INTEGRABLE COUPLING SYSTEMS OF HAMILTONIAN LATTICE EQUATIONS BY SEMI-DIRECT SUMS OF LIE ALGEBRAS

2010 ◽  
Vol 24 (07) ◽  
pp. 681-694
Author(s):  
LI-LI ZHU ◽  
JUN DU ◽  
XIAO-YAN MA ◽  
SHENG-JU SANG
Keyword(s):  
Eigenvalue Problem ◽  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Liouville Integrability ◽  
Direct Sums ◽  
Direct Sum ◽  
Hamiltonian Lattice ◽  
Integrable Couplings ◽  
Type Lattice ◽  
Integrable Coupling

By considering a discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations are derived. The relation to the Toda type lattice is achieved by variable transformation. With the help of Tu scheme, the Hamiltonian structure of the resulting lattice hierarchy is constructed. The Liouville integrability is then demonstrated. Semi-direct sum of Lie algebras is proposed to construct discrete integrable couplings. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

A FAMILY OF DISCRETE INTEGRABLE COUPLING SYSTEMS AND ITS LIOUVILLE INTEGRABILITY

2009 ◽  
Vol 23 (13) ◽  
pp. 1671-1685
Author(s):  
XI-XIANG XU ◽  
HONG-XIANG YANG
Keyword(s):  
Spectral Problem ◽  
Lie Algebras ◽  
Integrable Systems ◽  
Liouville Integrability ◽  
Discrete Integrable Systems ◽  
Direct Sum ◽  
The Family ◽  
Matrix Spectral Problem ◽  
Hamiltonian Forms ◽  
Integrable Coupling

A discrete matrix spectral problem and corresponding family of discrete integrable systems are discussed. A semi-direct sum of Lie algebras of four-by-four matrices is introduced, and the related integrable coupling systems of resulting discrete integrable systems are derived. The obtained discrete integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, Liouville integrability of the family of obtained integrable coupling systems is demonstrated.

SOME EXPANDING SEMI-DIRECT SUMS OF LIE ALGEBRA $\bar{G}$ AND ITS APPLICATION

2008 ◽  
Vol 22 (32) ◽  
pp. 3215-3225 ◽  
Author(s):  
HUI CHANG ◽  
SHUANGLING CHANG ◽  
YU ZHANG
Keyword(s):  
Lie Algebra ◽  
Direct Sums ◽  
Direct Sum ◽  
Higher Dimensional ◽  
Integrable Couplings ◽  
Type Lattice

Based on the fundamental special Lie algebra [Formula: see text], some semi-direct sums are presented. Then, a new semi-direct sum Lie algebra [Formula: see text] is constructed. The method presented in this paper can be used to obtain other higher-dimensional Lie algebra. Making use of the above Lie algebra, three integrable couplings of the new Bargmann type lattice hierarchy are obtained.

scholarly journals On (2+1)-dimensional expanding integrable model of the Davey-Stewartson hierarchy

2021 ◽  
Vol 25 (6 Part B) ◽  
pp. 4431-4439
Author(s):  
Xiu-Rong Guo ◽  
Fang-Fang Ma ◽  
Juan Wang
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Integrable Model ◽  
Coupling System ◽  
Integrable Coupling ◽  
Type Abstraction Hierarchy

This paper mainly investigates the reductions of an integrable coupling of the Levi hierarchy and an expanding model of the (2+1)-dimensional Davey-Stewartson hierarchy. It is shown that the integrable coupling system of the Levi hierarchy possesses a quasi-Hamiltonian structure under certain constraints. Based on the Lie algebras construct, The type abstraction hierarchy scheme is used to gener?ate the (2+1)-dimensional expanding integrable model of the Davey-Stewartson hierarchy.

A SOLITON HIERARCHY FROM THE LEVI SPECTRAL PROBLEM AND ITS TWO INTEGRABLE COUPLINGS, HAMILTONIAN STRUCTURE

2009 ◽  
Vol 23 (05) ◽  
pp. 731-739
Author(s):  
YONGQING ZHANG ◽  
YAN LI
Keyword(s):  
Spectral Problem ◽  
Lie Algebras ◽  
Hamiltonian Structure ◽  
Soliton Equation ◽  
Curvature Equation ◽  
Loop Algebras ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Time Part

A soliton-equation hierarchy from the D. Levi spectral problem is obtained under the framework of zero curvature equation. By employing two various multi-component Lie algebras and the loop algebras, we enlarge the Levi spectral problem and the corresponding time-part isospectral problems so that two different integrable couplings are produced. Using the quadratic-form identity yields the Hamiltonian structure of one of the two integrable couplings.

scholarly journals A Hierarchy of Discrete Integrable Coupling System with Self-Consistent Sources

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yuqing Li ◽  
Huanhe Dong ◽  
Baoshu Yin
Keyword(s):  
Quadratic Form ◽  
Hamiltonian Structure ◽  
Liouville Integrability ◽  
Soliton Equation ◽  
Coupling System ◽  
Self Consistent ◽  
Integrable Coupling

Integrable coupling system of a lattice soliton equation hierarchy is deduced. The Hamiltonian structure of the integrable coupling is constructed by using the discrete quadratic-form identity. The Liouville integrability of the integrable coupling is demonstrated. Finally, the discrete integrable coupling system with self-consistent sources is deduced.

THE MULTIPLIER AND THE COVER OF DIRECT SUMS OF LIE ALGEBRAS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250026 ◽  
Author(s):  
Ali Reza Salemkar ◽  
Behrouz Edalatzadeh
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Schur Multiplier ◽  
Direct Sums ◽  
Schur Multipliers ◽  
Direct Sum ◽  
Group Case

In this paper, we prove that the Schur multiplier of the direct sum of two arbitrary Lie algebras is isomorphic to the direct sum of the Schur multipliers of the direct factors and the usual tensor product of the Lie algebras, which is similar to the work of Miller (1952) in the group case. Also, a cover for the direct sum of two Lie algebras in terms of given covers of them will be constructed.

AN ALGEBRAIC STRUCTURE OF ZERO CURVATURE REPRESENTATIONS ASSOCIATED WITH COUPLED INTEGRABLE COUPLINGS AND APPLICATIONS TO τ-SYMMETRY ALGEBRAS

2011 ◽  
Vol 25 (23n24) ◽  
pp. 3237-3252 ◽  
Author(s):  
LIN LUO ◽  
WEN-XIU MA ◽  
ENGUI FAN
Keyword(s):  
Lie Algebras ◽  
Algebraic Structure ◽  
Algebraic Structures ◽  
Direct Sums ◽  
Zero Curvature ◽  
Integrable Couplings ◽  
Symmetry Algebras

We establish an algebraic structure for zero curvature representations of coupled integrable couplings. The adopted zero curvature representations are associated with Lie algebras possessing two sub-Lie algebras in form of semi-direct sums of Lie algebras. By applying the presented algebraic structures to the AKNS systems, we give an approach for generating τ-symmetry algebras of coupled integrable couplings.

scholarly journals Semi-direct sums of Lie algebras and continuous integrable couplings

2006 ◽  
Vol 351 (3) ◽  
pp. 125-130 ◽  
Author(s):  
Wen-Xiu Ma ◽  
Xi-Xiang Xu ◽  
Yufeng Zhang
Keyword(s):  
Lie Algebras ◽  
Direct Sums ◽  
Integrable Couplings
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