Lie Algebras for Constructing Nonlinear Integrable Couplings

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

Matrix Lie Algebras and Integrable Couplings

Three kinds of higher-dimensional Lie algebras are given which can be used to directly construct integrable couplings of the soliton integrable systems. The relations between the Lie algebras are discussed. Finally, the integrable couplings and the Hamiltonian structure of Giachetti-Johnson hierarchy and a new integrable system are obtained, respectively.

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.

Two Kinds of New Lie Algebras for Producing Integrable Couplings

Two types of Lie algebras are constructed, which are directly used to deduce the two resulting integrable coupling systems with multi-component potential functions. Many other integrable couplings of the known integrable systems may be obtained by the approach.

Upper Triangular Matrix of Lie Algebra and a New Discrete Integrable Coupling System

The upper triangular matrix of Lie algebra is used to construct integrable couplings of discrete solition equations. Correspondingly, a feasible way to construct integrable couplings is presented. A nonlinear lattice soliton equation spectral problem is obtained and leads to a novel hierarchy of the nonlinear lattice equation hierarchy. It indicates that the study of integrable couplings using upper triangular matrix of Lie algebra is an important step towards constructing integrable systems.

Multi-component KN Hierarchy and Associated two Integrable Couplings as Well as Their Hamiltonian Structure

Firstly, a vector loop algebra G3 is constructed, by use of it multi-component KN hierarchy is obtained. Further, by taking advantage of the extending vector loop algebras G6 and G9 of G3 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.

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.

New DLW Hierarchy of an Integrable Coupling and Its Hamiltonian Structure

A type of higher-dimensionaJ loop algebra is constructed from which an isospectral problem is established. It follows that an integrable coupling, actually an extended integrable model of the existed solitary hierarchy of equations, is obtained by taking use of the zero curvature equation, whose Hamiltonian structure is worked out by employing the constructed quadratic 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.

Integrable Couplings of Classical-Boussinesq Hierarchy and Its Hamiltonian Structure

By using a Lie algebra, an integrable couplings of the classicai-Boussinesq hierarchy is obtained. Then, the Hamiltonian structure of the integrable couplings of the classical-Boussinesq is obtained by the quadratic-form identity.

Integrable Coupling of KN Hierarchy and Its Hamiltonian Structure

The Hamiltonian structure of.the integrable couplings obtained by our method has not been solved. In this paper, the Hamiltonian structure of the KN hierarchy is obtained by making use of the quadratlc-form identity.

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.

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.

A Higher Dimensional Loop Algebra and Integrable Couplings System of Evolution Equations Hierarchy

An extension of the Lie algebra A_~n-1 has been proposed [Phys. Lett. A, 2003, [STHZ]310:19-24]. In this paper, the new Lie algebra was used to construct a new higher dimensional loop algebra [AKG～]. Based on the loop algebra [AKG～], the integrable couplings system of the NLS-MKdV equations hierarchy was obtained. As its reduction case, generalized nonlinear NLS-MKdV equations were obtained. The method proposed in this letter can be applied to other hierarchies of evolution equations.

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).

Two New Expanding Lie Algebras and Their Integrable Models

A new Lax integrable hierarchy is obtained by constructing an isospectraJ problem with constrained conditions. Two kinds of integrable couplings are obtained by constructing two new expanding Lie algebras of the Lie algebra Be, respectively.

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.

Double Integrable Couplings and Their Constructing Method

To extend the study scopes of integrable couplings, the notion of double integrable couplings is proposed in the paper. The zero curvature equation appearing in the constructing method built in the paper consists of the elements of a new loop algebra which is obtained by using perturbation method. Therefore, the approach given in the paper has extensive applicable values, that is, it applies to investigate a lot of double integrable couplings of the known integrable hierarchies of evolution equations. As for explicit applications of the method proposed in the paper, the double integrable couplings of the AKNS hierarchy and the KN hierarchy are worked out, respectively.

Nonlinear Super Integrable Couplings of a Super Integrable Hierarchy

Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. And its super Hamiltonian structures were established by using super trace identity. As its reduction, special cases of this nonlinear super integrable coupling were obtained.