General Solution and Localized Coherent Soliton Structures of the (2+1)-Dimensional Generalized Davey-Stewarson Equations

In this paper, the variable separation approach is used to obtain localized coherent structures of the (2 1)-dimensional generalized Davey-Stewarson equations: iqt 1/2(qxx qyy) (R S)q = O, Rx=-σ/2|q|2y Sy = -σ/2|q|2/x.Applying a special Backlund transformation and introducing arbitrary functions of the seed solutions, an abundance of the localized structures of this model is derived. By selecting the arbitrary functions appropriately, some special typesof localized excitations such as dromions, dromion lattice, breathers, and instantons are constructed.     

Compacton, Peakon, and Foldon Structures in the (2+1)-Dimensional Nizhnik-Novikov-Veselov Equation

By the use of the extended homogenous balance method, the Backlund transformation for a (2+1)-dimensional integrable model, the(2+1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation, is obtained, and then the NNV equation is transformed into three equations of linear, bilinear, and tri-linear forms, respectively. From the above three equations, a rather general variable separation solution of the model is obtained. Three novel class localized structures of the model are founded by the entrance of two variable-separated arbitrary functions.     

事件空间中Birkhoff系统的参数方程及其第一积分

研究事件空间中Birkhoff系统动力学.在(2n+1)维事件空间中,建立了Birkhoff系统的Pfaff-Birkhoff-d'Alembert原理和Birkhoff参数方程,研究了方程的第一积分,给出了第一积分及其存在条件. 关键词： Birkhoff系统 事件空间 参数方程 第一积分     

New Exact Solution to （3＋1）-Dimensional Burgers Equation

In this Letter, using Ba^ecklund transformation and the new variable separation approach, we find a new general solution to the (3 1)-dimensional Burgers equation. The form of the universal formula obtained from many (2 1)-dimensional systems is extended. Abundant localized coherent structures can be found by seclecting corresponding functions appropriately.     

Solving Integrable Broer-Kaup Equations in （2＋1）-Dimensional Spaces via an Improved Variable Separation Approach

Starting from Baecklund transformation and using Cole-Hopf transformation, we reduce the integrable Broer-Kaup equations in (2 1)-dimensional spaces to a simple linear evolution equation with two arbitrary functions of {x, t} and {y, t} in this paper. And we can obtain some new solutions of the original equations by investigating the simple nonlinear evolution equation, which include the solutions obtained by the variable separation approach.     

Exact Excitation and Abundant Localized Coherent Soliton Structures of (2+1)-Dimensional Perturbed AKNS System

A simple and direct method is applied to solving the (2+1)-dimensional perturbed Ablowitz-Kaup-Newell-Segur system (PAKNS). Starting from a special Backlund transformation and the variable separation approach, we convert the PAKNS system into the simple forms, which are four variable separation equations, then obtain a quite generalsolution. Some special localized coherent structures like fractal dromions and fractal lumps of this model are constructed by selecting some types of lower-dimensional fractal patterns.     

New explicit and exact travelling wave solutions for a system of dispersive long wave equations

A method to construct the new exact solutions of nonlinear partial differential equations (NLPDEs) in a unified way is presented, which is named an improved sine-cosine method. This method is more powerful than the sine-cosine method. Systems of dispersive long wave equations in (1+1) and (2+1) dimensions are chosen to illustrate the method and several types of explicit and exact travelling wave solutions are obtained. These solutions contain Wang's results and other types of solitary wave solutions and new solutions. The method presented here is general and can also be applied to solve more systems of nonlinear partial differential equations, such as the coupled KdV equations.     

A Hierarchy of Integrable Nonlinear Lattice Equations and New Integrable Symplectic Map

A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related tothis spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are givenby nonlinearization method. The binary Bargmann constraint gives rise to a Backlund transformation for the resultingintegrable lattice equations.     

Gauge and integrable theories in loop spaces

We propose an integral formulation of the equations of motion of a large class of field theories which leads in a quite natural and direct way to the construction of conservation laws. The approach is based on generalized non-abelian Stokes theorems for p-form connections, and its appropriate mathematical language is that of loop spaces. The equations of motion are written as the equality of a hyper-volume ordered integral to a hyper-surface ordered integral on the border of that hyper-volume. The approach applies to integrable field theories in (1+1) dimensions, Chern-Simons theories in (2+1) dimensions, and non-abelian gauge theories in (2+1) and (3+1) dimensions. The results presented in this paper are relevant for the understanding of global properties of those theories. As a special byproduct we solve a long standing problem in (3+1)-dimensional Yang-Mills theory, namely the construction of conserved charges, valid for any solution, which are invariant under arbitrary gauge transformations.     

Some Special Types of Solitary Wave Solutions for the (3+1)-Dimensional Kadomt sev-Petviashvilli Equation

Using the standard truncated Painlevé analysis, we have obtained some special types of exact explicit solitary wave solutions of the (3+1)-dimensional Kadomtsev-Petviashvilli equations. Usually, one obtains the single solitary wave solution from the Backlund transformation related to the truncated Painlevé analysis starting from the trivial vacuum solution. In this letter, we find some special types of the multi-solitary wave solution from the truncated Painlevé analysis and the trivial vacuum solution.     

Multisoliton Solutions of the （3＋1）—Dimensional Nizhnik—Novikov—Veselov Equation

Using the standard truncated Painleve analysis,we can obtain a Backlund transformation of the (3 1)-dimensional Nizhnik-Novikov-Veselov (NNV) equation and get some(3 1)-dimensional single-,two- and three-soliton solutions and some new types of multisoliton solutions of the (3 1)-dimensional NNV system from the Backlund transformation and the trivial vacuum solution.     

B?cklund Transformation and Solutions for the Long-Wave and Short-Wave Resonance Equations and Their Extending System

By using Painlevé analysis, we derive the Backlund transformation for the longwave and short-wave resonance equations and their extending systems. Then we take some constraints of the B?ckJund transformation and find some solutions.     

Non-abelian traveling wave solutions for massless QCD2

The field equations for quantum chromodynamics in 1 + 1 dimensions (QCD₂) with massless fermions are shown to admit classical non-abelian traveling wave solutions. In this case, the field equations reduce to the linear Frenet-Serret equations for a curve in the three-space corresponding to an SU(2) subalgebra of the SU(N) gauge group.     

The Matrix Kadomtsev–Petviashvili Equation as a Source of Integrable Nonlinear Equations

Abstract

A new integrable class of Davey–Stewartson type systems of nonlinear partial differential equations (NPDEs) in 2+1 dimensions is derived from the matrix Kadomtsev– Petviashvili equation by means of an asymptotically exact nonlinear reduction method based on Fourier expansion and spatio-temporal rescaling. The integrability by the inverse scattering method is explicitly demonstrated, by applying the reduction technique also to the Lax pair of the starting matrix equation and thereby obtaining the Lax pair for the new class of systems of equations. The characteristics of the reduction method suggest that the new systems are likely to be of applicative relevance. A reduction to a system of two interacting complex fields is briefly described.     

Similarity Reductions for the Davey-Stewartson Equations Using a Direct Method

By the direct method that involves no group theory, the Davey-Stewartson equations (DSE's)in (2+1) dimensions have been reduced to some (1+1)-or 2-dimensional partial differentialequations which have included all the types of symmetry reductions using the standard Liegroup theory. As a special case we obtained the 1-dimensional reducing equations fromDSE's in (2+1) dimensions directly.     

Traversible wormholes in (2 + 1) dimensions

We investigate traversible wormhole solutions to the Einstein field equations in (2 + 1) dimensions. The constraints on the field equations to obtain a wormhole solution are presented and further constraints for traversibility of the wormhole are also given. We show that there is no analog of the (3 + 1)-dimensional Schwarzschild wormhole in (2 + 1) dimensions. For general wormholes, the radial tension and lateral pressure at the throat of the wormhole must be zero, and the energy density must be negative. Two specific wormhole solutions are presented. We perform a stability analysis on the solutions.     

Solitary wave solutions to nonlinear evolution equations in mathematical physics

This paper obtains solitons as well as other solutions to a few nonlinear evolution equations that appear in various areas of mathematical physics. The two analytical integrators that are applied to extract solutions are tan–cot method and functional variable approaches. The soliton solutions can be used in the further study of shallow water waves in (1+1) as well as (2+1) dimensions.     

On Triply Periodic Wave Solutions for(2+1)-Dimensional Boussinesq Equation

By employing Hirota bilinear method and Riemann theta functions of genus one,explicit triply periodic wave solutions for the(2+1)-dimensional Boussinesq equation are constructed under the Backlund transformation u =(1 /6)(u0 1) + 2[ln f(x,y,t)] xx,four kinds of triply periodic wave solutions are derived,and their long wave limit are discussed.The properties of one of the solutions are shown in Fig.1.     

Localized coherent structures of (2+1) dimensional generalizations of soliton systems

We briefly review the recent progress in obtaining (2+1) dimensional integrable generalizations of soliton equations in (1+1) dimensions. Then, we develop an algorithmic procedure to obtain interesting classes of solutions to these systems. In particular using a Painlevé singularity structure analysis approach, we investigate their integrability properties and obtain their appropriate Hirota bilinearized forms. We identify line solitons and from which we introduce the concept of ghost solitons, which are patently boundary effects characteristic of these (2+1) dimensional integrable systems. Generalizing these solutions, we obtain exponentially localized solutions, namely the dromions which are driven by the boundaries. We also point out the interesting possibility that while the physical field itself may not be localized, either the potential or composite fields may get localized. Finally, the possibility of generating an even wider class of localized solutions is hinted by using curved solitons.     

An eighth-order KdV-type equation in (1+1) and (2+1) dimensions: multiple soliton solutions

In this work we study an eighth-order KdV-type equations in (1+1) and (2+1) dimensions. The new equations are derived from the KdV6 hierarchy. We show that these equations give multiple soliton solutions the same as the multiple soliton solutions of the KdV6 hierarchy except for the dispersion relations.