Liouville integrability of perturbation systems of finite-dimensional integrable Hamiltonian systems

2000 ◽  
Vol 276 (1-4) ◽  
pp. 73-78 ◽  
Author(s):  
Wen-Xiu Ma ◽  
Ruguang Zhou
Keyword(s):  
Hamiltonian Systems ◽  
Liouville Integrability ◽  
Integrable Hamiltonian Systems ◽  
Finite Dimensional

AN INTEGRABLE DECOMPOSITION OF THE SYMMETRIC MATRIX KdV EQUATION

2008 ◽  
Vol 22 (13) ◽  
pp. 1307-1315
Author(s):  
RUGUANG ZHOU ◽  
ZHENYUN QIN
Keyword(s):  
Hamiltonian Systems ◽  
Matrix Method ◽  
Kdv Equation ◽  
Symmetric Matrix ◽  
Lax Pair ◽  
Integrable Hamiltonian Systems ◽  
R Matrix ◽  
Finite Dimensional ◽  
Higher Rank ◽  
Gaudin Models

A technique for nonlinearization of the Lax pair for the scalar soliton equations in (1+1) dimensions is applied to the symmetric matrix KdV equation. As a result, a pair of finite-dimensional integrable Hamiltonian systems, which are of higher rank generalization of the classic Gaudin models, are obtained. The integrability of the systems are shown by the explicit Lax representations and r-matrix method.

DEFORMATIONS OF LOOP ALGEBRAS AND CLASSICAL INTEGRABLE SYSTEMS: FINITE-DIMENSIONAL HAMILTONIAN SYSTEMS

2004 ◽  
Vol 16 (07) ◽  
pp. 823-849 ◽  
Author(s):  
T. SKRYPNYK
Keyword(s):  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Spectral Parameter ◽  
Integrable Systems ◽  
Integrable Hamiltonian Systems ◽  
Infinite Dimensional ◽  
Loop Algebras ◽  
Finite Dimensional ◽  
Quasigraded Lie Algebras ◽  
Classical Integrable

We construct a family of infinite-dimensional quasigraded Lie algebras, that could be viewed as deformation of the graded loop algebras and admit Kostant–Adler scheme. Using them we obtain new integrable hamiltonian systems admitting Lax-type representations with the spectral parameter.

scholarly journals Complete commutative subalgebras in polynomial poisson algebras: A proof of the Mischenko-Fomenko conjecture

2016 ◽  
Vol 43 (2) ◽  
pp. 145-168 ◽  
Author(s):  
Alexey Bolsinov
Keyword(s):  
Poisson Bracket ◽  
Lie Algebras ◽  
Hamiltonian Systems ◽  
Dual Space ◽  
State University ◽  
Integrable Hamiltonian Systems ◽  
Finite Dimensional ◽  
Complete Set ◽  
Commutative Subalgebras ◽  
Moscow State University

The Mishchenko-Fomenko conjecture says that for each real or complex finite-dimensional Lie algebra g there exists a complete set of commuting polynomials on its dual space g*. In terms of the theory of integrable Hamiltonian systems this means that the dual space g* endowed with the standard Lie-Poisson bracket admits polynomial integrable Hamiltonian systems. This conjecture was proved by S. T. Sadetov in 2003. Following his idea, we give an explicit geometric construction for commuting polynomials on g* and consider some examples. (This text is a revised version of my paper published in Russian: A. V. Bolsinov, Complete commutative families of polynomials in Poisson?Lie algebras: A proof of the Mischenko?Fomenko conjecture in book: Tensor and Vector Analysis, Vol. 26, Moscow State University, 2005, 87?109.)

INTEGRABLE HAMILTONIAN SYSTEMS RELATED TO THE AKNS SYSTEM WITH MATRIX POTENTIALS

2008 ◽  
Vol 22 (29) ◽  
pp. 2831-2842 ◽  
Author(s):  
ZHENYUN QIN ◽  
ZIXIANG ZHOU ◽  
RUGUANG ZHOU
Keyword(s):  
Hamiltonian Systems ◽  
Complete Integrability ◽  
Integrable Hamiltonian Systems ◽  
Finite Dimensional ◽  
Akns System ◽  
Independent Motion

In this paper, we extend the nonlinearization method to the AKNS system with matrix potentials for obtaining new finite-dimensional Hamiltonian systems. The Liouville complete integrability of these Hamiltonian systems is further shown by using involutive and independent motion integrals.

scholarly journals Binary Nonlinearization for AKNS-KN Coupling System

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Xiangrong Wang ◽  
Xiaoen Zhang ◽  
Peiyi Zhao
Keyword(s):  
Hamiltonian Systems ◽  
Integrable Hamiltonian Systems ◽  
Symmetry Constraint ◽  
Curvature Equation ◽  
Finite Dimensional ◽  
Zero Curvature ◽  
Coupling System ◽  
Spectral Equation ◽  
Liouville Integrable

The AKNS-KN coupling system is obtained on the base of zero curvature equation by enlarging the spectral equation. Under the Bargmann symmetry constraint, the AKNS-KN coupling system is decomposed into two integrable Hamiltonian systems with the corresponding variablesx,tnand the finite dimensional Hamiltonian systems are Liouville integrable.

Sign in / Sign up

Export Citation Format

Share Document