Kadomtsev-Petviashvili System Research Articles

On the line-soliton solutions of a coupled modified Kadomtsev–Petviashvili system in two-layer shallow water

In this paper, the line-soliton solutions of a coupled modified Kadomtsev–Petviashvili (MKP) equation are studied. Based on the tau-functions and Grassmannian Gr(N,M), we present and study the coupled modified Kadomtsev–Petviashvili equation and its coupled soliton solutions v and vˆ which can describe two layers shallow water. By giving examples and theoretical analysis, we give a general classification of the structures of coupled soliton solutions v and vˆ. At the same time, we describe the solutions v, vˆ by Le-diagram, chord diagram and analyze the influence of parameter ϵ on the solutions v, vˆ.

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.

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 a coupled Kadomtsev–Petviashvili system associated with an elliptic curve

AbstractThe coupled Kadomtsev–Petviashvili system associated with an elliptic curve, proposed by Date, Jimbo, and Miwa [J. Phys. Soc. Jpn., 52:766–771, 1983], is reinvestigated within the direct linearization framework, which provides us with more insights into the integrability of this elliptic model from the perspective of a general linear integral equation. As a result, we successfully construct for the elliptic coupled Kadomtsev–Petviashvili system not only a Lax pair composed of differential operators in 2 × 2 matrix form but also multisoliton solutions with phases parametrized by points on the elliptic curve. Dimensional reductions based on the direct linearization, to the elliptic coupled Korteweg–de Vries and Boussinesq systems, are also discussed. In addition, a novel class of solutions is obtained for the ‐type Kadomtsev–Petviashvili equation with nonzero constant background as a by‐product.

Integrability, similarity reductions and solutions for a (3+1)-dimensional modified Kadomtsev–Petviashvili system with variable coefficients

In this study, a (3+1)-dimensional modified Kadomtsev–Petviashvili system with variable-coefficients (vcmKP) arising in many fields of science like plasma physics, optics, and fluid mechanics is studied. The consistent Riccati expansion solvability technique is applied to the vcmKP system to obtain necessary integrability conditions between the variable coefficients. Moreover, many novel different types of travelling wave solutions are given. Additionally, the vcmKP system is reduced to a nonlinear third order ordinary differential equation and many other new solutions are obtained. Finally, some plots are given to illustrate how the arbitrary function choices affect the propagation of the wave solutions.

Reflecting upon some electromagnetic waves in a ferromagnetic film via a variable-coefficient modified Kadomtsev-Petviashvili system

Kadomtsev-Petviashvili-type models are employed in magneto-optics, ferromagnetism, plasma physics, fluid mechanics, etc. For a variable-coefficient modified Kadomtsev-Petviashvili system modelling the electromagnetic waves in an isotropic charge-free infinite ferromagnetic thin film, exciting the magnetization and with the potential application in magneto-optic recording, we symbolically compute out two sets of the similarity reductions, each of which can be from that system to a known ordinary differential equation. Both sets rely on the constant/variable coefficients for that system.

The discrete Burgers equation

In this paper I give a short review on the continuous, semi-discrete and discrete Burgers equations, which are featured as integrable equations that are linearisable. The review focuses more on connections of these three kinds of equations and connections with (2+1)-dimensional systems. The continuous Burgers hierarchy is a reduction of the derivative nonlinear Schrödinger hierarchy. The later is a result of the squared eigenfunction symmetry constraint of the modified Kadomtsev–Petviashvili system. The semi-discrete Burgers hierarchy can be considered as Bäcklund transformations of the continuous Burgers hierarchy. They are also reductions of the relativistic Toda hierarchy, which is a consequence of the squared eigenfunction symmetry constraint of the semi-discrete modified Kadomtsev–Petviashvili system. The fully discrete Burgers equation is the Bianchi identity of the Bäcklund transformation for the continuous Burgers hierarchy. It is 3-point lattice equation, three-dimensionally consistent, and linearisable.

The Sylvester Equation and Kadomtsev–Petviashvili System

In this paper, we seek connections between the Sylvester equation and Kadomtsev–Petviashvili system. By introducing Sylvester equation LM are bold, please chekc if bold neceaasry, if not, please remove all bold of equation −MK = rsT together with an evolution equation set of r and s, master function S(i,j)=sTKjC(I + MC)−1Lir is used to construct the Kadomtsev–Petviashvili system, including the Kadomtsev–Petviashvili equation, modified Kadomtsev–Petviashvili equation and Schwarzian Kadomtsev–Petviashvili equation. The matrix M provides τ-function by τ = |I + MC|. With the help of some recurrence relations, the reductions to the Korteweg–de Vries and Boussinesq systems are discussed.

Tilted Optical-Terahertz Solitons of the Yajima–Oikawa–Kadomtsev–Petviashvili System

Tilted Optical-Terahertz Solitons of the Yajima–Oikawa–Kadomtsev–Petviashvili System

Bilinear auto-Bäcklund transformation, soliton and periodic-wave solutions for a (2+1)-dimensional generalized Kadomtsev–Petviashvili system in fluid mechanics and plasma physics

Fluid mechanics and plasma physics are currently interesting. In fluid mechanics and plasma physics, a (2+1)-dimensional generalized Kadomtsev–Petviashvili system is hereby investigated via symbolic computation. Based on the Hirota method, a bilinear auto-Bäcklund transformation is determined, along with some soliton solutions. Via the Hirota–Riemann method, periodic-wave solutions are derived. We also disscuss the relation between the periodic-wave and soliton solutions, i.e., how the periodic-wave solutions approach the soliton solutions under a limiting procedure. Our results might be useful in understanding certain nonlinear phenomena in fluid mechanics and plasma physics.

Optical Terahertz Lumps of a Yajima–Oikawa–Kadomtsev–Petviashvili System

Optical Terahertz Lumps of a Yajima–Oikawa–Kadomtsev–Petviashvili System

Certain electromagnetic waves in a ferromagnetic film

Electromagnetic-wave/ferromagnetic-material investigations with magneto-optics are carried out in magneto-optic recording, computer data-storage, waveguides, etc. In this paper, we investigate a variable-coefficient modified Kadomtsev–Petviashvili system for certain electromagnetic waves in an isotropic charge-free infinite ferromagnetic thin film with the potential application in magneto-optic recording. With respect to the electromagnetic waves in the ferromagnetic film, symbolic computation helps us work out (1) two sets of the variable-coefficient-dependent auto-Bäcklund transformations via the generalized Laurent series, along with some solitonic features, (2) one set of the variable-coefficient-dependent bilinear forms and (3) two sets of the variable-coefficient-dependent N-solitonic solutions with N being a positive integer. We note that the electromagnetic field can excite the magnetization of a ferromagnetic medium in the form of solitons with the applications in magneto-optic recording. Relevant constraints on the variable coefficients are presented. We hope that the results could help the studies on magneto-optic recording, computer data-storage and waveguides.

Invariance Analysis, Exact Solution and Conservation Laws of (2 + 1) Dim Fractional Kadomtsev-Petviashvili (KP) System

In this work, a Lie group reduction for a (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the Lie symmetry method with Riemann Liouville derivative. After reducing the system into a two-dimensional nonlinear fractional partial differential system (NLFPDEs), the power series (PS) method is applied to obtain the exact solution. Further the obtained power series solution is analyzed for convergence. Then, using the new conservation theorem with a generalized Noether’s operator, the conservation laws of the KP system are obtained.

Magneto-optical/ferromagnetic-material computation: Bäcklund transformations, bilinear forms and [formula omitted] solitons for a generalized (3+1)-dimensional variable-coefficient modified Kadomtsev–Petviashvili system

Of current interest, magneto-optics deals with the phenomena associated with magnetic effects on matter as it emits light, having the potential applications in computer data-storage and waveguides, while ferromagnetic materials are the ones displaying ferromagnetism, such as the various forms of iron, steel, cobalt, nickel, and their alloys. In this paper, in magneto-optics, ferromagnetism, fluid mechanics and plasma physics, we investigate a generalized (3+1)-dimensional variable-coefficient modified Kadomtsev–Petviashvili system for the electromagnetic waves in a ferromagnetic material, or water waves, or dust-acoustic/ion-acoustic/dust–ion-acoustic waves in a plasma. Special cases of the system in those fields are listed out, such as one special case in magneto-optics, which describes the electromagnetic waves in an isotropic charge-free ferromagnetic thin film with the potential application in magneto-optic recording. With symbolic computation, we work out (1) two sets of the variable-coefficient-dependent auto-Bäcklund transformations along with some solitonic features, (2) the variable-coefficient-dependent bilinear forms with the Hirota method and (3) two branches of the variable-coefficient-dependent N-soliton solutions with N being a positive integer. Relevant constraints on the variable coefficients are presented.

Symmetry Breaking Soliton, Breather, and Lump Solutions of a Nonlocal Kadomtsev–Petviashvili System

The Kadomtsev–Petviashvili equation is one of the well-studied models of nonlinear waves in dispersive media and in multicomponent plasmas. In this paper, the coupled Alice-Bob system of the Kadomtsev–Petviashvili equation is first constructed via the parity with a shift of the space variable x and time reversal with a delay. By introducing an extended Bäcklund transformation, symmetry breaking soliton, symmetry breaking breather, and symmetry breaking lump solutions for this system are presented through the established Hirota bilinear form. According to the corresponding constants in the involved ansatz function, a few fascinating symmetry breaking structures of the presented explicit solutions are shown.

Modulation instability analysis and soliton solutions of an integrable coupled nonlinear Schrödinger system

In this paper, an integrable coupled nonlinear Schrodinger system is investigated, which is derived from the integrable Kadomtsev–Petviashvili system, and can be used to describe the stability of soliton in a nonlinear media with weak dispersion. By using the Hirota bilinear method, we derive the exact bilinear formalism and soliton solutions of the system, respectively. Furthermore, we also obtain the linear stability analysis via analyzing the stability condition of the system. Finally, we discuss two kinds of interaction phenomena between solitary wave solutions. It is hoped that our results can be used to enrich the dynamical of the nonlinear Schrodinger system.

Integrable hierarchies of Heisenberg ferromagnet equation

In this paper we consider the coupled Kadomtsev-Petviashvili system. From compatibility conditions we obtain the form of matrix operators. After using a gauge transformation, obtained a new type of Lax representation for the hierarchy of Heisenberg ferromagnet equation, which is equivalent to the gauge coupled Kadomtsev-Petviashvili system.

Nonlinear ion acoustic waves scattered by vortexes

The Kadomtsev–Petviashvili (KP) hierarchy is the archetype of infinite-dimensional integrable systems, which describes nonlinear ion acoustic waves in two-dimensional space. This remarkably ordered system resides on a singular submanifold (leaf) embedded in a larger phase space of more general ion acoustic waves (low-frequency electrostatic perturbations). The KP hierarchy is characterized not only by small amplitudes but also by irrotational (zero-vorticity) velocity fields. In fact, the KP equation is derived by eliminating vorticity at every order of the reductive perturbation. Here, we modify the scaling of the velocity field so as to introduce a vortex term. The newly derived system of equations consists of a generalized three-dimensional KP equation and a two-dimensional vortex equation. The former describes ‘scattering’ of vortex-free waves by ambient vortexes that are determined by the latter. We say that the vortexes are ‘ambient’ because they do not receive reciprocal reactions from the waves (i.e., the vortex equation is independent of the wave fields). This model describes a minimal departure from the integrable KP system. By the Painlevé test, we delineate how the vorticity term violates integrability, bringing about an essential three-dimensionality to the solutions. By numerical simulation, we show how the solitons are scattered by vortexes and become chaotic.

New exact solutions and conservation laws of a coupled Kadomtsev–Petviashvili system

This paper obtains exact solutions of a new coupled Kadomtsev–Petviashvili system, which arises in the analysis of various problems in fluid mechanics, theoretical physics and many scientific applications. Lie symmetry method along with the (G′/G)-expansion method is employed to find the travelling wave solutions of the underlying system. In addition, we derive the conservation laws of the coupled Kadomtsev–Petviashvili system using the multiplier method.

Pfaffianized systems for a generalized Kadomtsev–Petviashvili equation

A one-parameter class of Pfaffian extensions is made for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (KP) equation by the Pfaffianization procedure. Wronski-type Pfaffian and Gramm-type Pfaffian solutions of the resulting Pfaffianized bilinear systems of the generalized KP equation are constructed. Under a dimensional reduction, our results yield a class of Pfaffianized bilinear systems for the KP equation, which contains the standard Pfaffianized bilinear KP system as a particular example. Two kinds of Pfaffian identities provide the basis for our analysis.