Two Types of Expanding Lie Algebra and New Expanding Integrable Systems

From a new Lie algebra proposed by Zhang, two expanding Lie algebras and its corresponding loop algebrasare obtained.Two expanding integrable systems are produced with the help of the generalized zero curvature equation.One of them has complex Hamiltion structure with the help of generalized Tu formula (GTM).

The Liouville integrable coupling system of the m-AKNS hierarchy and its Hamiltonian structure

In this paper a type of 9-dimensional vector loop algebra F is constructed, which is devoted to establish an isospectral problem. It follows that a Liouville integrable coupling system of the m-AKNS hierarchy is obtained by employing the Tu scheme, whose Hamiltonian structure is worked out by making use of constructed quadratic identity. The method given in the paper can be used to obtain many other integrable couplings and their Hamiltonian structures.

A New Lie Algebra and a Way to Generate Multiple Integrable Couplings

A new higher-dimensional Lie algebra is constructed,which is used to generate multiple integrable couplingssimultaneously.From this,we come to a general approach for seeking multi-integrable couplings of the known integrablesoliton equations.

Nonlinear Super Integrable Couplings of A Super Integrable Hierarchy and Its Super Hamiltonian Structures

Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction,special cases of this nonlinear super integrable couplings were obtained.

Lie Algebras for Constructing Nonlinear Integrable Couplings

Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti-Johnson (GJ)hierarchy and the Yang hierarchy are obtained,respectively.By employing the variational identity theirHamiltonian structures are also generated.The approach presented in the paper can also provide nonlinear integrablecouplings of other soliton hierarchies of evolution equations.

Integrable Couplings of Classical-Boussinesq Hierarchy and Its Hamiltonian Structure

由使用谎言代数学， classical-Boussinesq 层次的 integrable couplings 被获得。然后， classical-Boussinesq 的 integrable couplings 的 Hamiltonian 结构被二次形式的身份获得。

A Complex Higher-Dimensional Lie Algebra with Real and Imaginary Structure Constants as Well as Its Decomposition

A new Lie algebra G of the Lie algebra sl(2) is constructed with complex entries whose structure constantsare real and imaginary numbers.A loop algebra G corresponding to the Lie algebra G is constructed,for which itis devoted to generating a soliton hierarchy of evolution equations under the framework of generalized zero curvatureequation which is derived from the compatibility of the isospectral problems expressed by Hirota operators.Finally,wedecompose the Lie algebra G to obtain the subalgebras G_1 and G_2.Using the G_2 and its one type of loop algebra (?)_2,aLiouville integrable soliton hierarchy is obtained,furthermore,we obtain its bi-Hamiltonian structure by employing thequadratic-form identity.

Nonlinear Super Integrable Couplings of Super Yang Hierarchy and Its Super Hamiltonian Structures

Nonlinear super integrable couplings of the super Yang hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity. As its reduction, nonlinear integrable couplings of Yang hierarchy were obtained.

A NEW HIERARCHY OF INTEGRABLE SYSTEMS AND ITS HAMILTONIAN STRUCTURE

On the basis of an analysis of the loop algebra ?1 a new hierarchy of integrable sys-tems is proposed. The Hamiltonian structure of the new hierarchy of equations is established by means of constrained formal variational calculus. This hierarchy of evolution equations is demonstrated to possess an infinite number of conserved densities, which are ininvolution in pairs.

Two new discrete integrable systems

In this paper,we focus on the construction of new（1＋1）-dimensional discrete integrable systems according to a subalgebra of loop algebra A 1.By designing two new（1＋1）-dimensional discrete spectral problems,two new discrete integrable systems are obtained,namely,a 2-field lattice hierarchy and a 3-field lattice hierarchy.When deriving the two new discrete integrable systems,we find the generalized relativistic Toda lattice hierarchy and the generalized modified Toda lattice hierarchy.Moreover,we also obtain the Hamiltonian structures of the two lattice hierarchies by means of the discrete trace identity.

Induced Lie Algebras of a Six-Dimensional Matrix Lie Algebra

By using a six-dimensional matrix Lie algebra [Y.F.Zhang and Y.Wang,Phys.Lett.A 360 (2006) 92], three induced Lie algebras are constructed.One of them is obtained by extending Lie bracket,the others are higher- dimensional complex Lie algebras constructed by using linear transformations.The equivalent Lie algebras of the later two with multi-component forms are obtained as well.As their applications,we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.

A New Three-Dimensional Lie Algebra and a Modified AKNS Hierarchy of Soliton Equations

新三维的谎言代数学和它的相应的环代数学被构造，从哪个一个修改 AKNS soliton 方程层次被获得。

An Eight Component Integrable Hamiltonian Hierarchy from a Reduced Seventh-Order Matrix Spectral Problem

We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.

INTEGRABLE COUPLINGS OF THE TB HIERARCHY AND ITS HAMILTONIAN STRUCTURE

In this paper,we obtain integrable couplings of the TB hierarchy using the new subalgebra of the loop algebra A.Then the Hamiltonian structure of the above system is given by the quadratic-form identity.

Symplectic feedback using Hamiltonian Lie algebra and its applications to an inverted pendulum

From the symplectic representation of an autonomous nonlinear dynamical system with holonomic con- straints, i.e., those that can be represented through a symplectic form derived from a Hamiltonian, we present a new proof on the realization of the symplectic feedback action, which has several theoretical advantages in demonstrating the uniqueness and existence of this type of solution. Also, we propose a technique based on the interpretation, construction and character- ization of the pull-back differential on the symplectic manifold as a member of a one-parameter Lie group. This allows one to synthesize the control law that governs a certain system to achieve a desired behavior; and the method developed from this is applied to a classical system such as the inverted pendulum.

An integrable Hamiltonian hierarchy and associated integrable couplings system

This paper establishes a new isospectral problem. By making use of the Tu scheme, a new intcgrablc system is obtained. It gives integrable couplings of the system obtained. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.

Nonlinear Super Integrable Couplings of Super Dirac Hierarchy and Its Super Hamiltonian Structures

We construct nonlinear super integrable couplings of the super integrable Dirac hierarchy based on an enlarged matrix Lie superalgebra.Then its super Hamiltonian structure is furnished by super trace identity.As its reduction,we gain the nonlinear integrable couplings of the classical integrable Dirac hierarchy.

A New Liouville Integrable Hamiltonian System

在谎言代数学的帮助下， isospectral 宽松的对为进化方程的一个新 Liouville integrable 层次为哪个被产生被介绍。它的 Hamiltonian 结构被二次形式的身份的使用也得出。

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented,and a discrete four-by-fourmatrix spectral problem is introduced.A hierarchy of discrete integrable coupling systems is derived.The obtainedintegrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity.Finally,we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discreteHamiltonian systems.

Super-KN Hierarchy and Its Super-Hamiltonian Structure

根据构造谎言基于超级代数学， KN 层次的 super-isospectral 问题被考虑。在零个弯曲方程的框架下面， super-KN 层次被获得。而且，它的 super-Hamiltonian 结构被使用超级踪迹的身份介绍，它有 super-bi-Hamiltonian 结构。