A trick loop algebra and a corresponding Liouville integrable hierarchy of evolution equations

2004 ◽  
Vol 21 (2) ◽  
pp. 445-456 ◽  
Author(s):  
Yufeng Zhang ◽  
Xixiang Xu
Keyword(s):  
Evolution Equations ◽  
Loop Algebra ◽  
Integrable Hierarchy ◽  
Liouville Integrable

A NEW LOOP ALGEBRA AND TWO NEW LIOUVILLE INTEGRABLE HIERARCHY

2007 ◽  
Vol 21 (11) ◽  
pp. 663-673 ◽  
Author(s):  
HUAN-HE DONG
Keyword(s):  
Quadratic Form ◽  
Loop Algebra ◽  
Integrable Hierarchy ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Liouville Integrable

A new loop algebra containing four arbitrary constants is presented, and the corresponding computing formula of constant γ in the quadratic-form identity is obtained in this paper, which can be reduced to a computing formula of constant γ in the trace identity. As application, two new Liouville integrable hierarchy and Hamiltonian structures are derived.

scholarly journals A New Approach for Generating the TX Hierarchy as well as Its Integrable Couplings

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Guangming Wang
Keyword(s):  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Loop Algebra ◽  
Integrable Hierarchy ◽  
New Approach ◽  
Zero Curvature ◽  
Integrable Couplings ◽  
Isospectral Problem ◽  
Integrable Coupling

Tu Guizhang and Xu Baozhi once introduced an isospectral problem by a loop algebra with degree beingλ, for which an integrable hierarchy of evolution equations (called the TX hierarchy) was derived under the frame of zero curvature equations. In the paper, we present a loop algebra whose degrees are2λand2λ+1to simply represent the above isospectral matrix and easily derive the TX hierarchy. Specially, through enlarging the loop algebra with 3 dimensions to 6 dimensions, we generate a new integrable coupling of the TX hierarchy and its corresponding Hamiltonian structure.

A HIERARCHY OF SOLITON EQUATIONS ASSOCIATED WITH A HIGHER-DIMENSIONAL LOOP ALGEBRA AND ITS TRI-HAMILTONIAN STRUCTURE

2008 ◽  
Vol 22 (19) ◽  
pp. 1837-1850 ◽  
Author(s):  
YUFENG ZHANG ◽  
YAN LI
Keyword(s):  
Compatibility Condition ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Loop Algebra ◽  
Trace Identity ◽  
Higher Dimensional ◽  
Soliton Hierarchy ◽  
Isospectral Problem ◽  
Set Up ◽  
Liouville Integrable

A new higher-dimensional loop algebra is given for which a Lax isospectral problem is set up whose compatibility condition gives rise to a Liouville integrable soliton hierarchy along with eight-component potential functions. Specially, the hierarchy of evolution equations has a tri-Hamiltonian structure obtained by the trace identity.

The Multi-Component Jaulent-Miodek Hierarchy and its Multi-Component Integrable Coupling System with Two Arbitrary Functions

2012 ◽  
Vol 442 ◽  
pp. 124-128
Author(s):  
Jian Ya Ge ◽  
Tie Cheng Xia
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Loop Algebra ◽  
Simple Loop ◽  
Coupling System ◽  
Integrable Couplings ◽  
Isospectral Problem ◽  
Integrable Coupling

We devise a new simple loop algebra GM and an isospectral problem. By making use of Tu scheme, the multi-component Jaulent-Miodek (JM) hierarchy is obtained. Furthermore, an expanding loop algebra FM of the loop algebra GM is presented. Based on FM the multi-component integrable couplings system with two arbitrary functions of the multi-component Jaulent-Miodek (JM) hierarchy are worked out. The method can be applied to other nonlinear evolution equations hierarchies.

INTEGRABLE HIERARCHIES FROM BRST COHOMOLOGY

2005 ◽  
Vol 20 (01) ◽  
pp. 51-59
Author(s):  
V. CALIAN ◽  
G. STOENESCU
Keyword(s):  
Field Theory ◽  
Hamiltonian System ◽  
Hamiltonian Structure ◽  
Integrable Hierarchies ◽  
Evolution Equations ◽  
Classical Field Theory ◽  
Classical Field ◽  
Integrable Hierarchy ◽  
Brst Cohomology ◽  
Gauge Fixing

We demonstrate that the Hamiltonian structure and the integrability of a system of evolution equations can be formulated in terms of a classical field theory using BRST and anti-BRST symmetries. We derive the field theory action and explicitly generate the integrable hierarchy associated to a bi-Hamiltonian system based on cohomological arguments and gauge-fixing deformations.

A nonisospectral integrable model of AKNS hierarchy and KN hierarchy, as well as its extended system

2021 ◽  
pp. 2150156
Author(s):  
Haifeng Wang ◽  
Yufeng Zhang
Keyword(s):  
Hamiltonian System ◽  
Integrable Model ◽  
Loop Algebra ◽  
Integrable Hierarchy ◽  
Trace Identity ◽  
Extended System ◽  
Akns Hierarchy ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Higher Dimensional

In this paper, we first introduce a nonisospectral problem associate with a loop algebra. Based on the nonisospectral problem, we deduce a nonisospectral integrable hierarchy by solving a nonisospectral zero curvature equation. It follows that the standard AKNS hierarchy and KN hierarchy are obtained by reducing the resulting nonisospectral hierarchy. Then, the Hamiltonian system of the resulting nonisospectral hierarchy is investigated based on the trace identity. Additionally, an extended integrable system of the resulting nonisospectral hierarchy is worked out based on an expanded higher-dimensional Loop algebra.

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