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.

INTEGRABLE COUPLINGS OF THE TC HIERARCHY AND ITS HAMILTONIAN STRUCTURE

2007 ◽  
Vol 21 (10) ◽  
pp. 595-602 ◽  
Author(s):  
ZHU LI ◽  
YUJUAN ZHANG ◽  
HUANHE DONG
Keyword(s):  
Quadratic Form ◽  
Hamiltonian Structure ◽  
Loop Algebra ◽  
Integrable Couplings

Integrable couplings of the TC hierarchy is obtained by use of the new subalgebra of the loop algebra Ã3, then the Hamiltonian structure of the above system is given by the quadratic-form identity.

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.

TWO (2+1)-DIMENSIONAL INTEGRABLE COUPLINGS OF THE KN HIERARCHY AND CORRESPONDING HAMILTONIAN STRUCTURES

2009 ◽  
Vol 23 (30) ◽  
pp. 3643-3658
Author(s):  
CHAO YUE ◽  
ZHAOJUN LIU ◽  
JIADONG YU
Keyword(s):  
Quadratic Form ◽  
Burgers Equation ◽  
The Self ◽  
Hamiltonian Structures ◽  
Yang Mills ◽  
Curvature Equation ◽  
Reduced Equations ◽  
Zero Curvature ◽  
Integrable Couplings

A (2+1) zero curvature equation is generated from one of the reduced equations of the self-dual Yang–Mills equations. As its applications, two (2+1)-dimensional integrable couplings of the famous KN hierarchy are obtained with the help of a subalgebra of the Lie subalgebra R9, which can be reduced to the Burgers equation. Furthermore, their Hamiltonian structures are worked out by taking use of the quadratic-form identity and the variational identity, respectively.

A LIOUVILLE INTEGRABLE MULTI-COMPONENT INTEGRABLE SYSTEM AND ITS INTEGRABLE COUPLINGS

2010 ◽  
Vol 24 (08) ◽  
pp. 1021-1046
Author(s):  
LI ZHU ◽  
HONGWEI YANG ◽  
HUANHE DONG
Keyword(s):  
Integrable System ◽  
Hamiltonian Structure ◽  
Loop Algebra ◽  
Integrable Couplings ◽  
Liouville Integrable

A Liouville integrable multi-component integrable system is obtained by the vector loop algebra. Then, the integrable couplings of the above system are presented by using the expanding vector loop algebra [Formula: see text] of the [Formula: see text]. Finally, the bi -Hamiltonian structure of the obtained system is given, respectively, by the variational identity.

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 Fractional Quadratic-Form Identity and Hamiltonian Structure of an Integrable Coupling of the Fractional Broer-Kaup Hierarchy

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Chao Yue ◽  
Tiecheng Xia ◽  
Guijuan Liu ◽  
Jianbo Liu
Keyword(s):  
Quadratic Form ◽  
Fractional Order ◽  
Hamiltonian Structure ◽  
Integrable Couplings ◽  
Isospectral Problem ◽  
Integrable Coupling

A fractional quadratic-form identity is derived from a general isospectral problem of fractional order, which is devoted to constructing the Hamiltonian structure of an integrable coupling of the fractional BK hierarchy. The method can be generalized to other fractional integrable couplings.

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.

NEW INTEGRABLE LATTICE HIERARCHY AND ITS INTEGRABLE COUPLING

2009 ◽  
Vol 23 (23) ◽  
pp. 4791-4800 ◽  
Author(s):  
ZHU LI ◽  
HUANHE DONG
Keyword(s):  
Spectral Problem ◽  
Hamiltonian Structure ◽  
Lattice Equation ◽  
Integrable Lattice ◽  
Integrable Couplings ◽  
Liouville Integrable ◽  
Integrable Coupling

New hierarchy of Liouville integrable lattice equation and their Hamiltonian structure are generated by use of the Tu model. Then, integrable couplings of the obtained system is worked out by the extending spectral problem.

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