The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems

1989 ◽  
Vol 30 (2) ◽  
pp. 330-338 ◽  
Author(s):  
Gui‐zhang Tu
Keyword(s):  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Trace Identity

Some 2+1 Dimensional Super-Integrable Systems

2015 ◽  
Vol 70 (10) ◽  
pp. 791-796 ◽  
Author(s):  
Yufeng Zhang ◽  
Honwah Tam ◽  
Jianqin Mei
Keyword(s):  
Diffusion Equation ◽  
Integrable Systems ◽  
Schrodinger Equation ◽  
Hamiltonian Structure ◽  
Nonlinear Schrodinger Equation ◽  
The Other ◽  
Trace Identity ◽  
Akns Hierarchy ◽  
Dimensional Diffusion ◽  
Integrable Couplings

AbstractIn the article, we make use of the binormial-residue-representation (BRR) to generate super 2+1 dimensional integrable systems. Using these systems, we can deduce a super 2+1 dimensional AKNS hierarchy, which can be reduced to a super 2+1 dimensional nonlinear Schrödinger equation. In particular, two main results are obtained. One of them is a set of super 2+1 dimensional integrable couplings. The other one is a 2+1 dimensional diffusion equation. The Hamiltonian structure of the super 2+1 dimensional hierarchy is derived by using the super-trace identity.

(1+1)-dimensional m-cKdV, g-cKdV integrable systems, and (2+1)-dimensional m-cKdV hierarchy

2008 ◽  
Vol 86 (12) ◽  
pp. 1367-1380 ◽  
Author(s):  
Y Zhang ◽  
H Tam
Keyword(s):  
Integrable Systems ◽  
Linear Functional ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Integrable Model ◽  
Lax Pair ◽  
The Self ◽  
Hamiltonian Structures ◽  
Yang Mills ◽  
Higher Dimensional

A few isospectral problems are introduced by referring to that of the cKdV equation hierarchy, for which two types of integrable systems called the (1 + 1)-dimensional m-cKdV hierarchy and the g-cKdV hierarchy are generated, respectively, whose Hamiltonian structures are also discussed by employing a linear functional and the quadratic-form identity. The corresponding expanding integrable models of the (1 + 1)-dimensional m-cKdV hierarchy and g-cKdV hierarchy are obtained. The Hamiltonian structure of the latter one is given by the variational identity, proposed by Ma Wen-Xiu as well. Finally, we use a Lax pair from the self-dual Yang–Mills equations to deduce a higher dimensional m-cKdV hierarchy of evolution equations and its Hamiltonian structure. Furthermore, its expanding integrable model is produced by the use of a enlarged Lie algebra.PACS Nos.: 02.30, 03.40.K

Matrix factorization method for the Hamiltonian structure of integrable systems

Pramana ◽  
2003 ◽  
Vol 61 (1) ◽  
pp. 161-165 ◽  
Author(s):  
S. Ghosh ◽  
B. Talukdar ◽  
S. Chakraborti
Keyword(s):  
Integrable Systems ◽  
Matrix Factorization ◽  
Hamiltonian Structure ◽  
Factorization Method

Generalised (2+1)-dimensional Super MKdV Hierarchy for Integrable Systems in Soliton Theory

2015 ◽  
Vol 5 (3) ◽  
pp. 256-272 ◽  
Author(s):  
Huanhe Dong ◽  
Kun Zhao ◽  
Hongwei Yang ◽  
Yuqing Li
Keyword(s):  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Lie Superalgebra ◽  
Lax Pair ◽  
Mkdv Equation ◽  
Scientific Applications ◽  
Soliton Theory ◽  
De Vries ◽  
Pair Method ◽  
Mkdv Hierarchy

AbstractMuch attention has been given to constructing Lie and Lie superalgebra for integrable systems in soliton theory, which often have significant scientific applications. However, this has mostly been confined to (1+1)-dimensional integrable systems, and there has been very little work on (2+1)-dimensional integrable systems. In this article, we construct a class of generalised Lie superalgebra that differs from more common Lie superalgebra to generate a (2+1)-dimensional super modified Korteweg-de Vries (mKdV) hierarchy, via a generalised Tu scheme based on the Lax pair method where the Hamiltonian structure derives from a generalised supertrace identity. We also obtain some solutions of the (2+1)-dimensional mKdV equation using the G′/G2 method.

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.

SPECTRAL RADIUS ANALYSIS OF MATRICES AND THE ASSOCIATED WITH INTEGRABLE SYSTEMS

2009 ◽  
Vol 23 (24) ◽  
pp. 4855-4879 ◽  
Author(s):  
HONWAH TAM ◽  
YUFENG ZHANG
Keyword(s):  
Lie Algebra ◽  
Spectral Radius ◽  
Integrable Systems ◽  
Hamiltonian Structure ◽  
Soliton Equation ◽  
Spectral Matrix ◽  
Akns Hierarchy ◽  
Integrable Couplings ◽  
Isospectral Problem ◽  
Set Up

An isospectral problem is introduced, a spectral radius of the corresponding spectral matrix is obtained, which enlightens us to set up an isospectral problem whose compatibility condition gives rise to a zero curvature equation in formalism, from which a Lax integrable soliton equation hierarchy with constraints of potential functions is generated along with 5 parameters, whose reduced cases present three integrable systems, i.e., AKNS hierarchy, Levi hierarchy and D-AKNS hierarchy. Enlarging the above Lie algebra into two bigger ones, the two integrable couplings of the hierarchy are derived, one of them has Hamiltonian structure by employing the quadratic-form identity or variational identity. The corresponding integrable couplings of the reduced systems are obtained, respectively. Finally, as comparing study for generating expanding integrable systems, a Lie algebra of antisymmetric matrices and its corresponding loop algebra are constructed, from which a great number of enlarging integrable systems could be generated, especially their Hamiltonian structure could be computed by the trace identity.

SUPER-BURGERS SOLITON HIERARCHY AND ITS SUPER-HAMILTONIAN STRUCTURE

2009 ◽  
Vol 23 (24) ◽  
pp. 2907-2914 ◽  
Author(s):  
ZHU LI ◽  
HONGWEI YANG ◽  
HUANHE DONG
Keyword(s):  
Hamiltonian Structure ◽  
Trace Identity ◽  
Soliton Hierarchy ◽  
Burgers Hierarchy

A super-Burgers hierarchy and its super-Hamiltonian structure is obtained respectively based on Lie super-algebra and is associated with super-trace identity.

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