Two new integrable couplings of the soliton hierarchies with self-consistent sources

A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with sl(4) has been presented (Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613). Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra sl(4). In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.     

High-Order Binary Symmetry Constraints of a Liouville Integrable Hierarchy and Its Integrable Couplings

A 3 × 3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra A^-2. Furthermore, high-order binary symmetry constraints of the corresponding hierarchy are obtained by using the binary nonlinearization method. Finally, according to another new subalgebra of loop algebra A^-2, its integrable couplings are established.     

A New Lie Algebra and Its Related Liouville Integrable Hierarchies

A new Lie algebra G and its two types of loop algebras \tilde{G₁} and \tilde{G₂} are constructed. Basing on \tilde{G₁} and \tilde{G₂}, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obtained respectively under the framework of zero curvature equation, which is derived from the compatibility of the isospectral problems expressed by Hirota operators. At the same time, we obtain the Hamiltonian structure of the first hierarchy and the bi-Hamiltonian structure of the second one with the help of the quadratic-form identity.     

Multi-component Dirac equation hierarchy and its multi-component integrable couplings system

A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M-dimensional loop algebra \tilde{X} is produced. By taking advantage of \tilde{X}, a new isospectral problem is established and then by making use of the Tu scheme the multi-component Dirac equation hierarchy is obtained. Finally, an expanding loop algebra \tilde{F}_M of the loop algebra \tilde{X} is presented. Based on the \tilde{F}_M, the multi-component integrable coupling system of the multi-component Dirac equation hierarchy is investigated. The method in this paper can be applied to other nonlinear evolution equation hierarchies.     

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.     

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.     

Some Evolution Hierarchies Derived from Self-dual Yang-Mills Equations

We develop in this paper a new method to construct two explicit Lie algebras E and F. By using a loop algebra \bar{E} of the Lie algebra E and the reduced self-dual Yang-Mills equations, we obtain an expanding integrable model of the Giachetti-Johnson (GJ) hierarchy whose Hamiltonian structure can
also be derived by using the trace identity. This provides a much simplier construction method in comparing with the tedious variational identity approach. Furthermore, the nonlinear integrable coupling of the GJ hierarchy is readily obtained by introducing the Lie algebra g_N. As an application, we apply the loop algebra \tilde{E} of the Lie algebra E to obtain a kind of expanding integrable model of the Kaup-Newell (KN) hierarchy which, consisting of two arbitrary parametersα andβ, can be reduced to two nonlinear evolution equations. In addition, we use a loop algebra \tilde{F} of the Lie algebra F to obtain an
expanding integrable model of the BT hierarchy whose Hamiltonian structure is the same as using the trace identity. Finally, we deduce five integrable systems in R³ based on the self-dual Yang-Mills equations, which include Poisson structures, irregular lines, and the reduced equations.     

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 F^-M of the loop algebra G^-M is presented. Based on F^-M , 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.     

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.     

A generalized SHGI integrable hierarchy and its expanding integrable model

An anti-symmetric loop algebra \overline{A}_2 is constructed. It follows that an integrable system is obtained by use of Tu's scheme. The eminent feature of this integrable system is that it is reduced to a generalized Schr?dinger equation, the well-known heat-conduction equation and a Gerdjkov－Ivanov (GI) equation. Therefore, we call it a generalized SHGI hierarchy. Next, a new high-dimensional subalgebra \tilde{G} of the loop algebra ?_2 is constructed. As its application, a new expanding integrable system with six potential functions is engendered.     

New Integrable Couplings of Generalized Kaup--Newell Hierarchy and Its Hamiltonian Structures

A new isospectral problem is firstly presented, then we derive integrable system of soliton hierarchy. Also we obtain new integrable couplings of the eneralized Kaup--Newell soliton equations hierarchy and its Hamiltonian structures by using Tu scheme and the quadratic-form identity. The method can be
generalized to other soliton hierarchy.     

Rosochatius Deformed Soliton Hierarchy with Self-Consistent Sources

Integrable Rosochatius deformations of finite-dimensional integrable systems are generalized to the soliton hierarchy with self-consistent sources. The integrable Rosochatius deformations of the Kaup-Newell hierarchy with self-consistent sources, of the TD hierarchy with self-consistent sources, and of the Jaulent-Miodek hierarchy with self-consistent sources, together with their Lax representations are presented.     

Non-isospectral integrable couplings of Ablowitz-Kaup-Newell-Segur （AKNS） hierarchy with self-consistent sources

A hierarchy of non-isospectral Ablowitz-Kaup-Newell-Segur （AKNS） equations with self-consistent sources is derived. As a general reduction case, a hierarchy of non-isospectral nonlinear SchrSdinger equations （NLSE） with selfconsistent sources is obtained. Moreover, a new non-isospectral integrable coupling of the AKNS soliton hierarchy with self-consistent sources is constructed by using the Kronecker product.     

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.     

Multi-component Harry--Dym hierarchy and its integrable couplings as well as their Hamiltonian structures

This paper obtains the multi-component Harry--Dym (H--D) hierarchy and its integrable couplings by using two kinds of vector loop algebras \widetilde{G}₃ and \widetilde{G}₆. The Hamiltonian structures of the above system are given by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.     

Non-isospectral integrable couplings of Ablowitz-Ladik hierarchy with self-consistent sources

A kind of new non-isospectral integrable couplings of discrete soliton equations hierarchy with self-consistent sources associated with is presented. As an application example, the integrable coupling hierarchy of non-isospectral Ablowitz-Ladik with self-consistent sources is derived by using of the loop algebra .     

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.     

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.     

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 Loop Algebra and Its Corresponding Multi-component Integrable Hierarchy

A type of new loop algebra $\tilde{G}_M$ is constructed by making use of the concept of cycled numbers. As its application, an isospectral problem is designed and a new multi-component integrable hierarchy with multi-potential functions is worked out, which can be reduced to the famous KN hierarchy.