40 research outputs found
Locality of symmetries generated by nonhereditary, inhomogeneous, and time-dependent recursion operators: a new application for formal symmetries
Using the methods of the theory of formal symmetries, we obtain new easily
verifiable sufficient conditions for a recursion operator to produce a
hierarchy of local generalized symmetries. An important advantage of our
approach is that under certain mild assumptions it allows to bypass the
cumbersome check of hereditariness of the recursion operator in question, what
is particularly useful for the study of symmetries of newly discovered
integrable systems. What is more, unlike the earlier work, the homogeneity of
recursion operators and symmetries under a scaling is not assumed as well. An
example of nonhereditary recursion operator generating a hierarchy of local
symmetries is presented.Comment: 11 pages, LaTeX 2e, submitted to Acta Appl. Mat
Weakly Nonlocal Hamiltonian Structures: Lie Derivative and Compatibility
We show that under certain technical assumptions any weakly nonlocal
Hamiltonian structure compatible with a given nondegenerate weakly nonlocal
symplectic structure can be written as the Lie derivative of along
a suitably chosen nonlocal vector field. Moreover, we present a new description
for local Hamiltonian structures of arbitrary order compatible with a given
nondegenerate local Hamiltonian structure of zero or first order, including
Hamiltonian operators of the Dubrovin-Novikov type.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on
Integrable Systems and Related Topics, published in SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
Exact solvability of superintegrable Benenti systems
We establish quantum and classical exact solvability for two large classes of
maximally superintegrable Benenti systems in dimensions with arbitrarily
large . Namely, we solve the Hamilton--Jacobi and Schr\"odinger equations
for the systems in question. The results obtained are illustrated for a model
with the cubic potential.Comment: 15 pages, LaTeX 2e, no figures; in the updated version a number of
typos were fixed and other minor changes were mad
Central extensions of cotangent universal hierarchy: (2+1)-dimensional bi-Hamiltonian systems
We introduce the cotangent universal hierarchy that extends the so-called
universal hierarchy (as for the latter, see e.g. arXiv:nlin/0202008,
arXiv:nlin/0312043 and arXiv:nlin/0310036). Then we construct a
(2+1)-dimensional double central extension of the cotangent universal hierarchy
and show that this extension is bi-Hamiltonian. This yields, as a byproduct,
the central extension of the original universal hierarchy.Comment: 12 pages, LaTeX 2e, minor changes (typos fixed, English improved,
etc.
A remark on nonlocal symmetries for the Calogero-Degasperis-Ibragimov-Shabat equation
We consider the Calogero-Degasperis-Ibragimov-Shabat (CDIS) equation and find
the complete set of its nonlocal symmetries depending on the local variables
and on the integral of the only local conserved density of the equation in
question. The Lie algebra of these symmetries turns out to be a central
extension of that of local generalized symmetries.Comment: arxiv version is already officia