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.     

Liouville Integrable System and Associated Integrable Coupling

Liouville integrable discrete integrable system is derived based on discrete isospectral problem. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.     

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 integrable 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.     

Discrete integrable system and its integrable coupling

This paper derives new discrete integrable system based on discrete isospectral problem. It shows that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.     

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+1)-dimensional Guo 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.     

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.     

Lie Algebras Associated with Group U（n）

Starting from the subgroups of the group U（n）, the corresponding Lie algebras of the Lie algebra Al are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.     

Discrete integrable couplings associated with modified Korteweg——de Vries lattice and two hierarchies of discrete soliton equations

A direct way to construct integrable couplings for discrete systems is presented by use of two semi-direct sum Lie algebras. As their applications, the discrete integrable couplings associated with modified Korteweg--de Vries (m-KdV) lattice and two hierarchies of discrete soliton equations are developed. It is also indicated that the study of integrable couplings using semi-direct sums of Lie algebras is an important step towards the complete classification of integrable couplings.     

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 \tilde{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 Matrix Loop Algebra and Its Application

A new matrix Lie algebra and its corresponding Loop algebra are constructed firstly, as its application, the multi-component TC equation hierarchy is obtained, then by use of trace identity the Hamiltonian structure of the above system is presented. Finally, the integrable couplings of the obtained system is worked out by the expanding matrix Loop algebra.     

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.     

Lie Algebra and Lie Super Algebra for Integrable Couplings of C-KdV Hierarchy 1

Based on the constructed Lie algebra and Lie super algebra, the integrable couplings and super-integrable couplings of the C-KdV hierarchy are obtained respectively. Furthermore, its super-Hamiltonian structures are presented by using super-trace identity.     

Integrable Couplings of the Coupled Burgers Hierarchy

In this letter, a new loop algebra G is constructed, from which a new isospectral problem is established. It follows that integrable couplings of the well-known coupled Burgers hierarchy are obtained.     

Integrable Coupling System of JM Equations Hierarchy with Self-Consistent Sources

We propose a systematic method for generalizing the integrable couplings of soliton eqhations hierarchy with self-consistent sources associated with s/（4）. The JM equations hierarchy with self-consistent sources is derived. Furthermore, an integrable couplings of the JM soliton hierarchy with self-consistent sources is presented by using of the loop algebra sl（4）.     

A Bi-Hamiltonian Lattice System of Rational Type and Its Discrete Integrable Couplings

By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.     

A New Method to Construct Integrable Coupling System for Burgers Equation Hierarchy by Kronecker Product

It is shown that the Kronecker product can be applied to construct a new integrable coupling system of soliton equation hierarchy in this paper. A direct application to the Burgers spectral problem leads to a novel soliton equation hierarchy of integrable coupling system. It indicates that the Kronecker product is an efficient and straightforward method to construct the integrable couplings.     

Prolongation structure for nonlinear integrable couplings of a KdV soliton hierarchy

In this paper, a new nonlinear integrable coupling system of the soliton hierarchy is presented. From the Lax pairs, the coupled KdV equations are constructed successfully. Based on the prolongation method of Wahlquist and Estabrook, we study the prolongation structure of the nonlinear integrable couplings of the KdV equation.     

An integrable Hamiltonian hierarchy, a high-dimensional loop algebra and associated integrable coupling system

A subalgebra of loop algebra ?_2 is established. Therefore, a new isospectral problem is designed. By making use of Tu's scheme, a new integrable system is obtained, which possesses bi-Hamiltonian structure. As its reductions, a formalism similar to the well-known Ablowitz－Kaup－Newell－Segur (AKNS) hierarchy and a generalized standard form of the Schr?dinger equation are presented. In addition, in order for a kind of expanding integrable system to be obtained, a proper algebraic transformation is supplied to change loop algebra ?_2 into loop algebra ?_1. Furthermore, a high-dimensional loop algebra is constructed, which is different from any previous one. An integrable coupling of the system obtained is given. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.     

Constructing New Discrete Integrable Coupling System for Soliton Equation by Kronecker Product

It is shown that the Kronecker product can be applied to constructing new discrete integrable coupling system of soliton equation hierarchy in this paper. A direct application to the fractional cubic Volterra lattice spectral problem leads to a novel integrable coupling system of soliton equation hierarchy. It is also indicated that the study of discrete integrable couplings by using the Kronecker product is an efficient and straightforward method. This method can be used generally.     

A new multi-component integrable coupling system for AKNS equation hierarchy with sixteen-potential functions

It is shown in this paper that the upper triangular strip matrix of Lie algebra can be used to construct a new integrable coupling system of soliton equation hierarchy. A direct application to the Ablowitz--Kaup--Newell-- Segur(AKNS) spectral problem leads to a novel multi-component soliton equation hierarchy of an integrable coupling system with sixteen-potential functions. It is indicated that the study of integrable couplings when using the upper triangular strip matrix of Lie algebra is an efficient and straightforward method.