soliton, etc. Moreover, the direction and position of the solitons for the variation of other parameters are measured, from which a simple and comprehensive interpretation of the various aspects of water waves can be obtained. We think the obtained solution could be highly useful for studying nonlinear events in fluid dynamics, wind waves, and wind-generated waves.

Analysis of new soliton type solutions to generalized extended (2 + 1)-dimensional Kadomtsev-Petviashvili equation via two techniques

In this paper, we examine the solitary wave solutions of the generalised (2 + 1) extended Kadomtsev–Petviashvili (eKP) equation, which is used as model for the surface waves and internal waves in straits or channels and describes the dynamics of nonlinear waves in plasma physics and fluid dynamics. We secure distinct soliton solutions by employing two amelioration methodologies, namely the improved F-expansion method and the new Kudryashov method. These methods explore several unique forms of solitary wave solutions, including dark, rational, singular, periodic, and exponential ones. In order to illustrate propagation of some reported wave solutions, the graphs associated with these solutions are presented by choosing appropriate parametric values in 2D, 3D, contour, and density plots with the use of the symbolic software Mathematica 13.0. The presented solutions realize very significant fact for investigation the qualitative interpretations for various phenomena in our nature. The computed solutions revealed that the applied methods are influential, efficious and skilful and can be a best way to handle other higher order nonlinear evolution equations.

Periodic solitons of Davey Stewartson Kadomtsev Petviashvili equation in (4+1)-dimension

The aim of this paper is to study new exact solutions Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) equation. We use modified tanh method along associated with new Ricatti equation. The results are demonstrated for specific values of parameters, which show periodic singular as well as nonsingular solitons for specific values of parameters. Some solutions are plotted and presented in 3D and density graphs. The Mathematica are used for computation and simulations of the results, respectively.

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 R 2 , respectively. In our problem, the nonlinear term f is only continuous, which allows to consider larger classes of nonlinearities than usually assumed.

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.

Resonant collisions among multi-breathers in the Mel’nikov system

We study the resonant collisions among multi-breathers in the Mel’nikov system, which is the Kadomtsev–Petviashvili (KP) equation coupled to the nonlinear Schrödinger equation. The two-breather resonant collisions are classified into two different types: (i) the emergence of a Y–shaped pattern when the breathers are oblique on each other; and (ii) the occurrence of fission of one breather into two breathers or fusion of two breathers into a single one when the involving breathers are parallel to each other. The resonant collision of three breathers generates more complicated collision scenarios. They are divided into separate and mutual resonant collisions depending on whether the whole breathers are involved in the collision process. On this basis, each collision mode demonstrates two distinct forms: (i) diverse web–shaped waveforms with three to five infinitely long breathers when three breathers are obliquely intersecting, where some waveforms display a polygonal pattern under certain parametric constraints; and (ii) the combination of plentiful fission or fusion processes when the breathers are parallel to each other. The parameter constraints are discussed in detail for the classification of resonant collision scenarios.

Higher-order degenerations of Fay’s identities and applications to integrable equations

Higher-order degenerated versions of Fay’s trisecant identity are presented. It is shown that these lead to solutions for Schwarzian Kadomtsev–Petviashvili equations.

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.