A family of Liouville integrable lattice equations and its conservation laws

A new discrete spectral problem of matrix three-by-three with three potentials is introduced and by using the discrete trace identity a hierarchy of Liouville lattice equations is obtained. The infinite number of conservation laws of the hierarchy of Hamiltonian system are presented.     

Two families of Liouville integrable lattice equations

By virtue of zero curvature representations, we are successful to generate the Lax representations of two hierarchies of discrete lattice equations respectively, which are derived from two new and interesting 3 × 3 matrix spectral problems. Moreover, by using the trace identity, the bi-Hamiltonian structures of the above systems are given, and it is shown that they are integrable in the Liouville sense. Finally, infinitely many conservation laws for the second hierarchy of lattice equations are given by a direct method.     

An integrable coupling hierarchy of the Mkdv_integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy

A four-by-four matrix spectral problem is introduced, locality of solution of the related stationary zero curvature equation is proved. An integrable coupling hierarchy of the Mkdv_integrable systems is presented. The Hamiltonian structure of the resulting integrable coupling hierarchy is established by means of the variational identity. It is shown that the resulting integrable couplings are all Liouville integrable Hamiltonian systems. Ultimately, through the nonisospectral zero curvature representation, a nonisospectral integrable hierarchy associated with the resulting integrable couplings is constructed.     

An integrable lattice hierarchy, associated integrable coupling, Darboux transformation and conservation laws

A new integrable lattice hierarchy is constructed from a discrete matrix spectral problem, some related properties of the new hierarchy are discussed. The Hamiltonian structures and Liouville integrability of the new hierarchy are established by using the discrete trace identity. A kind of integrable coupling for the new hierarchy is constructed through enlarging spectral problems. A Darboux transformation (DT) with two variable parameters and the infinitely many conservation laws for a typical lattice equation in the new hierarchy are constructed based on its Lax representation, the explicit solutions are obtained via the DT, the structures for those solutions are graphically investigated. All these properties might be helpful to understanding some physical phenomena.     

A hierarchy of differential-difference equations,conservation laws and new integrable coupling system

Starting from a discrete spectral problem with two arbitrary parameters, a hierarchy of nonlinear differential-difference equations is derived. The new hierarchy not only includes the original hierarchy, but also the well-known Toda equation and relativistic Toda equation. Moreover, infinitely many conservation laws for a representative discrete equation are given. Further, a new integrable coupling system of the resulting hierarchy is constructed.     

Constructing integrable coupling systems of Hamiltonian lattice equations using semi-direct sums of Lie algebras

In this article, by considering a discrete isospectral problem, a hierarchy of Hamiltonian lattice equations are derived. Two types of semi-direct sums of Lie algebras are proposed, using which a practicable way to construct discrete integrable couplings is introduced. As an application, two kinds of discrete integrable couplings of the resulting system are worked out.     

The 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.     

The multi-component KdV hierarchy and its multi-component integrable coupling system

Construction of a type of simple loop algebra is devoted to establishing an isospectral problem. It follows that the multi-component KdV hierarchy of soliton equations is obtained. Further, an expanded loop algebra of the above algebra is presented, which is used to work out the multi-component integrable coupling system of the multi-component KdV hierarchy.     

The integrable coupling of the AKNS hierarchy and its Hamiltonian structure

The Hamiltonian structure of the integrable coupling of the AKNS hierarchy is obtained by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings.     

一族Liouville可积系及其3-Hamilton结构

构造了loop代数A↑～1的一个高阶子代数,设计了一个新的Lax对,利用屠格式获得了含8个位势的孤立子方程族;利用Gauteax导数直接验证了所得3个辛算子的线性组合仍为辛算子．因此该孤立族具有3-Hamilton结构,具有无穷多个对合的公共守恒密度,故Liouville可积．作为约化情形,得到了2个可积系,其中之一是著名的AKNS方程族．     

A three-parameter difference Hamiltonian operator,corresponding pair of Hamiltonian operators and a family of Liouville integrable lattice equations

A difference Hamiltonian operator involved three arbitrary real parameters is introduced. When these parameters in the difference Hamiltonian operator are properly chosen, we obtain a pair of difference Hamiltonian operators. Then, using Magri scheme of bi-Hamiltonian formulation, we construct a family of Liouville integrable lattice equations. Finally, the discrete zero curvature representation of obtained family is presented.     

The Liouville integrability of integrable couplings of Volterra lattice equation

A hierarchy of integrable couplings of Volterra lattice equations with three potentials is proposed, which is derived from a new discrete six-by-six matrix spectral problem. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebra, the two Hamiltonian forms are deduced for each lattice equation in the resulting hierarchy. A strong symmetry operator of the resulting hierarchy is given. Finally, we prove that the hierarchy of the resulting Hamiltonian equations are all Liouville integrable discrete Hamiltonian systems.     

Constructing super D-Kaup-Newell hierarchy and its nonlinear integrable coupling with self-consistent sources

How to construct new super integrable equation hierarchy is an important problem. In this paper, a new Lax pair is proposed and the super D-Kaup-Newell hierarchy is generated, then a nonlinear integrable coupling of the super D-Kaup-Newell hierarchy is constructed. The super Hamiltonian structures of coupling equation hierarchy is derived with the aid of the super variational identity. Finally, the self-consistent sources of super integrable coupling hierarchy is established. It is indicated that this method is a straight- forward and efficient way to construct the super integrable equation hierarchy.