154 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
A simple way of making a Hamiltonian system into a bi-Hamiltonian one
Given a Poisson structure (or, equivalently, a Hamiltonian operator) , we
show that its Lie derivative along a vector field defines
another Poisson structure, which is automatically compatible with , if and
only if , where is the Schouten bracket.
We further prove that if and is of locally constant
rank, then all Poisson structures compatible with a given Poisson structure
on a finite-dimensional manifold are locally of the form ,
where is a local vector field such that
for some other local vector field
. This leads to a remarkably simple construction of bi-Hamiltonian
dynamical systems. We also present a generalization of these results to the
infinite-dimensional case. In particular, we provide a new description for
pencils of compatible local Hamiltonian operators of Dubrovin--Novikov type and
associated bi-Hamiltonian systems of hydrodynamic type.
Key words: compatible Poisson structures, Hamiltonian operators,
bi-Hamiltonian systems (= bihamiltonian systems), integrability, Schouten
bracket, master symmetry, Lichnerowicz--Poisson cohomology, hydrodynamic type
systems.
MSC 2000: Primary: 37K10; Secondary: 37K05, 37J35Comment: 12 pages, LaTeX 2e, no figures, accepted for publication in Acta
Appl. Math. Major revision: In this version an important condition of local
constancy of rank of P is added (it is assumed that the vicinities where rank
P=const are of the same dimension as the underlying manifold M). Moreover,
this version contains Remarks 1 and 2, references
[14],[22],[23],[29],[30],[36],[41], and the discussion thereof that for
technical reasons were not included in the published version of the pape
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