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 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.

scholarly journals Nonlinear Super Integrable Couplings of Super Broer-Kaup-Kupershmidt Hierarchy and Its Super Hamiltonian Structures

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Sixing Tao ◽  
Tiecheng Xia
Keyword(s):  
Poisson Bracket ◽  
Integrable Hierarchy ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Integrable Couplings ◽  
Classical Integrable

Nonlinear integrable couplings of super Broer-Kaup-Kupershmidt hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction, nonlinear integrable couplings of the classical integrable hierarchy were obtained.

A HIERARCHY OF LIOUVILLE INTEGRABLE LATTICE EQUATION ASSOCIATED WITH A THREE-BY-THREE DISCRETE SPECTRAL PROBLEM AND ITS INFINITELY MANY CONSERVATION LAWS

2011 ◽  
Vol 25 (18) ◽  
pp. 2481-2492
Author(s):  
YU-QING LI ◽  
XI-XIANG XU
Keyword(s):  
Conservation Laws ◽  
Spectral Problem ◽  
Trace Identity ◽  
Lattice Equation ◽  
Hamiltonian Structures ◽  
Discrete Soliton ◽  
Soliton Equations ◽  
Integrable Lattice ◽  
Matrix Spectral Problem ◽  
Liouville Integrable

A discrete three-by-three matrix spectral problem is put forward and the corresponding discrete soliton equations are deduced. By means of the trace identity the Hamiltonian structures of the resulting equations are constructed, and furthermore, infinitely many conservation laws of the corresponding lattice system are obtained by a direct way.

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.

ON INTEGRABLE COUPLINGS OF THE DISPERSIVE LONG WAVE HIERARCHY AND THEIR HAMILTONIAN STRUCTURE

2007 ◽  
Vol 21 (01) ◽  
pp. 37-44 ◽  
Author(s):  
YUFENG ZHANG
Keyword(s):  
Quadratic Form ◽  
Lie Algebras ◽  
Compatibility Condition ◽  
Hamiltonian Structure ◽  
Loop Algebra ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Long Wave ◽  
Integrable Couplings

A new subalgebra of the loop algebra Ã3 is directly constructed and used to build a pair of Lax matrix isospectral problems. The resulting compatibility condition, i.e., zero curvature equation, gives rise to integrable couplings of the dispersive long wave hierarchy, as an application example. Through using a proper isomorphic map between two Lie algebras, two equivalent zero curvature equations are presented from which the Hamiltonian structure of the integrable couplings is obtained by the quadratic-form identity. The proposed method can be applied to the construction of integrable couplings and the corresponding Hamiltonian structures of other existing soliton hierarchies.

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.

scholarly journals Nonlinear Super Integrable Couplings of Super Classical-Boussinesq Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Xiuzhi Xing ◽  
Jingzhu Wu ◽  
Xianguo Geng
Keyword(s):  
Integrable Hierarchy ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Integrable Couplings ◽  
Classical Integrable

Nonlinear integrable couplings of super classical-Boussinesq hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then, its super Hamiltonian structures were established by using super trace identity. As its reduction, nonlinear integrable couplings of the classical integrable hierarchy were obtained.

scholarly journals A New Super Extension of Dirac Hierarchy

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Jiao Zhang ◽  
Fucai You ◽  
Yan Zhao
Keyword(s):  
Spectral Problem ◽  
Integrable Hierarchy ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature

We derive a new super extension of the Dirac hierarchy associated with a3×3matrix super spectral problem with the help of the zero-curvature equation. Super trace identity is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable hierarchy.

INTEGRABLE COUPLINGS OF THE (2+1)-DIMENSIONAL Li HIERARCHY AND ITS HAMILTONIAN STRUCTURE AS WELL AS THE MULTI-COMPONENT Li HIERARCHY

2008 ◽  
Vol 22 (14) ◽  
pp. 1401-1413
Author(s):  
MING SONG ◽  
HUANHE DONG ◽  
XIULI XU ◽  
HUIJIAN GAO
Keyword(s):  
Quadratic Form ◽  
Hamiltonian Structure ◽  
Loop Algebra ◽  
Yang Mills ◽  
Reduced Equations ◽  
Integrable Couplings ◽  
Liouville Integrable

A (2+1)-dimensional Li hierarchy is generated from one of the reduced equations of self-dual Yang–Mills equations. With the help of a proper loop algebra, the Hamiltonian structure of its expanding integrable couplings is worked out by using the quadratic-form identity, which is Liouville integrable. Moreover, a corresponding multi-component Li hierarchy is given. The method can be widely applied to other soliton hierarchies.

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