On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations

In the paper,we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly.By the approach the various loop algebras of the Lie algebra A_1are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained,respectively.A reduction of the later hierarchy is just right the famous Ablowitz-Ladik hierarchy.Finally,via two different enlarging Lie algebras of the Lie algebra A_1,we derive two resulting differential-difference integrable couplings of the Toda hierarchy,of course,they are all various discrete expanding integrable models of the Toda hierarchy.When the introduced spectral matrices are higher degrees,the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple.     

Upper Triangular Matrix of Lie Algebra and a New Discrete Integrable Coupling System

The upper triangular matrix of Lie algebra is used to construct integrable couplings of discrete solition equations. Correspondingly, a feasible way to construct integrable couplings is presented. A nonlinear lattice soliton equation spectral problem is obtained and leads to a novel hierarchy of the nonlinear lattice equation hierarchy. It indicates that the study of integrable couplings using upper triangular matrix of Lie algebra is an important step towards constructing integrable systems.     

密度矩阵重正化群的异构并行优化

密度矩阵重正化群方法（DMRG）在求解一维强关联格点模型的基态时可以获得较高的精度,在应用于二维或准二维问题时,要达到类似的精度通常需要较大的计算量与存储空间.本文提出一种新的DMRG异构并行策略,可以同时发挥计算机中央处理器（CPU）和图形处理器（GPU）的计算性能.针对最耗时的哈密顿量对角化部分,实现了数据的分布式存储,并且给出了CPU和GPU之间的负载平衡策略.以费米Hubbard模型为例,测试了异构并行程序在不同DMRG保留状态数下的运行表现,并给出了相应的性能基准.应用于4腿梯子时,观测到了高温超导中常见的电荷密度条纹,此时保留状态数达到10⁴,使用的GPU显存小于12 GB.     

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.     

The multicomponent discrete equation hierarchy with variable spectral parameters and a new integrable coupling system

This Letter reports on a study of the multicomponent discrete integrable equation hierarchy with variable spectral parameters. An instance of new multicomponent lattice equation is obtained. Correspondingly, a feasible way to construct the integrable couplings is presented. It indicates that the study of integrable couplings using the block matrix-form of Lie algebra is an important and effective method.     

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.     

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 Hierarchy of Lax Integrable Lattice Equations,Liouville Integrability and a New Integrable Symplectic Map

A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann constraint of the resulting hierarchy. Finally, an infinite set of conservation laws is given for the resulting hierarchy.     

A hierarchy of Liouville integrable discrete Hamiltonian equations

Based on a discrete four-by-four matrix spectral problem, a hierarchy of Lax integrable lattice equations with two potentials is derived. Two Hamiltonian forms are constructed for each lattice equation in the resulting hierarchy by means of the discrete variational identity. A strong symmetry operator of the resulting hierarchy is given. Finally, it is shown that the resulting lattice equations are all Liouville integrable discrete Hamiltonian systems.     

A Hierarchy of Integrable Lattice Soliton Equations and New Integrable Symplectic Map

Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure. A new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a Bäcklund transformation for the resulting integrable lattice equations. At last, conservation laws of the hierarchy are presented.     

The Application of Discrete Variational Identity on Semi-Direct Sums of Lie Algebras

With non-semisimple Lie algebras, the trace identity was generalized to discrete spectral problems. Then the corresponding discrete variational identity was used to a class of semi-direct sums of Lie algebras in a lattice hierarchy case and obtained Hamiltonian structures for the associated integrable couplings of the lattice hierarchy. It is a powerful tool for exploring Hamiltonian structures of discrete soliton equations.     

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.     

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.     

A Few Expanding Integrable Models,Hamiltonian Structures and Constrained Flows

Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced, for which a few expanding integrable models including the coupling integrable couplings of the Broer-Kaup (BK) hierarchy and the dispersive long wave (DLW) hierarchy as well as the TB hierarchy are obtained. From the reductions of the coupling integrable couplings, the corresponding coupled integrable couplings of the BK equation, the DLW equation, and the TB equation are obtained, respectively. Especially, thecoupling integrable coupling of the TB equation reduces to a few integrable couplings of the well-known mKdV equation. The Hamiltonian structures of the coupling integrable couplings of the three kinds of soliton hierarchies are worked out, respectively, by employing the variational identity. Finally, we decompose the BK hierarchy of evolution equations into x-constrained flows and t_n-constrained flows whose adjoint representations and the Lax pairs are given.     

A real nonlinear integrable couplings of continuous soliton hierarchy and its Hamiltonian structure

Some integrable coupling systems of existing papers are linear integrable couplings. In the Letter, beginning with Lax pairs from special non-semisimple matrix Lie algebras, we establish a scheme for constructing real nonlinear integrable couplings of continuous soliton hierarchy. A direct application to the AKNS spectral problem leads to a novel nonlinear integrable couplings, then we consider the Hamiltonian structures of nonlinear integrable couplings of AKNS hierarchy with the component-trace identity.     

Double Integrable Couplings and Their Constructing Method

To extend the study scopes of integrable couplings, the notion of double integrable couplings is proposed in the paper. The zero curvature equation appearing in the constructing method built in the paper consists of the elements of a new loop algebra which is obtained by using perturbation method. Therefore, the approach given in the paper has extensive applicablevalues, that is, it applies to investigate a lot of double integrable couplings of the known integrable hierarchies of evolution equations. As for explicit applications of the method proposed in the paper, the double integrable couplings of the AKNS hierarchy and the KN hierarchy are worked out, respectively.     

Higher-Dimensional Lie Algebra and New Integrable Coupling of Discrete KdV Equation

Based on semi-direct sums of Lie subalgebra \tilde{G}, a higher-dimensional 6 x 6 matrix Lie algebra sμ(6) is constructed. A hierarchy of integrable coupling KdV equation with three potentials is proposed, which is derivedfrom a new discrete six-by-six matrix spectral problem. Moreover, the Hamiltonian forms is deduced for lattice equation in the resulting hierarchy by means of the discrete variational identity --- a generalized trace identity. A strong symmetry operator of the resulting hierarchy is given. Finally, we provethat the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian systems.     

Semi-direct sums of Lie algebras and continuous integrable couplings

A relation between semi-direct sums of Lie algebras and integrable couplings of continuous soliton equations is presented, and correspondingly, a feasible way to construct integrable couplings is furnished. A direct application to the AKNS spectral problem leads to a novel hierarchy of integrable couplings of the AKNS hierarchy of soliton equations. It is also indicated that the study of integrable couplings using semi-direct sums of Lie algebras is an important step towards complete classification of integrable systems.     

Discrete integrable couplings associated with Toda-type lattice and two hierarchies of discrete soliton equations

With the help of two semi-direct sum Lie algebras, an efficient way to construct discrete integrable couplings is proposed. As its applications, the discrete integrable couplings of the Toda-type lattice equations are obtained. The approach can be devoted to establishing other discrete integrable couplings of the discrete lattice integrable hierarchies of evolution equations.     

A hierarchy of non-isospectral multi-component AKNS equations and its integrable couplings

A hierarchy of non-isospectral multi-component AKNS equations is derived from an arbitrary order matrix spectral problem. As a reduction, non-isospectral multi-component Schrödinger equations are obtained. Moreover, new non-isospectral integrable couplings of the resulting AKNS soliton hierarchy are constructed by enlarging the associated matrix spectral problem.