Finite-dimensional integrable systems through the decomposition of a modified Boussinesq equation

2003 ◽  
Vol 317 (5-6) ◽  
pp. 389-400 ◽  
Author(s):  
H.-H. Dai ◽  
Xianguo Geng
Keyword(s):  
Integrable Systems ◽  
Boussinesq Equation ◽  
Finite Dimensional ◽  
Modified Boussinesq Equation

scholarly journals ZN graded discrete Lax pairs and Yang–Baxter maps

2017 ◽  
Vol 473 (2201) ◽  
pp. 20160946 ◽  
Author(s):  
Allan P. Fordy ◽  
Pavlos Xenitidis
Keyword(s):  
Integrable Systems ◽  
Boussinesq Equation ◽  
Lax Pairs ◽  
Discrete Integrable Systems ◽  
New Family ◽  
Modified Boussinesq Equation ◽  
Lower Dimensional

We recently introduced a class of Z N graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). In this paper, we introduce the corresponding Yang–Baxter maps. Many well-known examples belong to this scheme for N =2, so, for N ≥3, our systems may be regarded as generalizations of these. In particular, for each N we introduce a class of multi-component Yang–Baxter maps, which include H B III (of Papageorgiou et al. 2010 SIGMA 6, 003 (9 p). (doi:10.3842/SIGMA.2010.033)), when N =2, and that associated with the discrete modified Boussinesq equation, for N =3. For N ≥5 we introduce a new family of Yang–Baxter maps, which have no lower dimensional analogue. We also present new multi-component versions of the Yang–Baxter maps F IV and F V (given in the classification of Adler et al. 2004 Commun. Anal. Geom. 12, 967–1007. (doi:10.4310/CAG.2004.v12.n5.a1)).

scholarly journals Asymptotic velocity for four celestial bodies

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170426
Author(s):  
Andreas Knauf
Keyword(s):  
Celestial Mechanics ◽  
Integrable Systems ◽  
Energy Surface ◽  
Initial Conditions ◽  
Theme Issue ◽  
Asymptotic Velocity ◽  
Pair Potentials ◽  
Finite Dimensional ◽  
Celestial Bodies ◽  
Almost All

Asymptotic velocity is defined as the Cesàro limit of velocity. As such, its existence has been proved for bounded interaction potentials. This is known to be wrong in celestial mechanics with four or more bodies. Here, we show for a class of pair potentials including the homogeneous ones of degree − α for α ∈(0, 2), that asymptotic velocities exist for up to four bodies, dimension three or larger, for any energy and almost all initial conditions on the energy surface. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Open problems, questions and challenges in finite- dimensional integrable systems

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170430 ◽  
Author(s):  
Alexey Bolsinov ◽  
Vladimir S. Matveev ◽  
Eva Miranda ◽  
Serge Tabachnikov
Keyword(s):  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Theme Issue ◽  
Open Problems ◽  
Finite Dimensional

The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference ‘Finite-dimensional Integrable Systems, FDIS 2017’ held at CRM, Barcelona in July 2017. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Convexity of the moment map image for torus actions on b m -symplectic manifolds

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170420 ◽  
Author(s):  
Victor W. Guillemin ◽  
Eva Miranda ◽  
Jonathan Weitsman
Keyword(s):  
Symplectic Manifold ◽  
Integrable Systems ◽  
Symplectic Manifolds ◽  
Moment Map ◽  
Torus Action ◽  
Convexity Theorem ◽  
Theme Issue ◽  
Torus Actions ◽  
Finite Dimensional ◽  
The Moment

We prove a convexity theorem for the image of the moment map of a Hamiltonian torus action on a b m -symplectic manifold. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

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.

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