Normal forms and the multiple-times expansion method for finite-dimensional integrable Hamiltonian systems

2007 ◽  
Vol 84 (12) ◽  
pp. 1819-1841 ◽  
Author(s):  
Mehmet Naciözer ◽  
Filiz Taşcan
Keyword(s):  
Hamiltonian Systems ◽  
Normal Forms ◽  
Expansion Method ◽  
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 Symplectic invariants for parabolic orbits and cusp singularities of integrable systems

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170424 ◽  
Author(s):  
Alexey Bolsinov ◽  
Lorenzo Guglielmi ◽  
Elena Kudryavtseva
Keyword(s):  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Normal Forms ◽  
Integrable Hamiltonian Systems ◽  
Theme Issue ◽  
New Techniques ◽  
Finite Dimensional ◽  
Parabolic Orbits ◽  
Symplectic Invariants ◽  
Cusp Singularities

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We also suggest some new techniques which apparently can be used for studying symplectic invariants of degenerate singularities of more general type. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

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