A multi-component matrix loop algebra and the multi-component Kaup–Newell (KN) hierarchy,as well as its integrable coupling system

Construction of a type of multi-component matrix loop algebra is devoted to establishing an isospectral problem. By making use of Tu scheme, the integrable multi-component KN hierarchy of soliton equation is obtained. Further, the Hamiltonian structure of the Liouville integrable multi-component hierarchy is worked. Finally, an expanding loop algebra of the above algebra is presented, which is used to work out the multi-component integrable coupling system of the multi-component KN hierarchy that contains an arbitrary positive integer M.     

A generalized AKNS hierarchy and its bi-Hamiltonian structures

First we construct a new isospectral problem with 8 potentials in the present paper. And then a new Lax pair is presented. By making use of Tu scheme, a class of new soliton hierarchy of equations is derived, which is integrable in the sense of Liouville and possesses bi-Hamiltonian structures. After making some reductions, the well-known AKNS hierarchy and other hierarchies of evolution equations are obtained. Finally, in order to illustrate that soliton hierarchy obtained in the paper possesses bi-Hamiltonian structures exactly, we prove that the linear combination of two-Hamiltonian operators admitted are also a Hamiltonian operator constantly. We point out that two Hamiltonian operators obtained of the system are directly derived from a recurrence relations, not from a recurrence operator.     

一族Liouville可积系及其3-Hamilton结构

构造了loop代数A↑～1的一个高阶子代数,设计了一个新的Lax对,利用屠格式获得了含8个位势的孤立子方程族;利用Gauteax导数直接验证了所得3个辛算子的线性组合仍为辛算子．因此该孤立族具有3-Hamilton结构,具有无穷多个对合的公共守恒密度,故Liouville可积．作为约化情形,得到了2个可积系,其中之一是著名的AKNS方程族．     

AN INTEGRABLE HIERARCHY AND ITS EXPANDING LAX INTEGRABLE MODEL

In general, Liouville integrable hierarchies of evolution equations were obtained by choosing proper U in zero curvature frame Ut - Vx + [U, V] = 0 first. But in the present paper, a new Liouville integrable hierarchy possessing bi-Hamiltonian structure is obtained by choosing V with derivatives in x and spectral potentials. Then integrable coupling, i.e. expanding Lax integrable model of the hierarchy obtained is presented by constructing a subalgebra of loop algebra A2.     

TWO CLASSES OF INTEGRABLE COUPLING FOR AKNS HIERARCHY

A set of new matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A 2M . Then we use the idea of enlarging spectral problems to make an enlarged spectral problems. It follows that the multi-component AKNS hierarchy is presented. Further, two classes of integrable coupling of the AKNS hierarchy are obtained by enlarging spectral problems.     

一个Lie代数的子代数及其相关的两类Loop代数

本文构造了Lie代数A2的一个子代数A2,通过选取恰当的基元阶数得到相应的一个loop代数A2,由此设计一个等谱问题,利用屠格式得到了一个新的Liouville可积的Hamilton方程族.作为其约化情形,得到了一个非线性有理分式型演化方程.再由一个矩阵变换,得到了换位运算与A2等价的Lie代数A1的一个子代数A1,将A1再扩展成一个新的高维loop代数G,利用G获得了所得方程族的一类扩展可积系统.     

一类孤子方程族及其多个Hamilton结构

本文建立了一个含11个位势的新的等谱问题,得到了一组新的Lax对,由此得到一类新的孤子方程族.该族是Liouville可积的,具有4-Hamilton结构,且循环算子的共轭算子是一个遗传对称算子.另外,为确切说明所得方程族是一个4-Hamilton结构,在附录中证明了所得的4个Hamilton算子的线性组合恒为Hamilton算子.     

A few new higher-dimensional Lie algebras and two types of coupling integrable couplings of the AKNS hierarchy and the KN hierarchy

Four higher-dimensional Lie algebras are introduced. With the help of their different loop algebras and the block matrices of Lax pairs for the zero curvature representations of two given integrable couplings, the two types of coupling integrable couplings of the AKNS hierarchy and the KN hierarchy are worked out, respectively, which fill up the consequences obtained by Ma and Gao (2009) [9]. The coupling integrable couplings of the AKNS hierarchy obtained in the paper again reduce to the coupling integrable couplings of the nonlinear Schrödinger equation and the modified KdV (mKdV) equation, which are different from the resulting results given by Ma and Gao.     

A general method for generating multicomponent integrable hierarchies

A Lie algebra, whose bases are the forms of M × 3 matrices, is defined. Subsequently two types of loop algebras are constructed, whose commutative operations are equivalent to known ones proposed before. By using the Tu scheme, the mulicomponent KN hierarchy and its integrable coupling system, as well as a generalized multicomponent AKNS integrable hierarchy with five potential functions are obtained. The procedure presented in this paper is simple and straightforward and can be used generally     

多分量的KN族与可积耦合及其它们的哈密顿结构

Firstly,a vector loop algebra (~G)3 is constructed,by use of it multi-component KN hierarchy is obtained.Further,by taking advantage of the extending vector loop algebras (~G)6 and (~G)9 of (~G)3 the double integrable couplings of the multi-component KN hierarchy are worked out respectively.Finally,Hamiltonian structures of obtained system are given by quadratic-form identity.     

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.     

Symmetries for the Ablowitz–Ladik Hierarchy: Part II. Integrable Discrete Nonlinear Schrödinger Equations and Discrete AKNS Hierarchy

In the paper, we continue to consider symmetries related to the Ablowitz–Ladik hierarchy. We derive symmetries for the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. The integrable discrete nonlinear Schrödinger hierarchy is in scalar form and its two sets of symmetries are shown to form a Lie algebra. We also present discrete AKNS isospectral flows, non‐isospectral flows and their recursion operator. In continuous limit these flows go to the continuous AKNS flows and the recursion operator goes to the square of the AKNS recursion operator. These discrete AKNS flows form a Lie algebra that plays a key role in constructing symmetries and their algebraic structures for both the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. Structures of the obtained algebras are different structures from those in continuous cases which usually are centerless Kac–Moody–Virasoro type. These algebra deformations are explained through continuous limit and degree in terms of lattice spacing parameter h.     

一个高维loop代数及其应用

本文构造了一个7维loop代数,由此获得两个已知的Liouville可积族．利用屠格式和可积耦合定义得到了相应的四个扩展可积模型．另外,利用马格式求得了上面已知可积系之一的非等谱流．     

一个新的高维向量Lie代数及其应用

构造了一个新的8维向量Lie代数,通过适当设计等谱问题,利用屠格式和扩展的迹恒等式得到了AKNS族的可积耦合及Hamilton结构.     

两个高维loop代数及应用

借助于循环数,构造了维数分别是5(s+1)和4(s+1)的两个高维loop代数．为了计算方便,本文只考虑s=1时的应用．利用第一个loop代数■_1~*得到了具有4-Hamilton结构的一个广义AKNS族,该方程族可约化为著名的AKNS族．利用第二个loop代数■_2~*,得到了具有4个分量位势函数的4-Hamilton结构方程族,该族可约化为一个非线性耦合Burgers方程和一个耦合的KdV方程．     

AN INTEGRABLE COUPLING OF THE GENERALIZED AKNS HIERARCHY

A direct method for establishing integrable couplings is proposed in this paper by constructing a new loop algebra G. As an illustration by example, an integrable coupling of the generalized AKNS hierarchy is given. Furthermore, as a reduction of the generalized AKNS hierarchy, an integrable coupling of the well-known G J hierarchy is presented. Again a simple example for the integrable coupling of the MKdV equation is also given. This method can be used generally.     

A new Lie algebra along with its induced Lie algebra and associated with applications

A 3 × 3 Lie algebra H is introduced whose induced Lie algebra by decomposition and linear combinations is obtained, which may reduce to the Lie algebra given by AP Fordy and J Gibbons. By employing the induced Lie algebra and the zero curvature equation, a kind of enlarged Boussinesq soliton hierarchy is produced. Again making use of a subalgebra of the induced Lie algebra leads to the well-known KdV hierarchy whose expanding integrable system is also worked out. As an applied example of the Lie algebra H, we obtain a new integrable coupling of the well-known AKNS hierarchy.     

A (2 + 1)-dimensional integrable hierarchy and its extending integrable model

A loop algebra is constructed, whose subalgebra is first used to present a Lax pair. By making use of the Tu scheme by Tu Guizhang, a generalized (2 + 1)-dimensional KN hierarchy is worked out. Further, based on the associated relations between the subalgebras in the above loop algebra, an extending integrable model of the generalized (2 + 1)-dimensional KN hierarchy as above is produced.     

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.