Super-KN Hierarchy and Its Super-Hamiltonian Structure

Based on the basis of the constructed Lie super algebra, the super-isospectral problem of KN hierarchy is considered. Under the frame of the zero curvature equation, the super-KN hierarchy is obtained. Furthermore, its super-Hamiltonian structure is presented by using super-trace identity and it has super-bi-Hamiltonian structure.     

A new eight-dimensional Lie superalgebra and two corresponding hierarchies of evolution equations

A new eight-dimensional Lie superalgebra is constructed and two isospectral problems with six potentials are designed. Corresponding hierarchies of nonlinear evolution equations, as well as super-AKNS and super-Levi, are derived. Their super-Hamiltonian structures are established by making use of the supertrace identity, and they are integrable in the sense of Liouville.     

The super-classical-Boussinesq hierarchy and its super-Hamiltonian structure

Based on the constructed Lie superalgebra,the super-classical-Boussinesq hierarchy is obtained.Then,its superHamiltonian structure is obtained by making use of super-trace identity.Furthermore,the super-classical-Boussinesq hierarchy is also integrable in the sense of Liouville.     

New DLW Hierarchy of an Integrable Coupling and Its Hamiltonian Structure

A type of higher-dimensional 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.     

Two Integrable Couplings of Modified Tu Hierarchy

Two different integrable couplings of the modified Tu hierarchy are obtained under the zero curvature equation by using two higher dimension Lie algebras. Furthermore, a complex Hamiltonian structures of the second integrable couplings is presented by taking use of the variational 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 quadratic-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.     

NLS-MKdV Hierarchy and Its Hamiltonian Structures

A new simple loop algebra is constructed, which is devote to establishing an isospectral problem. By making use of Tu scheme, NLS-MKdV hierarchy is obtained. Again via expanding the loop algebra above, another higher-dimensional loop algebra is presented. It follows that an integrable coupling of NLS-MkdV hierarchy is given. Also, the trace identity is extended to the quadratic-form identity and the Hamiltonian structures of the NLS-MKdV hierarchy and integrable coupling of NLS-MkdV hierarchy are obtained by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.     

Vector Loop Algebra and Its Applications to Tu Hierarchy

A vector loop algebra and its extended loop algebra are proposed, which are devoted to obtaining the Tu hierarchy. By making use of the extended trace identity, the Harniltonian structure of the Tu hierarchy is constructed. Furthermore, we apply the quadratic-form identity to the integrable coupling system of the Tu hierarchy.     

A New Integrable Lattice Hierarchy and Its Two Discrete Integrable Couplings

A new and efficient way is presented for discrete integrable couplings with the help of two semi-direct sum Lie algebras. As its applications, two discrete integrable couplings associated with the lattice equation are worked out. The approach can be used to study other discrete integrable couplings of the discrete hierarchies of solition equations.     

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.     

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

A new three-dimensional Lie algebra and its corresponding loop algebra are constructed, from which a modified AKNS soliton-equation hierarchy is obtained.     

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 constants are real and imaginary numbers. A loop algebra G corresponding to the Lie algebra G is constructed, for which it is devoted to generating a soliton hierarchy of evolution equations under the framework of generalized zero curvature equation which is derived from the compatibility of the isospectral problems expressed by Hirota operators. Finally, we decompose the Lie algebra G to obtain the subalgebras G1 and G2. Using the G2 and its one type of loop algebra G2, a Liouville integrable soliton hierarchy is obtained, furthermore, we obtain its bi-Hamiltonian structure by employing the quadratic-form identity.     

Two Integrable Couplings of Modified Tu Hierarchy

Two different integrable couplings of the modified Tu hierarchy are obtained under the zero curvature equation by using two higher dimension Lie algebras. Furthermore, a complex Hamiltonian structures of the second integrable couplings is presented by taking use of the variational identity.     

A New Liouville Integrable Hamiltonian System

With the help of a Lie algebra, an isospectral Lax pair is introduced for which a new Liouville integrable hierarchy of evolution equations is generated. Its Hamiltonian structure is also worked out by use of the quadratic-form identity.     

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 constants are real and imaginary numbers. A loop algebra ˜G corresponding to the Lie algebra G is constructed, for which it is devoted to generating a soliton hierarchy of evolution equations under the framework of generalized zero curvature equation which is derived from the compatibility of the isospectral problems expressed by Hirota operators. Finally, we decompose the Lie algebra G to obtain the subalgebras G₁ and G₂. Using the G₂ and its one type of loop algebra ˜G₂, a Liouville integrable soliton hierarchy is obtained, furthermore, we obtain its bi-Hamiltonian structure by employing the quadratic-form identity.     

A New Lie Algebra and Its Related Liouville Integrable Hierarchies

A new Lie algebra G and its two types of loop algebras G1 and G2 are constructed. Basing on G1 and G2, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obtained respectively under the framework of zero curvature equation, which is derived from the compatibility of the isospectral problems expressed by Hirota operators. At the same time, we obtain the Hamiltonian structure of the first hierarchy and the bi-Hamiltonian structure of the second one with the help of the quadratic-form identity.     

A New Lie Algebra and Its Related Liouville Integrable Hierarchies

A new Lie algebra G and its two types of loop algebras \tilde{G₁} and \tilde{G₂} are constructed. Basing on \tilde{G₁} and \tilde{G₂}, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obtained respectively under the framework of zero curvature equation, which is derived from the compatibility of the isospectral problems expressed by Hirota operators. At the same time, we obtain the Hamiltonian structure of the first hierarchy and the bi-Hamiltonian structure of the second one with the help of the quadratic-form identity.     

Simple Coherent State and New Form for Inhomogeneous Differential Realization of OSP(2,1) Superalgebra

The simple coherent state of the OSP(2,1) superalgebra is constructed and its properties are discussed in detail. The matrix elements of the OSP(2,1) generators in the coherent state space are calculated. The new form for inhomogeneous differential realization of the OSP(2,1) superalgebra is given.     

The Double Integrable Couplings of Tu Hierarchy

Two types of Lie algebras are presented, from which two integrable couplings associated with the Tu isospectral problem are obtained, respectively. One of them possesses the Hamiltonian structure generated by a linear isomorphism and the quadratic-form identity. An approach for working out the double integrable couplings of the same integrable system is presented in the paper.