Lie Algebra, Bi-Hamiltonian Structure and Reduction Problem for Integrable Nonlinear Systems

1990 ◽  
Vol 38 (11) ◽  
pp. 875-886 ◽  
Author(s):  
Swapna Roy ◽  
A. Roy Chowdhury
Keyword(s):  
Nonlinear Systems ◽  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Reduction Problem

INTEGRABLE HAMILTONIAN HIERARCHIES ASSOCIATED WITH THE EQUATION OF HEAT CONDUCTION

2010 ◽  
Vol 24 (14) ◽  
pp. 1573-1594 ◽  
Author(s):  
YUFENG ZHANG ◽  
HONWAH TAM ◽  
JIANQIN MEI
Keyword(s):  
Heat Conduction ◽  
Quadratic Form ◽  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Arbitrary Parameter ◽  
Higher Dimensional ◽  
Soliton Hierarchy ◽  
Equation Of Heat Conduction ◽  
Hamiltonian Functions

Using a 4-dimensional Lie algebra g, an isospectral Lax pair is introduced, whose compatibility condition is equivalent to a soliton hierarchy of evolution equations with three components of potential functions. Its Hamiltonian structure is obtained by employing the quadratic-form identity proposed by Guo and Zhang. In order to obtain explicit Hamiltonian functions, a detailed computing formula for the constant appearing in the quadratic-form identity is obtained. One type of reduction equations of the hierarchy is also produced, which is further reduced to the standard equation of heat conduction. By introducing a loop algebra of the Lie algebra g, we obtain a soliton hierarchy with an arbitrary parameter which can be reduced to the previous equation hierarchy obtained, whose quasi-Hamiltonian structure is also worked out by the quadratic-form identity. Finally, we extend the Lie algebra g into a higher-dimensional Lie algebra so that a new integrable Hamiltonian hierarchy, which comprise integrable couplings, is produced; its reduced equations in particular contain two arbitrary parameters.

HAMILTONIAN STRUCTURES OF TWO INTEGRABLE COUPLINGS OF THE MODIFIED AKNS HIERARCHY

2007 ◽  
Vol 21 (30) ◽  
pp. 2063-2074 ◽  
Author(s):  
YUFENG ZHANG ◽  
Y. C. HON
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Three Dimensional ◽  
Hamiltonian Structures ◽  
Akns Hierarchy ◽  
Higher Dimensional ◽  
Integrable Couplings

The extension of a three-dimensional Lie algebra into two higher-dimensional ones is used to deduce two new integrable couplings of the m-AKNS hierarchy. The Hamiltonian structures of the two integrable couplings are obtained, respectively. Specially, the complex Hamiltonian structure of the second integrable couplings is given.

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.

scholarly journals On the optimal control of affine nonlinear systems

2005 ◽  
Vol 2005 (4) ◽  
pp. 465-475 ◽  
Author(s):  
M. Popescu ◽  
A. Dumitrache
Keyword(s):  
Optimal Control ◽  
Nonlinear Systems ◽  
Lie Algebra ◽  
Closed Form Solution ◽  
Bilinear System ◽  
Open Loop ◽  
Form Solution ◽  
Loop Control ◽  
Nilpotent Operators ◽  
Affine Nonlinear Systems

The minimization control problem of quadratic functionals for the class of affine nonlinear systems with the hypothesis of nilpotent associated Lie algebra is analyzed. The optimal control corresponding to the first-, second-, and third-order nilpotent operators is determined. In this paper, we have considered the minimum fuel problem for the multi-input nilpotent control and for a scalar input bilinear system for such systems. For the multi-input system, usually an analytic closed-form solution for the optimal controlui∗(t)is not possible and it is necessary to use numerical integration for the set ofmnonlinear coupled second-order differential equations. The optimal control of bilinear systems is obtained by considering the Lie algebra generated by the system matrices. It should be noted that we have obtained an open-loop control depending on the initial value of the statex0.

SUBALGEBRAS OF THE LIE ALGEBRA R6 AND THEIR APPLICATIONS

2007 ◽  
Vol 21 (22) ◽  
pp. 3809-3824 ◽  
Author(s):  
YU-FENG ZHANG ◽  
EN-GUI FAN
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Systems ◽  
Hamiltonian Structure ◽  
Hamiltonian Structures ◽  
Soliton Theory ◽  
Soliton Equations ◽  
Integrable Couplings ◽  
Potential Component ◽  
Lagrange Mechanics ◽  
Important Significance

As we all know, the Hamiltonian systems are the same describing forms as Newton mechanics and Lagrange mechanics. Therefore, researching for a new Hamiltonian structure of the soliton equations has important significance. In the paper, firstly, with the help of the Lie algebra R6, a few types of subalgebras are constructed, from which the corresponding equivalent tensor systems are given. For their applications, two integrable couplings hierarchies along with the multi-potential component functions generated from the soliton theory and the Virasoro symmetric algebra are obtained. Secondly, the Hamiltonian structures of the above integrable couplings are worked out, which may become another describing expression for the Newton and Lagrange mechanics. In particular, one of the integrable couplings presented above reduces to the famous AKNS hierarchy of soliton equations.

Tri-integrable coupling of the Kaup-Newell soliton hierarchy and Liouville integrability

2016 ◽  
Vol 30 (21) ◽  
pp. 1650277 ◽  
Author(s):  
Shuimeng Yu ◽  
Yujian Ye ◽  
Jun Zhang ◽  
Junquan Song
Keyword(s):  
Lie Algebra ◽  
Hamiltonian Structure ◽  
Liouville Integrability ◽  
Block Matrices ◽  
Text Block ◽  
Soliton Hierarchy ◽  
Integrable Coupling

Based on a matrix Lie algebra consisting of [Formula: see text] block matrices, new tri-integrable coupling of the Kaup–Newell soliton hierarchy is constructed. Then, the bi-Hamiltonian structure which leads to Liouville integrability of this coupling is furnished by the variational identity.

Connectability-Based Controllability of Large Scale Nonlinear Dynamic Thermal Systems

2009 ◽  
Author(s):  
Rahmat Shoureshi ◽  
Virdi Permana
Keyword(s):  
Graph Theory ◽  
Nonlinear Systems ◽  
Lie Algebra ◽  
Large Scale ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rank Condition ◽  
Thermal Systems ◽  
Necessary And Sufficient ◽  
Controllability And Observability

A new approach using graph-theory to determine the controllability and observability of large scale nonlinear dynamic thermal systems is presented. The novelty of this method is in adapting graph theory for a nonlinear class and establishing graphic conditions that describe the necessary and sufficient conditions for a class of nonlinear systems to be controllable and observable which is equivalent to the analytical method of Lie algebra rank condition. Graph theory of directed graph (digraph) is utilized to model the system and its adaptation to nonlinear problems is defined. The necessary and sufficient conditions for controllability are investigated through the structural property of a digraph called connectability. In comparison to the Lie Algebra, this approach has proven to be easier, from a computational point of view, thus it is found to be useful when dealing with large scale systems. This paper presents the problem statement, properties of structured system, and analytical method of Lie algebra rank condition for controllability and observability of bilinear systems. The main results of graphical approach which describe the necessary and sufficient conditions for controllability of nonlinear systems are presented and applied to the problem of a coupled two heat exchangers, connected in an arbitrary fashion.

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