40 research outputs found
Quantized W-algebra of sl(2,1) and quantum parafermions of U_q(sl(2))
In this paper, we establish the connection between the quantized W-algebra of
and quantum parafermions of that a
shifted product of the two quantum parafermions of
generates the quantized W-algebra of
Gauge transformation and reciprocal link for (2+1)-dimensional integrable field systems
Appropriate restrictions of Lax operators which allows to construction of
(2+1)-dimensional integrable field systems, coming from centrally extended
algebra of pseudo-differential operators, are reviewed. The gauge
transformation and the reciprocal link between three classes of Lax hierarchies
are established.Comment: to appear in J. Nonl. Math. Phys., 12 page
Generalized St\"ackel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems
We present a multiparameter generalization of the St\"ackel transform (the
latter is also known as the coupling-constant metamorphosis) and show that
under certain conditions this generalized St\"ackel transform preserves the
Liouville integrability, noncommutative integrability and superintegrability.
The corresponding transformation for the equations of motion proves to be
nothing but a reciprocal transformation of a special form, and we investigate
the properties of this reciprocal transformation.
Finally, we show that the Hamiltonians of the systems possessing separation
curves of apparently very different form can be related through a suitably
chosen generalized St\"ackel transform.Comment: 21 pages, LaTeX 2e, no figures; major revision; Propositions 2 and 7
and several new references adde
On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian
It is shown that a class of dynamical systems (encompassing the one recently
considered by F. Calogero [J. Math. Phys. 37 (1996) 1735]) is both
quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the
separability of these systems; the second one is obtained trough a non
canonical map whose form is directly suggested by the associated Nijenhuis
tensor.Comment: 11 pages, AMS-LaTex 1.
From St\"{a}ckel systems to integrable hierarchies of PDE's: Benenti class of separation relations
We propose a general scheme of constructing of soliton hierarchies from
finite dimensional St\"{a}ckel systems and related separation relations. In
particular, we concentrate on the simplest class of separation relations,
called Benenti class, i.e. certain St\"{a}ckel systems with quadratic in
momenta integrals of motion.Comment: 24 page
Maximal superintegrability of Benenti systems
For a class of Hamiltonian systems naturally arising in the modern theory of
separation of variables, we establish their maximal superintegrability by
explicitly constructing the additional integrals of motion.Comment: 5 pages, LaTeX 2e, to appear in J. Phys. A: Math. Ge
Classification of integrable hydrodynamic chains and generating functions of conservation laws
New approach to classification of integrable hydrodynamic chains is
established. Generating functions of conservation laws are classified by the
method of hydrodynamic reductions. N parametric family of explicit hydrodynamic
reductions allows to reconstruct corresponding hydrodynamic chains. Plenty new
hydrodynamic chains are found
Dispersionless integrable equations as coisotropic deformations. Extensions and reductions
Interpretation of dispersionless integrable hierarchies as equations of
coisotropic deformations for certain algebras and other algebraic structures
like Jordan triple systInterpretation of dispersionless integrable hierarchies
as equations of coisotropic deformations for certain algebras and other
algebraic structures like Jordan triple systems is discussed. Several
generalizations are considered. Stationary reductions of the dispersionless
integrable equations are shown to be connected with the dynamical systems on
the plane completely integrable on a fixed energy level. ems is discussed.
Several generalizations are considered. Stationary reductions of the
dispersionless integrable equations are shown to be connected with the
dynamical systems on the plane completely integrable on a fixed energy level.Comment: 21 pages, misprints correcte
Notes on Exact Multi-Soliton Solutions of Noncommutative Integrable Hierarchies
We study exact multi-soliton solutions of integrable hierarchies on
noncommutative space-times which are represented in terms of quasi-determinants
of Wronski matrices by Etingof, Gelfand and Retakh. We analyze the asymptotic
behavior of the multi-soliton solutions and found that the asymptotic
configurations in soliton scattering process can be all the same as commutative
ones, that is, the configuration of N-soliton solution has N isolated localized
energy densities and the each solitary wave-packet preserves its shape and
velocity in the scattering process. The phase shifts are also the same as
commutative ones. Furthermore noncommutative toroidal Gelfand-Dickey hierarchy
is introduced and the exact multi-soliton solutions are given.Comment: 18 pages, v3: references added, version to appear in JHE
A new approach to deformation equations of noncommutative KP hierarchies
Partly inspired by Sato's theory of the Kadomtsev-Petviashvili (KP)
hierarchy, we start with a quite general hierarchy of linear ordinary
differential equations in a space of matrices and derive from it a matrix
Riccati hierarchy. The latter is then shown to exhibit an underlying 'weakly
nonassociative' (WNA) algebra structure, from which we can conclude, refering
to previous work, that any solution of the Riccati system also solves the
potential KP hierarchy (in the corresponding matrix algebra). We then turn to
the case where the components of the matrices are multiplied using a
(generalized) star product. Associated with the deformation parameters, there
are additional symmetries (flow equations) which enlarge the respective KP
hierarchy. They have a compact formulation in terms of the WNA structure. We
also present a formulation of the KP hierarchy equations themselves as
deformation flow equations.Comment: 25 page