A New Integrable Couplings of Classical-Boussinesq Hierarchy withSelf-Consistent Sources

A kind of integrable couplings of soliton equations hierarchy with self-consistent sources associated with \tilde{sl}(4) is presented by Yu. Based on this method, we construct a new integrable couplings of the classical-Boussinesq hierarchy with self-consistent sources by using of loop algebra \tilde{sl}(4). In this paper, we also point out that there exist some errors in Yu's paper and have corrected these errors and set up new formula. The method can be generalized other soliton hierarchy with self-consistent sources.     

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.     

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.     

A new extended q-deformed KP hierarchy

Abstract

A method is proposed in this paper to construct a new extended q-deformed KP (q-KP) hiearchy and its Lax representation. This new extended q-KP hierarchy contains two types of q-deformed KP equation with self-consistent sources, and its two kinds of reductions give the q-deformed Gelfand-Dickey hierarchy with self-consistent sources and the constrained q-deformed KP hierarchy, which include two types of q-deformed KdV equation with sources and two types of q-deformed Boussinesq equation with sources. All of these results reduce to the classical ones when q goes to 1. This provides a general way to construct (2+1)- and (1+1)-dimensional q-deformed soliton equations with sources and their Lax representations.     

The quasiclassical limit of the symmetry constraint of the KP hierarchy and the dispersionless KP hierarchy with self-consistent sources

Abstract

For the first time we show that the quasiclassical limit of the symmetry constraint of the Sato operator for the KP hierarchy leads to the generalized Zakharov reduction of the Sato function for the dispersionless KP (dKP) hierarchy which has been proved to be result of symmetry constraint of the dKP hierarchy recently. By either regarding the symmetry constrained dKP hierarchy as its stationary case or taking the dispersionless limit of the KP hierarchy with self-consistent sources directly, we construct a new integrable dispersionless hierarchy, i.e., the dKP hierarchy with self-consistent sources and find its associated conservation equations (or equations of Hamilton-Jacobi type). Some solutions of the dKP equation with self-consistent sources are also obtained by hodograph transformations.     

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.     

Integrability on generalized q-Toda equation and hierarchy

In this paper, we construct a new integrable equation which is a generalization of q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the generalized q-Toda equation and a whole integrable generalized q-Toda hierarchy are also constructed. To show the integrability, the Bi-Hamiltonian structure and tau symmetry of the generalized q-Toda hierarchy are given and this leads to the tau function.     

On generalized Lax equation of the Lax triple of KP Hierarchy

In terms of the operator Nambu 3-bracket and the Lax pair (L, B_n) of the KP hierarchy, we propose the generalized Lax equation with respect to the Lax triple (L, B_n, B_m). The intriguing results are that we derive the KP equation and another integrable equation in the KP hierarchy from the generalized Lax equation with the different Lax triples (L, B_n, B_m). Furthermore we derive some no integrable evolution equations and present their single soliton solutions.     

Factorized Weight Functions vs. Factorized Scattering

Starting from an extensive class of factorized weight functions W(p) on the N-dimensional torus ?, we construct an orthonormal base of symmetric N-variable polynomials for L ² _s (?,W(p)dp) via lexicographic ordering of the monomial symmetric functions (free boson states) and the Gram-Schmidt procedure. We show that the dominant asymptotics of these polynomials is factorized. As a corollary, we obtain a large class of quantum integrable soliton systems on the symmetric subspace of l ²(ℤ ^N ). The class of weight functions contains in particular the weight function yielding Macdonald polynomials. For that special case, the quantum soliton system can be viewed as the dual relativistic Calogero–Sutherland system. Received: 4 September 2001 / Accepted: 4 January 2002     

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.     

Conservation laws and self-consistent sources for a super-CKdV equation hierarchy

From the super-matrix Lie algebras, we consider a super-extension of the CKdV equation hierarchy in the present Letter, and propose the super-CKdV hierarchy with self-consistent sources. Furthermore, we establish the infinitely many conservation laws for the integrable super-CKdV hierarchy.     

A new extended KP hierarchy

A method is proposed to construct a new extended KP hierarchy, which includes two types of KP equation with self-consistent sources and admits reductions to k-constrained KP hierarchy and to Gelfand-Dickey hierarchy with sources. It provides a general way to construct soliton equations with sources and their Lax representations.     

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 .     

A Direct Method of Hamiltonian Structure

A direct method of constructing the Hamiltonian structure of the soliton hierarchy with self-consistent sources is proposed through computing the functional derivative under some constraints. The Hamiltonian functional is related with the conservation densities of the corresponding hierarchy. Three examples and their two reductions are given.     

Nonlinear integrable couplings of a nonlinear Schrdinger-modified Korteweg de Vries hierarchy with self-consistent sources

By means of the Lie algebra B 2,a new extended Lie algebra F is constructed.Based on the Lie algebras B 2 and F,the nonlinear Schro¨dinger-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.     

A new six-component super soliton hierarchy and its self-consistent sources and conservation laws

A new six-component super soliton hierarchy is obtained based on matrix Lie super algebras. Super trace identity is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable hierarchy. After that, the selfconsistent sources of the new six-component super soliton hierarchy are presented. Furthermore, we establish the infinitely many conservation laws for the integrable super 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.     

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.     

Bilinear Identities and Hirota's Bilinear Forms for an Extended Kadomtsev-Petviashvili Hierarchy

Abstract

In this paper, we construct the bilinear identities for the wave functions of an extended Kadomtsev-Petviashvili (KP) hierarchy, which is the KP hierarchy with particular extended flows. By introducing an auxiliary parameter, whose flow corresponds to the so-called squared eigenfunction symmetry of KP hierarchy, we find the tau-function for this extended KP hierarchy. It is shown that the bilinear identities will generate all the Hirota's bilinear equations for the zero-curvature forms of the extended KP hierarchy, which includes two types of KP equation with self-consistent sources (KPSCS). The Hirota's bilinear equations obtained in this paper for the KPSCS are in different forms by comparing with the existing results.