B-type Kadomtsev Petviashvili Research Articles

Dynamics of transformed nonlinear waves for the (2+1)-dimensional pKP-BKP equation: interactions and molecular waves

Abstract In this paper, the dynamical behaviors of transformed nonlinear waves for the (2+1)-dimensional combined potential Kadomtsev-Petviashvili and B-type Kadomtsev-Petviashvili (pKP-BKP) equation are investigated, which can be used to reveal the nonlinear wave phenomena in nonlinear optics, plasma physics and hydrodynamics. The breath-wave and the lump solutions are constructed by means of the soliton solutions. The conversion mechanism for the breath-waves is systematically analyzed, which leads to several new kink-shaped nonlinear waves. The gradient relationships of these transformed waves are revealed by a Riemannian circle. Through the analysis of the nonlinear superposition between the periodic wave component and the kink solitary wave component, the dynamical characteristics including the formation mechanism, oscillation and locality for the nonlinear waves are investigated. The time-varying properties of transformed waves are showed by the study of time variables. By virtue of the two breath-wave solution, several interactions including elastic and inelastic collisions between two nonlinear waves are studied. In particular, some transformed molecular waves encompassing the non-, semi- and full-transition modes are presented with the aid of velocity resonance. The results can help us further understand the complex nonlinear waves existing in the integrable systems.

Stability analysis and retrieval of new solitary waves of (2+1)- and (3+1)-dimensional potential Kadomtsev–Petviashvili and B-type Kadomtsev–Petviashvili equations using auxiliary equation technique

The Kadomtsev–Petviashvili (KP) equations are nonlinear partial differential equations which are widely used for the modeling of wave propagation in hydrodynamic and plasma systems. This study aims to make a valuable contribution to the literature by providing new solitary waves to the (2+1)- and (3+1)-dimensional potential Kadomtsev–Petviashvili (pKP)-B-type Kadomtsev–Petviashvili (BKP) equations. For this, the auxiliary equation method associated with Bernoulli equation is used and new solutions for the considered equations are obtained. The stability of obtained solutions is demonstrated using nonlinear analysis. It is shown that this method for the considered pKP–BKP equations is an important step forward in an overall mathematical framework for similar equations.

Decomposition solutions and Bäcklund transformations of the B-type and C-type Kadomtsev–Petviashvili equations

This paper introduces a modified formal variable separation approach, showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev–Petviashvili (BKP) equation. Through the application of this approach, we successfully ascertain decomposition solutions, Bäcklund transformations, the Lax pair, and the linear superposition solution associated with the aforementioned equation. Furthermore, we expand the utilization of this technique to the C-type Kadomtsev–Petviashvili (CKP) equation, leading to the derivation of decomposition solutions, Bäcklund transformations, and the Lax pair specific to this equation. The results obtained not only underscore the efficacy of the proposed approach, but also highlight its potential in offering a profound comprehension of integrable behaviors in nonlinear systems. Moreover, this approach demonstrates an efficient framework for establishing interrelations between diverse systems.

Breather wave solutions for an integrable (3+1)-dimensional combined pKP–BKP equation

This work is about the retrieval of breather wave solutions for the integrable (3+1)-dimensional combined potential Kadomtsev–Petviashvili (pKP) equation with B-type Kadomtsev–Petviashvili (BKP) equation, called pKP–BKP equation. We also provide breather wave solutions for two other (3+1)-dimensional reduced forms of the aforementioned pKP–BKP equation. Moreover, we furnish a variety of other solutions that include kink, periodic, and exponential solutions for these two reduced equations. The derived solutions for all examined models are all depicted through the Maple software.

N-soliton, soliton molecules, Y-type soliton, periodic lump and other wave solutions of the new reduced B-type Kadomtsev–Petviashvili equation for shallow water waves

N-soliton, soliton molecules, Y-type soliton, periodic lump and other wave solutions of the new reduced B-type Kadomtsev–Petviashvili equation for shallow water waves

The generalized (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation: resonant multiple soliton, N-soliton, soliton molecules and the interaction solutions

The generalized (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation: resonant multiple soliton, N-soliton, soliton molecules and the interaction solutions

Polynomial Tau-Functions of the n-th Sawada–Kotera Hierarchy

We give a review of the B-type Kadomtsev–Petviashvili (BKP) hierarchy and find all polynomial tau-functions of the n-th reduced BKP hierarchy (=n-th Sawada–Kotera hierarchy). The name comes from the fact that, for n=3, the simplest equation of the hierarchy is the famous Sawada–Kotera equation.

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.

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

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

Complexiton solutions, kink soliton and breather-wave solutions for the (2+1)-dimensional combined potential Kadomtsev-Petviashvili with B-type Kadomtsev-Petviashvili equation

In this paper, the (2+1)-dimensional combined potential Kadomtsev-Petviashvili with B-type Kadomtsev-Petviashvili equation is an object of the research. Based on the extended homogeneous balancing method, Auto-Bäcklund transformations are obtained in two cases. Subsequently, with the help of these transformations, we obtain various explicit solutions of this equation. We attain complexiton solutions consisting of exponential, hyperbolic and trigonometric solutions from the Hirota bilinear form of this equation through the extended transformed rational function method. Also, we derive one-kink and two-kink soliton solutions by Maple symbolic calculation and the breather-wave solution via the extended homoclinic test approach. In addition, 3D graphics and density plots of the obtained solutions are depicted to illustrate the dynamical features of these solutions.

Painlevé analysis, multi-kinks and hybrid nonlinear waves for an extended (3 + 1)-dimensional B-type Kadomtsev-Petviashvili equation

In this paper, we study an extended (3+1)-dimensional B-type Kadomtsev-Petviashvili equation having applications in diverse scientific fields. Painlevé anlysis is carried out to test the integrability of the model under consideration. Hirota’s simplified technique is used to investigate one, two and three kink-soliton solutions. Using a dependent variable transformation, bilinear form of the model is obtained which is then used to report lump and lump interaction solutions with periodic and kink waves. The dynamical properties and nature of obtained solutions is comprehensively studied using 3d and 2d graphs.

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.

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.

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.

Nonlinear Dynamic Behaviors of the (3+1)-Dimensional B-Type Kadomtsev—Petviashvili Equation in Fluid Mechanics

This paper provides an investigation on nonlinear dynamic behaviors of the (3+1)-dimensional B-type Kadomtsev—Petviashvili equation, which is used to model the propagation of weakly dispersive waves in a fluid. With the help of the Cole—Hopf transform, the Hirota bilinear equation is established, then the symbolic computation with the ansatz function schemes is employed to search for the diverse exact solutions. Some new results such as the multi-wave complexiton, multi-wave, and periodic lump solutions are found. Furthermore, the abundant traveling wave solutions such as the dark wave, bright-dark wave, and singular periodic wave solutions are also constructed by applying the sub-equation method. Finally, the nonlinear dynamic behaviors of the solutions are presented through the 3-D plots, 2-D contours, and 2-D curves and their corresponding physical characteristics are also elaborated. To our knowledge, the obtained solutions in this work are all new, which are not reported elsewhere. The methods applied in this study can be used to investigate the other PDEs arising in physics.

Pfaffian, soliton, hybrid and periodic-wave solutions for a ($$3+1$$)-dimensional B-type Kadomtsev–Petviashvili equation in fluid mechanics

Pfaffian, soliton, hybrid and periodic-wave solutions for a ($$3+1$$)-dimensional B-type Kadomtsev–Petviashvili equation in fluid mechanics

Resonant solitons of the B-type Kadomtsev-Petviashvili equation

In this letter, we study the resonant soliton interactions for the B-type Kadomtsev-Petviashvili equation. For the interaction between two solitons, we obtain two kinds of resonant interaction in terms of the value of Aij which determines the interaction intensity. Using the asymptotic analysis, we derive the exact expression of resonant soliton branch. One-resonant three soliton interaction is also discussed. Some graphic analysis are discussed to show these resonant interactions.

Variable separation solution and multi-valued soliton of an extended (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

Variable separation solution and multi-valued soliton of an extended (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

Dynamics of transformed nonlinear waves in the ($$3 + 1$$)-dimensional B-type Kadomtsev–Petviashvili equation II: interactions and molecular waves

Dynamics of transformed nonlinear waves in the ($$3 + 1$$)-dimensional B-type Kadomtsev–Petviashvili equation II: interactions and molecular waves