Kadomtsev-Petviashvili Hierarchy Research Articles

On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms

On the Cauchy problem for a Kadomtsev-Petviashvili hierarchy on non-formal operators and its relation with a group of diffeomorphisms

Rational and semi-rational solutions to the nonlocal Davey–Stewartson III equation

We utilize the bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction technique to obtain solutions for the nonlocal DS III equation. These solutions are elegantly expressed in terms of an N×N determinant, incorporating various parameter reduction constraints. It is important to note that these solutions are nontrivial and do not necessarily satisfy the local DS III equation. In the case of an even value of N, we derive solutions on constant backgrounds, while for odd values of N, solutions on periodic backgrounds are generated. To gain a deeper understanding of the dynamics involved, we conduct a thorough analysis of the derived solutions, accompanied by informative plots. Overall, our findings showcase the power of the bilinear method and KP hierarchy reduction technique in obtaining meaningful solutions for the nonlocal DS III equation, shedding light on its complex behavior and enriching our understanding of the system.

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.

General Soliton and (Semi-)Rational Solutions of a (2+1)-Dimensional Sinh-Gordon Equation

In this paper, we investigate solutions of a (2+1)-dimensional sinh-Gordon equation. General solitons and (semi-)rational solutions are derived by the combination of Hirota’s bilinear method and Kadomtsev-Petviashvili hierarchy reduction approach. General solutions are expressed as Ntimes N Gram-type determinants. When the determinant size N is even, we generate solitons, line breathers, and (semi-)rational solutions located on constant backgrounds. In particular, through the asymptotic analysis we prove that the collision of solitons are completely elastic. When N is odd, we derive exact solutions on periodic backgrounds. The dynamical behaviors of those derived solutions are analyzed with plots. For rational solutions, we display the interaction of lumps. For semi-rational solutions, we find the interaction solutions between lumps and solitons.

A Kadomtsev–Petviashvili Hierarchy Driven by Equation Manifolds

A Kadomtsev–Petviashvili Hierarchy Driven by Equation Manifolds

Hybrid-wave solutions for a (2 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics and plasma physics

In this paper, a (2 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics and plasma physics is studied. Gram-type solutions are derived via the bilinear Kadomtsev-Petviashvili hierarchy reduction method. Taking different parameter conditions in the Gram-type solutions, we construct the Y-shaped breather solutions and two types of the hybrid-wave solutions. Asymptotic forms for the aforementioned solutions are given. Based on the asymptotic forms, influences of the variable coefficients on the interactions of the breathers and solitons are studied. We obtain three types of the hybrid-wave solutions, which consist of several breathers and solitons. When those breathers and solitons interact, they form the evolving polyhedral arrangement. Changes of the entire arrangement of the breathers and solitons, and the processes of fission or fusion, are discussed and presented.

Gram-type, three-breather and hybrid solutions for a (3＋1)-dimensional generalized variable-coefficient shallow water wave equation

In this paper, a (3＋1)-dimensional generalized variable-coefficient shallow water wave equation is studied, which characterizes the flow below a pressure surface in a fluid. Through the Kadomtsev–Petviashvili hierarchy reduction, we construct some Gram-type solutions, three-breather solutions and three kinds of the hybrid solutions. Via the asymptotic analysis, we choose certain parameters to make the three connected breathers parallel to the x axis and reduce the first-order breather to the first-order homoclinic orbit on the x–z plane. Interaction among the three breathers is displayed and we observe that the one breather splits into the two breathers on the x–y plane, where x, y, z are the scaled spatial coordinates. For the three hybrid solutions, we reduce the V-shaped soliton to the one kink soliton on the y–z plane and show that the one breather converts into the one kink soliton on the y–z plane. Asymptotic analysis indicates that the interaction between the V-shaped soliton and kink soliton leads to the increase/decrease of the constant background of the V-shaped soliton as the constant background of the kink soliton decrease/increase. Furthermore, we present the two interactions among the V-shaped soliton and three breathers.

On the study of the higher-order and multiple lump/rogue waves to the (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation

In the text, rational solutions to a (3 + 1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in determinant form are solved by means of the Kadomtsev-Petviashvili hierarchy reduction method and Hirota’s bilinear form. The first-order, higher-order, multi-lump waves and multi-rogue waves on the basis of the solution in Gramian are presented in graphs. Periodic and parabolic line rogue waves are obtained by choosing different dispersion coefficient functions. The second-order rogue waves are also depicted, which can be seen as the composition of two first-order rogue waves. For multiple rogue waves, taking the periodic and parabolic line rogue waves as examples, it can be observed the change tendency of amplitudes of the intersection region is different from that in branches of line rogue waves. Besides, propagation of the first-order lump wave, interaction between one-lump-soliton and one-rogue-wave, and two kinds of collisions of two lump waves, i.e., elastic and inelastic, are also shown in the plane.

Solutions to the SU([formula omitted]) self-dual Yang–Mills equation

In this paper we aim to derive solutions for the SU(N) self-dual Yang–Mills (SDYM) equation with arbitrary N. A set of noncommutative relations are introduced to construct a matrix equation that can be reduced to the SDYM equation. It is shown that these relations can be generated from two different Sylvester equations, which correspond to the two Cauchy matrix schemes for the (matrix) Kadomtsev–Petviashvili hierarchy and the (matrix) Ablowitz–Kaup–Newell–Segur hierarchy, respectively. In each Cauchy matrix scheme we investigate the possible reductions that can lead to the SU(N) SDYM equation and also analyze the physical significance of some solutions, i.e. being Hermitian, positive-definite and of determinant being one.

Dark soliton solutions for a [formula omitted]-dimensional nonlocal Davey–Stewartson system

A (2+1)-dimensional nonlocal Davey–Stewartson system has been considered in this paper. Dark-soliton solutions in Gram determinant for such nonlocal Davey–Stewartson system have been demonstrated with the aid of the Kadomtsev–Petviashvili hierarchy reduction. According to the dark-soliton solutions, interactions between the dark–dark, antidark–antidark and dark–antidark solitons have been graphically presented. The interactions between two solitons (two dark, one dark and one anti-dark, two anti-dark) are all parallel. Furthermore, various types of the interactions among dark four solitons are also demonstrated: interactions between double dark two solitons, between double antidark two solitons, between dark two solitons and antidark two solitons, and between there mixed types. The interactions are all elastic.

General rogue waves in the AB system

In this work, general rogue wave solutions in the AB system are constructed by means of the Kadomtsev–Petviashvili hierarchy reduction method and explicit representations of these solutions are explored in terms of determinants whose matrix elements are fundamental Schur polynomials. Moreover, the dynamics of these rogue waves are investigated graphically by different choices of the free parameters. In particular, we observe that the rogue wave in the second component B is actually a four peaky-shaped rogue wave, different form the eye-shaped rogue wave in NLSE.

$$\varvec{N}$$-soliton, breathers, lumps and interaction solutions for a time-variable coefficients integrable equation in Kadomtsev–Petviashvili hierarchy

$$\varvec{N}$$-soliton, breathers, lumps and interaction solutions for a time-variable coefficients integrable equation in Kadomtsev–Petviashvili hierarchy

The generalized Giambelli formula and polynomial KP and CKP tau-functions

The first part of the paper is devoted to two descriptions of all polynomial tau-functions of the Kadomtsev-Petviashvili (KP) hierarchy: by a generalized Jacobi-Trudi formula, and a generalized Giambelli formula. We use the latter formula in the second part to obtain all polynomial tau-functions of the CKP hierarchy and its n-reductions. In particular, for we find all polynomial tau-functions of the Kaup-Kupershmidt hierarchy.

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.

Analysis of analytical solutions of fractional Date-Jimbo-Kashiwara-Miwa equation

In this study, the authors solved the (2+1)-dimensional fractional Date-Jimbo-Kashiwara-Miwa equation for nonlinear dispersive waves in an inhomogeneous medium, which is an extension of the Kadomtsev-Petviashvili hierarchy. A powerful analytical technique, the new auxiliary equation method, is employed to develop the class of soliton solutions to the considered equation which contains dark, bright, and periodic wave solutions. A comparative analysis is done by using two fractional derivatives, namely, β and Atangana-Baleanu. While using this technique, the nonlinear ordinary differential equation in integer order is formed using chain rule from the fractional Date-Jimbo-Kashiwara-Miwa equation. There is no more a requirement to perform any normalization or discretization because of the chain rule. These obtained results are graphically portrayed such as 3D and 2D using suitable parametric values. The exact solutions to this equation are necessary to comprehend wave behavior in physical models. Moreover, a sensitive analysis to the fractional dynamical system is performed and graphically shown the behavior on initial conditions.

Gelfand–Dickey hierarchy, generalized BGW tau-function, and W-constraints

Let be an integer. The generalized Brezin–Gross–Witten (BGW) tau-function for the Gelfand–Dickey hierarchy of dependent variables (aka the r-reduced Kadomtsev–Petviashvili hierarchy) is defined as a particular tau-function that depends on constant parameters . In this paper we show that this tau-function satisfies a family of linear equations, called the W-constraints of the second kind. The operators giving rise to the linear equations also depend on constant parameters. We show that there is a one-to-one correspondence between the two sets of parameters.

General high-order lump solutions and their dynamics in the Levi equations

General high-order lump solutions are derived for the Levi equations based on the Hirota bilinear method and Kadomtsev-Petviashvili (KP) hierarchy reduction technique. These lump solutions are given in terms of Gram determinants whose matrix elements are connected to Schur polynomials. Thus, our solutions have explicit algebraic expressions. Their dynamic behaviors are analyzed by using density maps. It is shown that when the absolute value of one group of these internal parameters in the lump solutions is very large, lump solutions exhibit obvious geometric structures. Interestingly, we have shown that their initial and middle state solutions possess various exciting geometric patterns, including hexagon, decagon, tetradecagon, etc and other quasi-structures in addition to the standard triangle, pentagon type patterns. Because the internal parameters are not complex conjugates of each other, the dynamic behaviors of solutions are richer. These results make several contributions to the current literature and have a number of important implications for further analysis of fluid dynamics in non-homogeneous media.

Lagrangian 3-form structure for the Darboux system and the KP hierarchy

A Lagrangian multiform structure is established for a generalisation of the Darboux system describing orthogonal curvilinear coordinate systems. It has been shown in the past that this system of coupled PDEs is in fact an encoding of the entire Kadomtsev–Petviashvili (KP) hierarchy in terms so-called Miwa variables. Thus, in providing a Lagrangian description of this multidimensionally consistent system amounts to a new Lagrangian 3-form structure for the continuous KP system. A generalisation to the matrix (also known as non-Abelian) KP system is discussed.

On (t2, t3)-Zakharov–Shabat equations of generalized Kadomtsev–Petviashvili hierarchies

We review the integration of the Kadomtsev–Petviashvili (KP) hierarchy in several non-standard contexts. Specifically, we consider KP in the following associative differential algebras: an algebra equipped with a nilpotent derivation, an algebra of functions equipped with a derivation that generalizes the gradient operator, an algebra of quaternion-valued functions, a differential Lie algebra, an algebra of polynomials equipped with the Pincherle differential, and a Moyal algebra. In all these cases, we can formulate and solve the Cauchy problem of the KP hierarchy. In addition, in each of these cases, we derive different Zakharov–Shabat (t2, t3)-equations—that is, different Kadomtsev–Petviashvili equations—and we prove the existence of solutions arising from solutions to the corresponding KP hierarchy.

Dynamic mechanism of nonlinear waves for the (3+1)-dimensional generalized variable-coefficient shallow water wave equation

Under investigation in this paper is a (3+1)-dimensional generalized variable-coefficient shallow water wave equation, which can be used to describe the flow below a pressure surface in oceanography and atmospheric science. Employing the Kadomtsev-Petviashvili hierarchy reduction, we obtain the breather and lump solutions in terms of Grammian. We investigate the generation mechanism and conversion of the breathers, lumps and rogue waves. We find that the breather is produced by the superposition of three parts: The soliton part, the periodic wave part and the background part. The angle between the soliton part and the periodic wave part affects the shape of the breather. Considering the influences of the variable coefficients, we observe the breathers propagating on the periodic backgrounds, with double peaks and the breathers propagating periodic with time, respectively. Taking the long-wave limits, we get the rational solutions which describe the lumps. We find that the characteristic lines keep unchanged on the x − y plane, which means that the lump is similar to a part of the breather. Linear rogue waves only appear on the y − z plane.