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.     

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 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.     

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.     

The multi-component Tu hierarchy of soliton equations and its multi-component integrable couplings system

A new simple loop algebra GM is constructed, which is devoted to the establishing of an isospectral problem. By making use of the Tu scheme, the multi-component Tu hierarchy is obtained.Furthermore, an expanding loop algebra FM of the loop algebra GM is presented. Based on the FM, the multi-component integrable coupling system of the multi-component Tu hierarchy has been worked out. The method can be applied to other nonlinear evolution equation hierarchies.     

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.     

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.     

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.     

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.     

A new generalized fractional Dirac soliton hierarchy and its fractional Hamiltonian structure

Based on the differential forms and exterior derivatives of fractional orders,Wu first presented the generalized Tu formula to construct the generalized Hamiltonian structure of the fractional soliton equation.We apply the generalized Tu formula to calculate the fractional Dirac soliton equation hierarchy and its Hamiltonian structure.The method can be generalized to the other fractional soliton hierarchy.     

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.     

A type of multi-component integrable hierarchy

A new isospectral problem is established by constructing a simple interesting loop algebra. A commutation operation of the loop algebra is as straightforward as the loop algebra ?_1. It follows that a type of multi-component integrable hierarchy is obtained. This can be used as a general 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.     

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.     

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.     

Nonlinear integrable couplings of a nonlinear Schrödinger—modified Korteweg de Vries hierarchy with self-consistent sources

By means of the Lie algebra B₂, a new extended Lie algebra F is constructed. Based on the Lie algebras B₂ and F, the nonlinear Schrödinger-modified Korteweg de Vries (NLS-mKdV) hierarchy with self-consistent sources as well as its nonlinear integrable couplings are derived. With the help of the variational identity, their Hamiltonian structures are generated.     

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.     

AKNS方程族的一类扩展可积模型

首先构造了loop代数A～2的一个新的子代数,设计了一个等谱问题.应用屠格式求出了著名的AKNS方程族的一类扩展可积模型,即可积耦合.然后将这种求可积耦合的方法一般化,可用于一大类方程族,如KN族、GJ族、WKI族等谱系的可积耦合.提出的方法具有普遍应用价值.最后作为AKNS方程族的特例,求得了KdV方程和MKdV方程的可积耦合 关键词： 可积耦合 loop代数 AKNS方程族     

一类S-mKdV方程族及其扩展可积模型

由loop代数的一个子代数出发，构造了一个线性等谱问题，再利用屠格式计算出了一类Liouvelle意义下的可积系统及其双Hamilton结构，作为该可积系统的约化，得到了著名的Schrdinger方程和mKdV方程，因此称该系统为S-mKdV方程族.根据已构造的的子代数，又构造了维数为5的loop代数的一个新的子代数，由此出发设计了一个线性等谱形式，再利用屠格式求得了S-mKdV方程族的一类扩展可积模型.利用这种方法还可以求BPT方程族、TB方程族等谱系的扩展可积模型.因此本方法具有普遍应用价值.最后作为特例，求得了著名的Schrdinger方程和mKdV方程的可积耦合系统.