Petviashvili Equation Research Articles

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.

Dynamic behaviors of the lump solutions and mixed solutions to a (2+1)-dimensional nonlinear model

In this paper, we propose a combined form of the bilinear Kadomtsev–Petviashvili equation and the bilinear extended (2+1)-dimensional shallow water wave equation, which is linked with a novel (2+1)-dimensional nonlinear model. This model might be applied to describe the evolution of nonlinear waves in the ocean. Under the effect of a novel combination of nonlinearity and dispersion terms, two cases of lump solutions to the (2+1)-dimensional nonlinear model are derived by searching for the quadratic function solutions to the bilinear form. Moreover, the one-lump-multi-stripe solutions are constructed by the test function combining quadratic functions and multiple exponential functions. The one-lump-multi-soliton solutions are derived by the test function combining quadratic functions and multiple hyperbolic cosine functions. Dynamic behaviors of the lump solutions and mixed solutions are analyzed via numerical simulation. The result is of importance to provide efficient expressions to model nonlinear waves and explain some interaction mechanism of nonlinear waves in physics.

The symmetry breaking solutions of the nonlocal Alice–Bob B-type Kadomtsev–Petviashvili system

The (2+1)-dimensional B-type Kadomtsev–Petviashvili equation is an integrable model, which can be used to describe the shallow water wave in a fluid. In this paper, the nonlocal Alice–Bob B-type Kadomtsev–Petviashvili system is induced via the principle of PˆsxPˆsyTˆd symmetry. An extended Bäcklund transformation introduced, the symmetry breaking solution, which contains the soliton, breather, lump and their hybrid structures for this system, is solved through the Hirota bilinear form.

Different solutions to the conformable generalized (3 + 1)-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation arising in shallow-water waves

This paper employed the [Formula: see text]-expansion, Riccati equation, [Formula: see text]-expansion, and modified Kudryashov methods to find new exact solution sets for the conformable generalized [Formula: see text]-dimensional Camassa–Holm–Kadomtsev–Petviashvili equation. The accuracy of the results has been demonstrated using a variety of graphical representations. These newly obtained solutions can be applied to further research and understand the dynamics of the Camassa–Holm–Kadomtsev–Petviashvili equation, which arises in ocean and water wave theory, hydrodynamics, plasma physics, nonlinear sciences, and engineering. The presented four methods are straightforward, robust, and successful in getting analytical solutions to nonlinear fractional differential equations, as the analytical results indicate.

A New Composite Technique to Obtain Non-traveling Wave Solutions of the (2+1)-dimensional Extended Variable Coefficients Bogoyavlenskii–Kadomtsev–Petviashvili Equation

A New Composite Technique to Obtain Non-traveling Wave Solutions of the (2+1)-dimensional Extended Variable Coefficients Bogoyavlenskii–Kadomtsev–Petviashvili Equation

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

Resonant collisions of high-order localized waves in the Maccari system

Exploring new nonlinear wave solutions to integrable systems has always been an open issue in physics, applied mathematics, and engineering. In this paper, the Maccari system, a two-dimensional analog of nonlinear Schrödinger equation, is investigated. The system is derived from the Kadomtsev–Petviashvili (KP) equation and is widely used in nonlinear optics, plasma physics, and water waves. A large family of semi-rational solutions of the Maccari system are proposed with the KP hierarchy reduction method and Hirota bilinear method. These semi-rational solutions reduce to the breathers of elastic collision and resonant collision under special parameters. In case of resonant collisions between breathers and rational waves, these semi-rational solutions describe lumps fusion into breathers, or lumps fission from breathers, or a mixture of these fusion and fission. The resonant collisions of semi-rational solutions are semi-localized in time (i.e., lumps exist only when t → +∞ or t → −∞), and we also discuss their dynamics and asymptotic behaviors.

Study of compressive and rarefactive lump solitons structures in dusty plasma with double spectral (r,q) distributed electrons

Nonlinear features are revealed by one of the most competent and powerful approaches, i.e. soliton theory. The present article starts with the dust collisionless magnetized plasma where electrons are following double spectral ( r , q ) distribution. From the hydrodynamical governing equations set, Kadomtsev–Petviashvili (KP) equation is derived using the reductive perturbation technique. The derived KP equations are converted to standard KP equations by a suitable transformation to use the Hirota bilinear method (HBM) more conveniently. By using HBM, an auxiliary function is constructed, and through symbolic computation, three arbitrary constant sets are formed. These sets associate with three lump soliton solution sets. Finally, all these lump soliton solutions are returned back by the inverse transformation that was used to make the derived KP equation to the standard KP equation and get the lump soliton solution of the associated system. It is investigated that associated plasma parameters have a significant impact on the lump soliton structures while tracing figures using these plasma parameters within the justified range of the system. While lump soliton features' are analysed for various parameters' effect on it, one most important aspect is revealed. Double spectral index r and q play a significant role in the lump solitons structures and depict compressive and rarefactive lump soliton structures with the associated system. It is found that these double spectral indices r and q are the decision-making parameters for the lump soliton structures to be compressive or rarefactive.

Multiple soliton and M-lump waves to a generalized B-type Kadomtsev–Petviashvili equation

In this study, we focus on the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili (gBKP) equation in fluid dynamics, which is useful for modeling weakly dispersive waves transmitted in quasi media and fluid mechanics. As a general matter, this paper examines the gBKP equation including variable coefficients of time that are widely employed in plasma physics, marine engineering, ocean physics, and nonlinear sciences to explain shallow water waves. Using Hirota’s bilinear approach, one-, two, and three-soliton solutions to the problem are constructed. By employing a long-wave method, 1-M-, 2-M, and 3-M-lump solutions are derived. In addition, interaction phenomena of one-, and two-soliton solutions with one-M-lump wave are revealed. Moreover, an interaction solution between a two-M-lump wave and a one-soliton solution is also offered. The planes that M-lump waves travel among them are derived. We believe that our findings will help improve the dynamical properties of (3+1)-dimensional BKP-type equation.

Lump, breather and interaction solutions to the (3+1)-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili equation

This paper analyzes the (3+1)-dimensional generalized Camassa–Holm Kadomtsev–Petviashvili(gCH-KP) equation, which has recently gained popularity in ocean physics and hydrodynamics engineering. Based on the Hirota bilinear method and symbolic computation, we explore the lump solution, breather solution and new interaction solutions. The maximum and minimum value of the lump solution are obtained by theoretical calculation. Dynamic characteristics of these solutions are depicted by presenting some three dimensional, two dimensional plots and density plots.

An extended Painlevé integrable Kadomtsev--Petviashvili equation with lumps and multiple soliton solutions

PurposeThis study aims to propose an extended (3 + 1)-dimensional integrable Kadomtsev–Petviashvili equation characterized by adding three new linear terms.Design/methodology/approachThis study formally uses Painlevé test to confirm the integrability of the new system.FindingsThe Painlevé analysis shows that the compatibility condition for integrability does not die away by adding three new linear terms with distinct coefficients.Research limitations/implicationsThis study uses the Hirota's bilinear method to explore multiple soliton solutions where phase shifts and phase variable are explored.Practical implicationsThis study also furnishes a class of lump solutions (LSs), which are rationally localized in all directions in space, using distinct values of the parameters via using the positive quadratic function method.Social implicationsThis study also shows the power of the simplified Hirota’s method in handling integrable equations.Originality/valueThis paper introduces an original work with newly developed Painlevé integrable model and shows new useful findings.

Creation of weakly interacting lumps by degeneration of lump chains in the KP1 equation

We present two different paths to degenerate normally interacting lump chains into anomalously interacting lump patterns within the Kadomtsev–Petviashvili (KP1) equation. In the first path, the periods of M lump chains with close velocities become infinite simultaneously; then, we obtain anomalously interacting M(M+1)/2 lumps directly in one step. The second path consists of two steps: firstly, M normally interacting lump chains degenerate into M anomalously interacting lump chains having the same parameters. Then, when the periods of anomalously interacting lump chains tend to infinity, we obtain M(M+1)/2 weakly interacting lumps. In the weakly interacting lump chains, the distance between them varies with time proportional to ln|t|.

New (3+1)-dimensional conformable KdV equation and its analytical and numerical solutions

In this research, we used the Riccati equation approach to establish new exact solution sets for a new [Formula: see text]-dimensional Korteweg–de Vries (KdV) equation, which is an enlarged version of the time-fractional Kadomtsev–Petviashvili equation. Additionally, using the Mathematica symbolic computation program, approximate solutions to the equation, applying the residual power series method (RPSM), have been obtained. A comparison table and various graphical representations have been presented to demonstrate the validity of the solutions. The numerical findings indicate the effectiveness of both approaches in finding exact and approximate solutions to nonlinear fractional differential equations.

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.

Direct separation approach and multi-valued localized excitation for (M+N)-dimensional nonlinear system

In this paper, we propose a new variable separation method that does not need the Hirota’s bilinear form and directly gives the analytic form of the solution u instead of its potential uy. This new method covers the N-soliton solution obtained by the Hirota’s direct method and the multi-valued solution of the multi-linear variable separation approach(MLVSA), and is applicable to the (M+N)-dimensional nonlinear model. We would like to call it the “direct separation approach” (DSA). Taking the extended (3+1)-dimensional Kadomtsev–Petviashvili equation as an example, we introduce an entirely fresh variable separation ansatz and substitute it into the equation, resulting in a new variable separation solution. With the help of this variable separation solution, we construct two types of N-soliton solutions via the single-valued functions. Then, the rogue waves and four typical folded solitary waves are obtained by using the multi-valued functions. In addition, we study the head-on and chase-after collision between two, three, and four folded solitary waves and systematically analyze their dynamic behaviors, such as the phase shifts and their difference values of the interaction. Especially, we detect a multi-directional movement in the collision between four folded solitary waves, which significantly differs from the others. The obtained results may help us simulate complex folded appearance in real life and better understand the wave motion of the solitary waves.

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.

Integrability, breather, lump and quasi-periodic waves of non-autonomous Kadomtsev–Petviashvili equation based on Bell-polynomial approach

This article describes the infinite conservation law, quasi-periodic wave, breather, lump, and characteristic of integrability via bilinear Bäcklund and lax pairs of the non-autonomous Kadomtsev–Petviashvili (NKP) equation (in the presence of an external force and damping). Bell-polynomial is deduced from Bäcklund transformation from which infinite conservation law for NKP equation is derived. Obtaining well-known quasi-periodic solutions is an important aspect of this investigation. A limiting procedure is used for analyzing the asymptotic behavior of multi-periodic waves. This method proves rigorously that periodic waves tend toward soliton solutions under a small amplitude limit, which results in a relationship between periodic waves and soliton solutions. For the first time, the effect of external force in a quasi-periodic wave is demonstrated explicitly from the numerical understanding. Further, some complicated solution structures of the NKP equation such as lump and breather-type solitons are explored from the bilinear form of the said equation with the appropriate choice of polynomial functions. Important characteristics of lump and breather waves in different forcing backgrounds are graphically described. Finally, the lump wave of the NKP equation is explored from the breather wave in the limiting stage. It is also confirmed from the analytical results of the relevant motions that the velocity, maximum altitude, and interacting natures of the wave quantities are all influenced by the damping and forcing terms.

Geometrical patterns of time variable Kadomtsev–Petviashvili (I) equation that models dynamics of waves in thin films with high surface tension

Geometrical patterns of time variable Kadomtsev–Petviashvili (I) equation that models dynamics of waves in thin films with high surface tension