A new loop algebra and a corresponding integrable hierarchy, as well as its integrable coupling

2003 ◽  
Vol 44 (12) ◽  
pp. 5793 ◽  
Author(s):  
Fukui Guo ◽  
Yufeng Zhang
Keyword(s):  
Loop Algebra ◽  
Integrable Hierarchy ◽  
Integrable Coupling

A NEW EXPANDING INTEGRABLE HIERARCHY OF KAUP–NEWELL HIERARCHY

2006 ◽  
Vol 20 (20) ◽  
pp. 1241-1246 ◽  
Author(s):  
HUANHE DONG
Keyword(s):  
Lax Pair ◽  
Mkdv Equation ◽  
Loop Algebra ◽  
Integrable Hierarchy ◽  
Direct Sum ◽  
Reduction Case ◽  
Integrable Coupling

A higher loop algebra is constructed which is devoted to establish an isospectral Lax pair. Integrable coupling of the well-known Kaup–Newell hierarchy is obtained by using a relation of direct sum between two sub-algebras. As in the reduction case, an integrable coupling of a generalized MKdV equation is presented. The approach presented in the paper can be used generally.

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.

scholarly journals The Semidirect Sum of Lie Algebras and Its Applications to C-KdV Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xia Dong ◽  
Tiecheng Xia ◽  
Desheng Li
Keyword(s):  
Quadratic Form ◽  
Lie Algebras ◽  
Integrable Systems ◽  
The Other ◽  
Loop Algebra ◽  
Hamiltonian Structures ◽  
Kdv Hierarchy ◽  
Integrable Coupling

By use of the loop algebraG-~, integrable coupling of C-KdV hierarchy and its bi-Hamiltonian structures are obtained by Tu scheme and the quadratic-form identity. The method can be used to produce the integrable coupling and its Hamiltonian structures to the other integrable systems.

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.

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.

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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