（2＋1）-DIMENSIONAL Tu HIERARCHY AND ITS INTEGRABLE COUPLINGS AS WELL AS THE MULTI-COMPONENT INTEGRABLE HIERARCHY

Under the frame of the(2 1)-dimensional zero curvature equation and Tu model,(2 1)-dimensional Tu hierarchy is obtained.Again by employing a subalgebra of the loop algebra ■_2 the integrable coupling system of the above hierarchy is presented.Finally,A multi-component integrable hierarchy is ob- tained by employing a multi-component loop algebra ■_M.     

一个有限维Hamilton系统的Liouville可积性研究（英文）

Based on a 2 × 2 eigenvalue problem,a set of（1 ＋ 1）-dimensional soliton equations are proposed.Moreover,we obtain a finite dimensional Hamilton system with the help of nonlinearization approach.Then the generating function approach and the way to straighten out of Fm-flow are used to prove the involutivity and the functional independence of conserved integrals for the finite-dimensional Hamilton system,hence,we can verify it is completely integrable in Liouville sense.     

Generalized dromion solutions of a new higher dimensional soliton equation

Introduction Since the pioneering work of Boiti et al.II], the study of soliton-laal structures in highdimensions has attracted much more attation. In particular, for some (2+l)-dimensional integrable models such as the Davey-Stewartson (DS) equationlZ], Kadomtsev- Petviashvill (K P )equation['], Nizhnili- Novikov- VeSelov (NNV) equationI4], the (2 + 1 ) -dimensional breajdng soliton equ&tionl'1, the (2+1)-dimensional long dispersive wave equationI6] and the scalar nonlinearSchrsdinger…     

ON THE USE OF THE PAINLEV PROPERTY FOR OBTAINING MIURA TYPE TRANSFORMATION

We present a method for obtaining an associated hierarchy of evolution equations possessingthe Painleve property from a given hierarchy which possesses the Painleve property. This method isapplied to the classical Boussinesq hierarchy to obtain the Miura type transformation and themodified classical Boussinesq hierarchy. It is also used to construct a large hierarchy of evolutionequations which possess the Paiuleve property and include the classical Boussinesq the JaulentMiodek, the dispersive long wave hierarchy as special cases. All these hierarchies have the samemodified hierarchy.     

A NEW DISCRETE INTEGRABLE COUPLING SYSTEM AND ITS HAMILTONIAN STRUCTURE FOR THE MODIFIED TODA LATTICE HIERARCHY

We present a new discrete integrable coupling system by using the matrix Lax pair U, V ∈ sl(4). A novel spectral problem of modified Toda lattice soliton hierarchy is considered. Then, a new discrete integrable coupling equation hierarchy is obtained through the method of the enlarged Lax pair. Finally, we obtain the Hamiltonian structure of the integrable coupling system of the soliton equation hierarchy using the matrix-form trace identity. This discrete integrable coupling system includes a kind of a modified Toda lattice hierarchy.     

(2+1)维Burgers方程新的复合解

In this paper, a new generalized compound Riccati equations rational expansion method （GCRERE） is proposed. Compared with most existing rational expansion methods and other sophisticated methods, the proposed method is not only recover some known solutions, but also find some new and general complexiton solutions. Being concise and straightforward, it is applied to the （2＋1）-dimensional Burgers equation. As a result, eight families of new exact analytical solutions for this equation are found. The method can also be applied to other nonlinear partial differential equations.     

扩展的可积模型及其约化与达布变换

In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras?H and?T, respectively. By using the loop algebra?H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra?T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra?H to obtain an integrable hierarchy with variable coefficients.     

Generations of integrable hierarchies and exact solutions of related evolution equations with variable coefficients

We first propose a way for generating Lie algebras from which we get a few kinds of reduced 6 6 Lie algebras, denoted by R6, R8 and R1,R6/2, respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra R6 whose compatibility gives rise to an integrable hierarchy with 4- potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra R6 to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again,via using the Lie algebra R62, we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.     

一类非线性演化方程族的可积耦合

Starting from a Tu Guizhang‘s isospectral‘problem, a Lax pair is obtained by means of Tu scheme ( we call it Tu Lax pair ). By applying a gauge transformation between matrices, the Tu Lax pair is changed to its equivalent Lax pair with the traces of spectral matrices being zero, whose compatibility gives rise to a type of Tu hierarchy of equations. By making use of a high order loop algebra constructed by us, an integrable coupling system of the Tu hierarchy of equations are presented. Especially, as reduction cases, the integrable couplings of the celebrated AKNS hierarchy, TD hierarchy and Levi hierarchy are given at the same time.     

On the Cauchy problem and peakons of a two-component Novikov system

We study a two-component Novikov system, which is integrable and can be viewed as a twocomponent generalization of the Novikov equation with cubic nonlinearity. The primary goal of this paper is to understand how multi-component equations, nonlinear dispersive terms and other nonlinear terms affect the dispersive dynamics and the structure of the peaked solitons. We establish the local well-posedness of the Cauchy problem in Besov spaces B_(p,r)~s with 1 p, r +∞, s max{1 + 1/p, 3/2} and Sobolev spaces Hs(R)with s 3/2, and the method is based on the estimates for transport equations and new invariant properties of the system. Furthermore, the blow-up and wave-breaking phenomena of solutions to the Cauchy problem are studied. A blow-up criterion on solutions of the Cauchy problem is demonstrated. In addition, we show that this system admits single-peaked solitons and multi-peaked solitons on the whole line, and the single-peaked solitons on the circle, which are the weak solutions in both senses of the usual weak form and the weak Lax-pair form of the system.     

Integrable couplings of the (2 + 1)-dimensional dispersive long wave hierarchy and its Hamiltonian structure

A higher loop algebra is constructed, from which the integrable couplings associated with the (2 + 1)-dimensional dispersive long wave hierarchy is obtained with the help of the (2 + 1) zero curvature equation generated from one of reduced equations of the self-dual Yang–Mills equations. Furthermore, the Hamiltonian structure of the integrable couplings is worked out by taking use of the variational identity.     

Solitons, chaos and fractals in the (2 + 1)-dimensional dispersive long wave equation

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of a projective Riccati equation approach, the paper obtains several types of exact solutions to the (2 + 1)-dimensional dispersive long wave (DLW) equation which include multiple soliton solution, periodic soliton solution and Weierstrass function solution. Subsequently, several multisolitons are derived and some novel features are revealed by introducing lower-dimensional patterns.     

The ( \frac{G'}{G}) -expansion method and its applications to some nonlinear evolution equations in the mathematical physics

In the present paper, we construct the traveling wave solutions involving parameters for some nonlinear evolution equations in the mathematical physics via the (2+1)-dimensional Painlevé integrable Burgers equations, the (2+1)-dimensional Nizhnik-Novikov-Vesselov equations, the (2+1)-dimensional Boiti-Leon-Pempinelli equations and the (2+1)-dimensional dispersive long wave equations by using a new approach, namely the ( $\frac{G'}{G})$ -expansion method, where G=G(ξ) satisfies a second order linear ordinary differential equation. When the parameters are taken special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by hyperbolic, trigonometric and rational functions.     

Solitons,chaos and fractals in the (2 + 1)-dimensional dispersive long wave equation

For a higher-dimensional integrable nonlinear dynamical system, there are abundant coherent soliton excitations. With the aid of a projective Riccati equation approach, the paper obtains several types of exact solutions to the (2 + 1)-dimensional dispersive long wave (DLW) equation which include multiple soliton solution, periodic soliton solution and Weierstrass function solution. Subsequently, several multisolitons are derived and some novel features are revealed by introducing lower-dimensional patterns.     

分离变量法在变系数(2+1)维方程求解中的应用

分离变量法是求解具有局域相干结构解的有效解析方法.考虑到传播介质的非均匀性和边界的不一致性,变系数(2+1)色散长波方程可以实际地描述宽广的河道或有限深的远海中非线性波的传播.解析研究了变系数(2+1)维色散长波方程.通过分离变量法,得到了该方程组的具有丰富结构的分离变量解.     

新(2+1)-维超可积系统的生成

本文利用二项式残数表示方法生成(2+1)-维超可积系统. 由这些系统得到了一个新的(2+1)-维超孤子族,它能约化为(2+1)-维超非线性Schrodinger方程. 特别地,我们得到两个具有重要物理应用的结果,一个是(2+1)-维超可积耦合方程,另一个是(2+1)-维的扩散方程. 最后借助超迹恒等式给出了新(2+1)-维超可积系统的Hamilton结构.     

Model equation of the theory of solitons

We consider the hierarchy of integrable (1+2)-dimensional equations related to the Lie algebra of vector fields on the line. We construct solutions in quadratures that contain n arbitrary functions of a single argument. A simple equation for the generating function of the hierarchy, which determines the dynamics in negative times and finds applications to second-order spectral problems, is of main interest. Considering its polynomial solutions under the condition that the corresponding potential is regular allows developing a rather general theory of integrable (1+1)-dimensional equations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 29–45, October, 2007.     

和高阶约束相联系的可积系统

对（１＋１）维可积系统，本文在零曲率方程表示理论框架内，给出统一的方法去构造和高阶约束相联系的有限维可积系统，导出这些系统的守恒积分的生成函数，证明它们的可积性，并进而把一族（１＋１）维可积系统中的每一个方程分解为两个可交换的有限维可积的Ｈａｍｉｌｔｏｎ系统。     

广义射影Riccati方程方法与(2+1)维色散长波方程新的精确行波解

助于符号计算软件Maple,通过一种构造非线性偏微分方程更一般形式行波解的直接方法,即改进的广义射影Ricccati方程方法, 求解(2+1)维色散长波方程, 得到该方程的新的更一般形式的行波解, 包括扭状孤波解, 钟状解,孤子解和周期解. 并对部分新形式孤波解画图示意.     

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings.