Continuous limits for an integrable coupling system of Toda equation hierarchy

In this Letter, we present an integrable coupling system of lattice hierarchy and its continuous limits by using of Lie algebra sl(4). By introducing a complex discrete spectral problem, the integrable coupling system of Toda lattice hierarchy is derived. It is shown that a new complex lattice spectral problem converges to the integrable couplings of discrete soliton equation hierarchy, which has the integrable coupling system of C-KdV hierarchy as a new kind of continuous limit.     

An integrable coupling system of lattice hierarchy and its continuous limits

In [E.G. Fan, Phys. Lett. A 372 (2008) 6368], Fan present a lattice hierarchy and its continuous limits. In this Letter, we extend this method, by introducing a complex discrete spectral problem, a coupling lattice hierarchy is derived. It is shown that a new sequence of combinations of complex lattice spectral problem converges to the integrable coupling couplings of soliton equation hierarchy, which has the integrable coupling system of AKNS hierarchy as a continuous limit.     

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.     

A new method to construct the integrable coupling system for discrete soliton equation with the Kronecker product

It is shown that the Kronecker product can be applied to construct a new integrable coupling system of discrete soliton equation hierarchy in this Letter. A direct application to the generalized Toda lattice spectral problem leads to a novel integrable coupling system. It is also indicated that the study of integrable couplings by using of the Kronecker product is an efficient and straightforward method.     

New Matrix Lie Algebra and Its Application

A set of new matrix Lie algebra and its corresponding loop algebra are constructed. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equation is generated. As its reduction cases, the multi-component Tu hierarchy is given. Finally, the multi-component integrable coupling system of Tu hierarchy is presented through enlarging matrix spectral problem.     

R-Matrix of the Coupling AKNS Equation Hierarchy

A new r-matrix of the integrable coupling system is obtained by nonlineaxization of the coupling eigenvalue problem. As a reduction, we present the r-matrix of the coupling Ablowitz-Kaup Newell Segur （AKNS） equation hierarchy via enlarged matrix Lie algebra.     

New Matrix Lie Algebra, a Powerful Tool for Constructing Multi-component C-KdV Equation Hierarchy

A set of new multi-component matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A-2M. It follows that an isospectral problem is established. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equations is generated, which possesses the multi-component Hamiltonian structures. As its reduction cases, the multi-component C-KdV hierarchy is given. Finally, the multi-component integrable coupling system of C-KdV hierarchy is presented through enlarging matrix spectral problem.     

A hierarchy of Hamiltonian lattice equations associated with the relativistic Toda type system

By considering a discrete iso-spectral problem, a hierarchy of bi-Hamiltonian relativistic Toda type lattice equations are revisited. After introducing a semi-direct sum Lie algebras of four by four matrices, integrable coupling system associated with the relativistic Toda type lattice are derived. It is shown that the resulting lattice soliton hierarchy possesses Hamiltonian structures and infinitely many common commuting symmetries as well infinitely many conserved functions. The Liouville integrability of the resulting system is then demonstrated.     

On （2＋1）-Dimensional Non-isospectral Toda Lattice Hierarchy and Integrable Coupling System

By considering （2＋1）-dimensional non-isospectral discrete zero curvature equation, the （2＋1）-dimensional non-isospectral Toda lattice hierarchy is constructed in this article. It follows that some reductions of the （2＋1）- dimensional Toda lattice hierarchy are given. Finally, the （2＋1）-dimensional integrable coupling system of the Toda lattice hierarchy is obtained through enlarging spectral problem.     

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.     

Multi-component SC Lax Integrable Hierarchy of Soliton Equations and Its Multi-component Integrable Coupling System with Two Arbitrary Functions

A new simple loop algebra GM is constructed, which is devoted to establishing an isospectral problem.By making use of generalized Tu scheme, the multi-component SC hierarchy is obtained. Furthermore, an expanding loop algebra FM of the loop algebra GM is presented. Based on FM, the multi-component integrable coupling system of the multi-component SC hierarchy of soliton equations is worked out. How to design isospectral problem of mulitcomponent hierarchy of soliton equations is a technique and interesting topic. The method can be applied to other nonlinear evolution equations hierarchy.     

Multi-component Levi Hierarchy and Its Multi-component Integrable Coupling System

A simple 3M-dimensional loop algebra X is produced, whose commutation operation defined by us is as simple and straightforward as that in the loop algebra A1. It follows that a general scheme for generating multicomponent integrable hierarchy is proposed. By taking advantage of X, a new isospectral problem is established, and then by making use of the Tu scheme the well-known multi-component Levi hierarchy is obtained. Finally, an expanding loop algebra FM of the loop algebra X is presented, based on the FM, the multi-component integrable coupling system of the multi-component Levi hierarchy is worked out. The method in this paper can be applied to other nonlinear evolution equation hierarchies.     

Multi-component Levi Hierarchy and Its Multi-component Integrable Coupling System

A simple 3M-dimensional loop algebra X is produced, whose commutation operation defined by us is A1 as simple and straightforward as that in the loop algebra A1. It follows that a general scheme for generating multi-component integrable hierarchy is proposed. By taking advantage of X, a new isospectral problem is established, and then by making use of the Tu scheme the well-known multi-component Levi hierarchy is obtained. Finally, an expanding loop algebra FM of the loop algebra .X is presented, based on the FM, the multi-component integrable coupling system of the multi-component Levi hierarchy is worked out. The method in this paper can be applied to other nonlinear evolution equation hierarchies.     

Multi-component C-KdV Hierarchy of Soliton Equations and Its Multi-component Integrable Coupling System

A new simple loop algebra G M is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component C-KdV hierarchy is obtained. Further, an expanding loop algebra FM of the loop algebra G M is presented. Based on FM , the multi-component integrable coupling system of the multi-component C-KdV hierarchy is worked out. The method can be used to other nonlinear evolution equations hierarchy.     

Noncommutative AKNS Equation Hierarchy and Its Integrable Couplings with Kronecker Product

We present a noncommutative version of the Ablowitz-Kaup-Newell-Segur （AKNS） equation hierarchy, which possesses the zero curvature representation. Furthermore, we derive the noncommutative AKNS equation from the noncommutative （anti-）self-dual Yang-Mills equation by reduction, which is an evidence for the noncommutative Ward conjecture. Finally, the integrable coupling system of the noncommutative AKNS equation hierarchy is constructed by using the Kronecker product.     

The multicomponent （2＋1）-dimensional Glachette-Johnson （GJ） equation hierarchy and its super-integrable coupling system

This paper presents a set of multicomponent matrix Lie algebra, which is used to construct a new loop algebra A^-M. By using the Tu scheme, a Liouville integrable multicomponent equation hierarchy is generated, which possesses the Hamiltonian structure. As its reduction cases, the multicomponent （2＋1）-dimensional Glachette-Johnson （G J） hierarchy is given. Finally, the super-integrable coupling system of multicomponent （2＋1）-dimensional GJ hierarchy is established through enlarging the spectral problem.     

A Hierarchy of Differential-Difference Equations and Their Integrable Couplings

Starting from a discrete spectral problem, the corresponding hierarchy of nonlinear differential-difference equation is proposed. It is shown that the hierarchy possesses the bi-Hamiltionian structures. Further, two integrable coupling systems for the hierarchy are constructed through enlarged Lax pair method.     

A nonlinear discrete integrable coupling system and its infinite conservation laws

We construct a nonlinear integrable coupling of discrete soliton hierarchy,and establish the infinite conservation laws(CLs) for the nonlinear integrable coupling of the lattice hierarchy.As an explicit application of the method proposed in the paper,the infinite conservation laws of the nonlinear integrable coupling of the Volterra lattice hierarchy are presented.     

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.