A New Hierarchy of Lax and Liouville Integrable Evolution Equations Associated with an Isospectral Problem in the Loop Algebra Ã2

2006 ◽  
Vol 19 (3) ◽  
pp. 301-306
Author(s):  
Zhenya Yan
Keyword(s):  
Evolution Equations ◽  
Loop Algebra ◽  
Integrable Evolution ◽  
Isospectral Problem ◽  
Liouville Integrable

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.

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 NEW METHOD FOR GENERATING A NEW INTEGRABLE SYSTEM

2011 ◽  
Vol 25 (11) ◽  
pp. 1553-1558
Author(s):  
XIURONG GUO
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Heat Conduction Equation ◽  
Evolution Equations ◽  
Integrable Model ◽  
Loop Algebra ◽  
Heat Condition ◽  
Higher Dimensional ◽  
Isospectral Problem ◽  
Constrained Conditions

With the help of the known Lie algebra given by Zhang,2 a new higher-dimensional Lie algebra G is obtained by generalizing the commutative operations in the Lie algebras. Using a subalgebra [Formula: see text] of a loop algebra [Formula: see text] which corresponds to the Lie algebra G, a new heat-conduction equation hierarchy with some constrained conditions, is obtained. We again consider the constrained conditions as new evolution equations, the new scheme for generating soliton equations are given. Then we use the loop algebra [Formula: see text] to further establish an isospectral problem and derive an extending integrable model of the above heat-condition hierarchy, we also obtain a corresponding extending constrained condition which is thought as a type of evolution equations.

A HIERARCHY OF HAMILTONIAN EQUATIONS ASSOCIATED WITH THE MODIFIED JAULENT–MIODEK HIERARCHY

2010 ◽  
Vol 24 (02) ◽  
pp. 183-193
Author(s):  
HAI-YONG DING ◽  
HONG-XIANG YANG ◽  
YE-PENG SUN ◽  
LI-LI ZHU
Keyword(s):  
Eigenvalue Problem ◽  
Evolution Equations ◽  
Trace Identity ◽  
Liouville Integrability ◽  
Matrix Eigenvalue Problem ◽  
Hamiltonian Structures ◽  
Hamiltonian Equations ◽  
Matrix Eigenvalue ◽  
Integrable Evolution

By considering a new four-by-four matrix eigenvalue problem, a hierarchy of Lax integrable evolution equations with four potentials is derived. The Hamiltonian structures of the resulting hierarchy are established by means of the generalized trace identity. The Liouville integrability for the hierarchy of the resulting Hamiltonian equations is presented.

A FOUR-BY-FOUR MATRIX EIGENVALUE PROBLEM, RELATED HIERARCHY OF LAX INTEGRABLE EQUATIONS AND THEIR HAMILTONIAN STRUCTURES

2008 ◽  
Vol 22 (23) ◽  
pp. 4027-4040 ◽  
Author(s):  
XI-XIANG XU ◽  
HONG-XIANG YANG ◽  
WEI-LI CAO
Keyword(s):  
Eigenvalue Problem ◽  
Evolution Equations ◽  
Trace Identity ◽  
Liouville Integrability ◽  
Matrix Eigenvalue Problem ◽  
Hamiltonian Structures ◽  
Hamiltonian Equations ◽  
Integrable Equations ◽  
Matrix Eigenvalue ◽  
Integrable Evolution

Starting from a new four-by-four matrix eigenvalue problem, a hierarchy of Lax integrable evolution equations with four potentials is derived. The Hamiltonian structures of the resulting hierarchy are established by means of the generalized trace identity. The Liouville integrability for the hierarchy of the resulting Hamiltonian equations is proved.

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