TWO FAMILIES GENERALIZATION OF AKNS HIERARCHIES AND THEIR HAMILTONIAN STRUCTURES

2010 ◽  
Vol 24 (08) ◽  
pp. 791-805 ◽  
Author(s):  
YUNHU WANG ◽  
XIANGQIAN LIANG ◽  
HUI WANG
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature

By means of the Lie algebra G1, we construct an extended Lie algebra G2. Two different isospectral problems are designed by the two different Lie algebra G1 and G2. With the help of the variational identity and the zero curvature equation, two families generalization of the AKNS hierarchies and their Hamiltonian structures are obtained, respectively.

scholarly journals Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with SO(3)

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Jian Zhang ◽  
Chiping Zhang ◽  
Yunan Cui
Keyword(s):  
Lie Algebra ◽  
Three Dimensional ◽  
Hamiltonian Structures ◽  
Spectral Problems ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Orthogonal Lie Algebra

Based on the three-dimensional real special orthogonal Lie algebra SO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities.

scholarly journals A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ning Zhang ◽  
Xi-Xiang Xu
Keyword(s):  
Spectral Problem ◽  
Lie Algebra ◽  
Difference Equations ◽  
Vector Fields ◽  
Hamiltonian Structure ◽  
Trace Identity ◽  
Zero Curvature Representation ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Liouville Integrable

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.

scholarly journals On Construction of a Super Hierarchy of the Wadati-Konno-Ichikawa Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Xuemei Li ◽  
Lutong Li
Keyword(s):  
Spectral Problem ◽  
Spectral Parameter ◽  
Trace Identity ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Matrix Spectral Problem

In this paper, a super Wadati-Konno-Ichikawa (WKI) hierarchy associated with a 3×3 matrix spectral problem is derived with the help of the zero-curvature equation. We obtain the super bi-Hamiltonian structures by using of the super trace identity. Infinitely, many conserved laws of the super WKI equation are constructed by using spectral parameter expansions.

SU(N) LIE ALGEBRA, EXTENDED TODA LATTICE EQUATION AND THE EXACT SOLUTIONS

1994 ◽  
Vol 09 (06) ◽  
pp. 525-534
Author(s):  
A. ROY CHOWDHURY ◽  
A. GHOSE CHOUDHURY
Keyword(s):  
Exact Solutions ◽  
Lie Algebra ◽  
Toda Lattice ◽  
Two Dimensional ◽  
Lattice Equation ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Dimensional Gravity ◽  
Differential Generalization ◽  
Toda Lattice Equation

An integro-differential generalization of the Toda lattice equation is proposed via the zero curvature equation belonging to SU(N) Lie algebra. It is shown that the exact solutions for this equation can be constructed by the method of chiral vectors. Explicit results are given for SU(2) and SU(3). We also demonstrate that these equations are connected to the constrained WZW theory and hence Polyakov’s two-dimensional gravity.

scholarly journals Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)

2017 ◽  
Vol 15 (1) ◽  
pp. 203-217
Author(s):  
Jian Zhang ◽  
Chiping Zhang ◽  
Yunan Cui
Keyword(s):  
Lie Algebra ◽  
Discrete Case ◽  
Hamiltonian Structures ◽  
Spectral Problems ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Integrable Couplings ◽  
Orthogonal Lie Algebra

Abstract In our paper, the theory of bi-integrable and tri-integrable couplings is generalized to the discrete case. First, based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct bi-integrable and tri-integrable couplings associated with SO(4) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by the variational identities.

SUPER INTEGRABLE HIERARCHY AND SUPER HAMILTONIAN STRUCTURES ASSOCIATED WITH GUO HIERARCHY

2009 ◽  
Vol 23 (29) ◽  
pp. 3491-3496 ◽  
Author(s):  
NING ZHANG ◽  
HUANHE DONG
Keyword(s):  
Integrable Hierarchies ◽  
Lie Superalgebra ◽  
Integrable Hierarchy ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Zero Curvature

A Lie superalgebra is constructed from which establishes an isospectral problems. By solving the zero curvature equation, a resulting super hierarchies of the Guo hierarchy are obtained. By making use of the super identity, the Hamiltonian structures of the above super integrable hierarchies are generated, this method can be used to other superhierarchy.

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 BLASZAK–MARCINIAK LATTICE HIERARCHY WITH SELF-CONSISTENT SOURCES

2011 ◽  
Vol 25 (25) ◽  
pp. 3371-3379 ◽  
Author(s):  
FAJUN YU ◽  
LI LI
Keyword(s):  
Lie Algebra ◽  
Discrete Soliton ◽  
Curvature Equation ◽  
Zero Curvature ◽  
Soliton Hierarchy ◽  
Self Consistent

In this paper, we consider the discrete soliton hierarchy with self-consistent sources by using Lie algebra sl(3). Then a Blaszak–Marciniak equations hierarchy with self-consistent sources is obtained through the discrete zero curvature equation.

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.

Three-Component Generalisation of Burgers Equation and Its Bi-Hamiltonian Structures

2017 ◽  
Vol 72 (5) ◽  
pp. 469-475
Author(s):  
Wei Liu ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Eigenvalue Problem ◽  
Conservation Laws ◽  
Spectral Parameter ◽  
Burgers Equation ◽  
Trace Identity ◽  
Matrix Eigenvalue Problem ◽  
Hamiltonian Structures ◽  
Curvature Equation ◽  
Matrix Eigenvalue ◽  
Zero Curvature

AbstractA hierarchy of three-component generalisation of Burgers equation, which is associated with a 3×3 matrix eigenvalue problem, is generated by using the zero-curvature equation. By means of the trace identity, the bi-Hamiltonian structures of this hierarchy are constructed. Moreover, the infinite conservation laws for the hierarchy are obtained with the aid of spectral parameter expansion.

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