Kadomtsev Petviashvili Research Articles

The partial-rogue ripple solutions of nonlocal Kadomtsev–Petviashvili equation

The derivation of new exact solutions for integrable systems has always been an open problem in mathematical physics and engineering. In this paper, a large class of partial-rogue ripple solutions of a nonlocal Kadomtsev–Petviashvili [KP] equation are derived, which means that these ripple waves are semi-localized in time. Firstly, the Wronskian and Grammian determinant expression of the nonlocal KP equation are constructed. Then, three kinds of partial-rogue ripplon solutions of nonlocal KP equations, including (i) partial-rogue ripple soliton solutions, (ii) partial-rogue ripple lump chain solutions, and (iii) partial-rogue ripple lump solutions, are proposed. These solutions approach to the zero background plane as t→∞, which is the characteristic of partial-rogue ripple waves. These solutions are rarely reported, and their dynamics have been analytically and numerically discussed. The results of this paper are more practical and closely match the attenuated ripple waves stirred up in water waves. Additionally, the research method in this paper provides new references for existing results.

Rogue wave solutions for the [formula omitted]-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili equation

In this paper, the 3+1-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili equation is investigated by Hirota bilinear method, from which the first, second and third order rogue wave solutions are derived. The relationships between the parameters and the amplitude, width and center of rogue wave are discussed. At the same time, several evolution and density plots are present to exhibit the dynamical structure of the rogue wave solutions. The results may be useful in nonlinear optics and other physically relevant integrable systems.

Solitons, one line rogue wave and breather wave solutions of a new extended KP-equation

The Kadomtsev–Petviashvili equation and its variants play a significant role in the description of several nonlinear phenomena arising in physics. In this paper, a new extension of the Kadomtsev–Petviashvili equation is considered. The main objective of this work is to study the traveling wave dynamics of the considered equation. Different traveling patterns are obtained for the governing equation in the presented work. Traveling wave solutions involving solitonic structures are derived by the application of the modified auxiliary equation method. The proposed method has provided solutions involving trigonometric, hyperbolic and rational functions. The wave profiles plotted for the obtained exact solutions exhibit the dark and bright solitons. Moreover, one line rogue wave and breather wave solutions are constructed for the governing model using combination of trigonometric and exponential functions as test functions. The dynamical characteristics of the earned solutions are depicted using interesting plots for deep understanding of obtained results.

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.

New Painlevé Integrable (3+1)-Dimensional Combined pKP-BKP Equation: Lump and Multiple Soliton Solutions

We introduce a new form of the Painlevé integrable (3+1)-dimensional combined potential Kadomtsev--Petviashvili equation incorporating the B-type Kadomtsev–Petviashvili equation (pKP–BKP equation). We perform the Painlevé analysis to emphasize the complete integrability of this new (3+1)-dimensional combined integrable equation. We formally derive multiple soliton solutions via employing the simplified Hirota bilinear method. Moreover, a variety of lump solutions are determined. We also develop two new (3+1)-dimensional pKP–BKP equations via deleting some terms from the original form of the combined pKP–BKP equation. We emphasize the Painlevé integrability of the newly developed equations, where multiple soliton solutions and lump solutions are derived as well. The derived solutions for all examined models are all depicted through Maple software.

Traveling Wave Solutions and Bifurcations of a New Generalized (3 + 1)-Dimensional Kadomtsev–Petviashvili Equation

Traveling Wave Solutions and Bifurcations of a New Generalized (3 + 1)-Dimensional Kadomtsev–Petviashvili Equation

Dynamical analysis of (4 + 1)-dimensional Davey Srewartson Kadomtsev Petviashvili equation by employing Lie symmetry approach

In this study, (4+1)-dimensional Davey Srewartson Kadomtsev Petviashvili equation is investigated, which can be employed to depict internal wave interactions encompassing both elastic and non-elastic behaviors. Initially, the translational symmetries of the equation are identified by employing Lie symmetry method. Then, by utilizing these symmetries, the discussed model is translated into an ordinary differential equation. The new auxiliary equation approach is then employed to obtain precise solutions, and the results are visually represented. Afterward, the dynamical behavior of the equation is analyzed from multiple perspectives, including bifurcation, chaos, and sensitivity analysis. Bifurcation is conducted at fixed points, while an external force is applied to the dynamical system, leading to the discovery of chaotic behavior using various tools such as 3D and 2D phase plots, time series plots, Poincaré maps, and multistability analysis. Furthermore, sensitive nature of the model is examined at different initial values, revealing the significant sensitivity of the proposed model, where even small changes in the initial conditions result in notable variations. At the end, conclusion is presented.

On the complex properties of the first equation of the Kadomtsev-Petviashvili hierarchy

Abstract This work studies the first equation of the Kadomtsev-Petviashvili (KP) hierarchy. The sine-Gordon expansion method (SGEM) and the rational SGEM (RSGEM) are applied to the governing model. RSGEM is the developed version of SGEM. New complex travelling wave solutions, logarithmic and complex function properties are obtained. Several simulations such as 2D, 3D and contour surfaces of the obtained results are plotted. Physical meanings of these solutions are also reported. Strain conditions are also extracted.

VC-PINN: Variable coefficient physics-informed neural network for forward and inverse problems of PDEs with variable coefficient

The paper proposes a deep learning method specifically dealing with the forward and inverse problem of variable coefficient partial differential equations – variable coefficient physics-informed neural network (VC-PINN). The shortcut connections (ResNet structure) introduced into the network alleviate the “vanishing gradient” problem and unify linear and nonlinear coefficients. The developed method was applied to four equations including the variable coefficient Sine–Gordon (vSG), the generalized variable coefficient Kadomtsev–Petviashvili equation (gvKP), the variable coefficient Korteweg–de Vries equation (vKdV), the variable coefficient Sawada–Kotera equation (vSK). Numerical results show that VC-PINN is successful in the case of high dimensionality, various variable coefficients (polynomials, trigonometric functions, fractions, oscillation attenuation coefficients), and the coexistence of multiple variable coefficients. We also conducted an in-depth analysis of VC-PINN in a combination of theory and numerical experiments, including four aspects: the necessity of ResNet; the relationship between the convexity of variable coefficients and learning; anti-noise analysis; the unity of forward and inverse problems/relationship with standard PINN.

New waveform solutions of Calogero–Degasperis (CD) and potential Kadomtsev–Petviashvili (pKP) equations

New waveform solutions of Calogero–Degasperis (CD) and potential Kadomtsev–Petviashvili (pKP) equations

Exploration of solitary wave solutions of highly nonlinear KDV–KP equation arise in water wave and stability analysis

For intellectual curiosity, in this manuscript, we study the sixth-order highly nonlinear generalized Korteweg–de Vries–Kadomtsev–Petviashvili dynamical model. The Kadomtsev–Petviashvili equation is used in conjunction with the popular fifth-order KDV model. The objective of this study is twofold: Firstly, we formulated the two analytical methods, namely, the unified method and the auxiliary method, to derive the differential families of soliton solutions under different conditions that have never been studied before. The scientific community interested in wave phenomena will find these results useful in understanding the dynamics of nonlinear phenomena in wave theory, optics, engineering, and plasma physics. Secondly, we examine the stability analysis of the selected model. For better understanding, we plotted 2D, 3D, contours and density graphs of these acquired solutions. We anticipate that this work will be a significant step forward in the analysis of highly nonlinear models.

Bilinear auto-Bäcklund transformations and the hybrid localized wave solutions for the ($$3+1$$)-dimensional B-type Kadomtsev–Petviashvili equation

Bilinear auto-Bäcklund transformations and the hybrid localized wave solutions for the ($$3+1$$)-dimensional B-type Kadomtsev–Petviashvili equation

Soliton interactions and Mach reflection in gas bubbles–liquid mixtures

In this study, we assume that blood is assumed to be a viscoelastic and incompressible homogeneous media in which several uniform sized oxygen bubbles are uniformly distributed. Based on this, we establish a (3 + 1)-dimensional modified Kadomtsev–Petviashvili (mKP) equation to describe the long nonlinear pressure waves in the gas bubbles–liquid mixtures. Using bell polynomials, a new bilinear form of the mKP equation is constructed, and then the one- and two-soliton solutions of the equation are obtained by the Hirota method. Via the one-soliton solutions, parametric conditions of the existence of shock wave, elevation and depression solitons, and the Mach reflection characters in the mixtures are discussed. Soliton interactions have been discussed on the basis of the two-soliton solutions. We find that the (i) parallel elastic interactions can exist between the shock and elevation solitons; (ii) oblique elastic interactions can exist between the (a) shock and depression solitons and (b) the elevation and depression solitons; and (iii) oblique inelastic interactions can exist between the two depression solitons.

Novel exact solutions to the $$(3 + 1)$$-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation by using BNNM

Novel exact solutions to the $$(3 + 1)$$-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation by using BNNM

Rare decaying ripple solutions within the KP equation

This paper studies ripple solutions for two versions of the Kadomtsev–Petviashvili (KP) equations. These ripples are classified as curve ripples, lump ripples, and lump chain ripples based on the waveforms. These three forms of ripples just mentioned are present in the KP1 equation, while only curve ripples have been found in the KP2 equation. The most significant feature of these ripples is that the amplitude is proportional to t−2/3 and the propagation distance is proportional to t1/3. In addition, a new curved lump chain is discovered for the first time, whose amplitude and wave width tend to be stable over time.

Rogue wave patterns of Newell type long-wave–short-wave model

The general rogue wave solutions of long-wave–short-wave model are obtained by using the Kadomtsev–Petviashvili (KP) hierarchy reduction method. Unlike previous studies, we have refined the differential operators involved in the solutions by eliminating the recursiveness. Based on the simplified expression, the rogue wave patterns from the second to fifth order are displayed. Specifically, there are N−1 (N≥2) polygonal patterns of Nth-order rogue waves, which are found to be connected to the Yablonskii–Vorob’ev polynomial hierarchy.

Periodic, n-soliton and variable separation solutions for an extended (3+1)-dimensional KP-Boussinesq equation

An extended (3+1)-dimensional Kadomtsev–Petviashvili–Boussinesq equation is studied in this paper to construct periodic solution, n-soliton solution and folded localized excitation. Firstly, with the help of the Hirota’s bilinear method and ansatz, some periodic solutions have been derived. Secondly, taking Burgers equation as an auxiliary function, we have obtained n-soliton solution and n-shock wave. Lastly, we present a new variable separation method for (3+1)-dimensional and higher dimensional models, and use it to derive localized excitation solutions. To be specific, we have constructed various novel structures and discussed the interaction dynamics of folded solitary waves. Compared with the other methods, the variable separation solutions obtained in this paper not only directly give the analytical form of the solution u instead of its potential u_y, but also provide us a straightforward approach to construct localized excitation for higher order dimensional nonlinear partial differential equation.

Conserved vectors and solutions of the two-dimensional potential KP equation

Abstract This article investigates the potential Kadomtsev–Petviashvili (pKP) equation, which describes the evolution of small-amplitude nonlinear long waves with slow transverse coordinate dependence. For the first time, we employ Lie symmetry methods to calculate the Lie point symmetries of the equation, which are then utilized to derive exact solutions through symmetry reductions and with the help of Kudryashov’s method. The solutions obtained include exponential, hyperbolic, elliptic, and rational functions. Furthermore, we provide one-parameter group of transformations for the pKP equation. To gain a better understanding of the nature of each solution, we present 3D, 2D, and density plots. These obtained solutions, along with their associated physical characteristics, offer valuable insights into the propagation of small yet finite amplitude waves in shallow water.In addition, the pKP equation conserved vectors are derived by utilizing the multiplier method and the theorems by Noether and Ibragimov.

Stabilization of the Kawahara–Kadomtsev–Petviashvili equation with time-delayed feedback

Results of stabilization for the higher order of the Kadomtsev-Petviashvili equation are presented in this manuscript. Precisely, we prove with two different approaches that under the presence of a damping mechanism and an internal delay term (anti-damping) the solutions of the Kawahara–Kadomtsev–Petviashvili equation are locally and globally exponentially stable. The main novelty of this work is that we present the optimal constant, as well as the minimal time, that ensures that the energy associated with this system goes to zero exponentially.

A semi-periodic initial-value problem for the Kadomtsev–Petviashvili II equation

We investigate the Cauchy problem on the cylinder, namely the semi-periodic problem where there is periodicity in the x-direction and decay in the y-direction, for the Kadomtsev–Petviashvili II equation by the inverse spectral transform method. For initial data with small L 1 and L 2 norms, assuming the zero mass constraint, this initial-value problem is reduced to a Riemann–Hilbert problem on the boundary of certain infinite strips with shift.