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.

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.

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.

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.

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.

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.

THE INTEGRABLE COUPLINGS OF THE GENERALIZED COUPLED mKdV HIERARCHY AND ITS HAMILTONIAN STRUCTURES

We construct a loop algebra 3 , then a new 4×4 isospectral problem is presented. By Tu scheme, the generalized coupled mKdV equation hierarchy is derived. Based on an expanding loop algebra F3 of the loop algebra 3 , the integrable couplings of the generalized coupled mKdV hierarchy is solved. Finally, the Hamiltonian structures of the integrable couplings of the generalized coupled mKdV hierarchy is obtained by the quadratic-form identity.

A Few Discrete Lattice Systems and Their Hamiltonian Structures,Conservation Laws

With the help of three shift operators and r-matrix theory, a few discrete lattice systems are obtained which can be reduced to the well-known Toda lattice equation with a constraint whose Hamiltonian structures are generated by Poisson tensors of some induced Lie–Poisson bracket. The recursion operators of these lattice systems are constructed starting from Lax representations. Finally, reducing the given shift operators to get a simpler one and its expanding shift operators, we produce a lattice system with three vector fields whose recursion operator is given. Furthermore,we reduce the lattice system with three vector fields to get a lattice system whose Lax pair and conservation laws are obtained, respectively.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Hamiltonian System of New Nonlinear Lattice Equations

A 3-dimensional Lie algebra sμ（3） is obtained with the help of the known Lie algebra. Based on the sμ（3）, a new discrete 3 × 3 matrix spectral problem with three potentials is constructed. In virtue of discrete zero curvature equations, a new matrix Lax representation for the hierarchy of the discrete lattice soliton equations is acquired. It is shown that the hierarchy possesses a Hamiltonian operator and a hereditary recursion operator, which implies that there exist infinitely many common commuting symmetries and infinitely many common commuting conserved functionals.