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.

The quadratic-form identity for constructing Hamiltonian structures of the Guo hierarchy

The trace identity is extended to the quadratic-form identity. The Hamiltonian structures of the multi-component Guo hierarchy, integrable coupling of Guo hierarchy and （2＋l）-dimensional Guo hierarchy are obtained by the quadraticform identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.

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 super- Hamiltonian structure is obtained by making use of super=trace identity. Furthermore, the super-classical-Boussinesq hierarchy is also integrable in the sense of Liouville.

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.

HAMILTONIAN STRUCTURE FOR RIGID BODY WITH FLEXIBLE ATTACHMENTS IN A CIRCULAR ORBIT

Hamiltonian structure of a rigid body in a circular orbit is established in this paper. With the reduction technique, the Hamiltonian structure of a rigid body in a circular orbit is derived from Lie-Poisson structure of semidirect product, and Hamiltonian is derived from Jacobi's integral. The above method can be generalized to establish the Hamiltonian structure of a rigid body with a flexible attachment in a circular or- bit. At last, an example of stability analysis is given.

A HIERARCHY OF NEW DISCRETE INTEGRABLE EQUATION AND ITS HAMILTONIAN STRUCTURE

A new discrete isospectral problem is introduced,from which a hierarchy of Lax i ntegrable lattice equation is deduced. By using the trace identity,the correspon ding Hamiltonian structure is given and its Liouville integrability is proved.

Multi-component Harry-Dym hierarchy and its integrable couplings as well as their Hamiltonian structures

This paper obtains the multi-component Harry-Dym （HI）） hierarchy and its integrable couplings by using two kinds of vector loop algebras G^-3 and G^-6. The Hamiltonian structures of the above system are given by the quadraticform identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.

A NEW COMPLETELY INTEGRABLE LIOUVILLE' S SYSTEM, ITS LAX REPRESENTATION AND BI-HAMILTONIAN STRUCTURE

A new isospectral problem and the corresponding hierarchy of nonlinear evolution equations is presented. As a reduction, the well-known MKdV equation is obtained. It is shown that the hierarchy of equations is integrable in Liouville' s sense and possesses Bi-Hamiltonian structure. Under the constraint between the potentials and eigenfunctions, the eigenvalue problem can be nonlinearized as a finite dimensional completely integrable system.

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.

Hamiltonian structure, Darboux transformation for a soliton hierarchy associated with Lie algebra so(4, C)

In this paper, we first introduce a Lie algebra of the special orthogonal group, g = so(4, C), whose elements are 4 × 4trace-free, skew-symmetric complex matrices. As its application, we obtain a new soliton hierarchy which is reduced to AKNS hierarchy and present its bi-Hamiltonian structure and Liouville integrability. Furthermore, for one of the equations in the resulting hierarchy, we construct a Darboux matrix T depending on the spectral parameter λ.

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.

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.

A Direct Method of Hamiltonian Structure

A direct method of constructing the Hamiltonian structure of the soliton hierarchy with self-consistent sources is proposed through computing the functional derivative under some constraints. The Hamiltonian functional is related with the conservation densities of the corresponding hierarchy. Three examples and their two reductions are given.

The Hamiltonian Structures of 3D ODE with Time-Independent Invariants

We have proved that any 3-dimensional dynamical system of ordinary differentialequations(in short, 3D ODE)With time-independent invariants can be rewritten asHaniltonian systems with respect to generalized Poisson brackets and theHamiltonians are these invariants. As an example,we discuss the Kermack-Mckendrick modelfor epidemics in detail. The results we obtained are generalizatioof those obtained by Y. Nutku.

The Second mKdV Equation and Its Hereditary Symmetry and Hamiltonian Structure

The isospectral problem of the second mKdV equation is found out firstly. It follows that the strong hereditary symmetry and the Hamiltonian structure of the second mKdV equation are presented.

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.

Integrable Coupling of (2+1)-Dimensional Multi-component DLW Integrable Hierarchy and Its Hamiltonian Structure

A new multi-component Lie algebra is constructed, and a type of new loop algebra is presented. A （2＋1）-dimensional multi-component DLW integrable hierarchy is obtained by using a （2＋1）-dimensional zero curvature equation. Furthermore, the loop algebra is expanded into a larger one and a type of integrable coupling system and its corresponding Hamiltonian structure are worked out.

A novel hierarchy of differential integral equations and their generalized bi-Hamiltonian structures

With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a 3 x 3 matrix spectral problem is proposed. By using the trace identity, the bi-Hamiltonian structures of the hierarchy are established with two skew-symmetric operators. Based on two linear spectral problems, we obtain the infinite many conservation laws of the first member in the hierarchy.

The Hamiltonian Structures and Algebro-geometric Solution of the Generalized Kaup-Newell Soliton Equations

Staring from a new spectral problem,a hierarchy of the generalized Kaup-Newell soliton equations is derived.By employing the trace identity their Hamiltonian structures are also generated.Then,the generalized Kaup-Newell soliton equations are decomposed into two systems of ordinary differential equations.The Abel-Jacobi coordinates are introduced to straighten the flows,from which the algebro-geometric solutions of the generalized KaupNewell soliton equations are obtained in terms of the Riemann theta functions.

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.