61 research outputs found
From bi-Hamiltonian geometry to separation of variables: stationary Harry-Dym and the KdV dressing chain
Separability theory of one-Casimir Poisson pencils, written down in arbitrary
coordinates, is presented. Separation of variables for stationary Harry-Dym and
the KdV dressing chain illustrates the theory.Comment: LaTex 14 pages, Proceedings of the Special Session on Integrable
Systems of the First Joint Meeting of the American Mathematical Society and
the Hong Kong Mathematical Society, to appear in J. Nonl. Math. Phy
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
Bi-presymplectic chains of co-rank one and related Liouville integrable systems
Bi-presymplectic chains of one-forms of co-rank one are considered. The
conditions in which such chains represent some Liouville integrable systems and
the conditions in which there exist related bi-Hamiltonian chains of vector
fields are derived. To present the construction of bi-presymplectic chains, the
notion of dual Poisson-presymplectic pair is used and the concept of
d-compatibility of Poisson bivectors and d-compatibility of presymplectic forms
is introduced. It is shown that bi-presymplectic representation of related flow
leads directly to the construction of separation coordinates in purely
algorithmic way. As an illustration bi-presymplectic and bi-Hamiltonian chains
in are considered in detail
Multi-component Painlev\'{e} ODE's and related non-autonomous KdV stationary hierarchies
First, starting from two hierarchies of autonomous St\"{a}ckel ODE's, we
reconstruct the hierarchy of KdV stationary systems. Next, we deform considered
autonomous St\"{a}ckel systems to non-autonomous Painlev\'{e} hierarchies of
ODE's. Finally, we reconstruct the related non-autonomous KdV stationary
hierarchies from respective Painlev\'{e} systems.Comment: 22 page
Meromorphic Lax representations of (1+1)-dimensional multi-Hamiltonian dispersionless systems
Rational Lax hierarchies introduced by Krichever are generalized. A
systematic construction of infinite multi-Hamiltonian hierarchies and related
conserved quantities is presented. The method is based on the classical
R-matrix approach applied to Poisson algebras. A proof, that Poisson operators
constructed near different points of Laurent expansion of Lax functions are
equal, is given. All results are illustrated by several examples.Comment: 28 page
R-matrix approach to integrable systems on time scales
A general unifying framework for integrable soliton-like systems on time
scales is introduced. The -matrix formalism is applied to the algebra of
-differential operators in terms of which one can construct infinite
hierarchy of commuting vector fields. The theory is illustrated by two
infinite-field integrable hierarchies on time scales which are difference
counterparts of KP and mKP. The difference counterparts of AKNS and Kaup-Broer
soliton systems are constructed as related finite-field restrictions.Comment: 21 page
Staeckel systems generating coupled KdV hierarchies and their finite-gap and rational solutions
We show how to generate coupled KdV hierarchies from Staeckel separable
systems of Benenti type. We further show that solutions of these Staeckel
systems generate a large class of finite-gap and rational solutions of cKdV
hierarchies. Most of these solutions are new.Comment: 15 page
Bi-Hamiltonian representation of St\"{a}ckel systems
It is shown that a linear separation relations are fundamental objects for
integration by quadratures of St\"{a}ckel separable Liouville integrable
systems (the so-called St\"{a}ckel systems). These relations are further
employed for the classification of St\"{a}ckel systems. Moreover, we prove that
{\em any} St\"{a}ckel separable Liouville integrable system can be lifted to a
bi-Hamiltonian system of Gel'fand-Zakharevich type. In conjunction with other
known result this implies that the existence of bi-Hamiltonian representation
of Liouville integrable systems is a necessary condition for St\"{a}ckel
separability.Comment: To appear in Physical Review
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