862 research outputs found
On the integrability of the shift map on twisted pentagram spirals
In this paper we prove that the shift map defined on the moduli space of
twisted pentagram spirals of type possesses a non-standard Lax
representation with an associated monodromy whose conjugation class is
preserved by the map. We prove this by finding a coordinate system in the
moduli space of twisted spirals, writing the map in terms of the coordinates
and associating a natural parameter-free non-standard Lax representation. We
then show that the map is invariant under the action of a -parameter group
on the moduli space of twisted spirals, which allows us to construct
the Lax pair. We also show that the monodromy defines an associated Riemann
surface that is preserved by the map. We use this fact to generate invariants
of the shift map
A nonlocal Poisson bracket of the sine-Gordon model
It is well known that the classical string on a two-sphere is more or less
equivalent to the sine-Gordon model. We consider the nonabelian dual of the
classical string on a two-sphere. We show that there is a projection map from
the phase space of this model to the phase space of the sine-Gordon model. The
corresponding Poisson structure of the sine-Gordon model is nonlocal with one
integration.Comment: 18 pages, LaTeX, v2,3: added Section 4 and reference
Geometric Poisson brackets on Grassmannians and conformal spheres
In this paper we relate the geometric Poisson brackets on the Grassmannian of
2-planes in R^4 and on the (2,2) Moebius sphere. We show that, when written in
terms of local moving frames, the geometric Poisson bracket on the Moebius
sphere does not restrict to the space of differential invariants of Schwarzian
type. But when the concept of conformal natural frame is transported from the
conformal sphere into the Grassmannian, and the Poisson bracket is written in
terms of the Grassmannian natural frame, it restricts and results into either a
decoupled system or a complexly coupled system of KdV equations, depending on
the character of the invariants. We also show that the biHamiltonian
Grassmannian geometric brackets are equivalent to the non-commutative KdV
biHamiltonian structure. Both integrable systems and Hamiltonian structure can
be brought back to the conformal sphere.Comment: 33 page
Geometric Realizations of Bi-Hamiltonian Completely Integrable Systems
In this paper we present an overview of the connection between completely
integrable systems and the background geometry of the flow. This relation is
better seen when using a group-based concept of moving frame introduced by Fels
and Olver in [Acta Appl. Math. 51 (1998), 161-213; 55 (1999), 127-208]. The
paper discusses the close connection between different types of geometries and
the type of equations they realize. In particular, we describe the direct
relation between symmetric spaces and equations of KdV-type, and the possible
geometric origins of this connection.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007,
Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
On integrable generalizations of the pentagram map
In this paper we prove that the generalization to of the
pentagram map defined in \cite{KS} is invariant under certain scalings for any
. This property allows the definition of a Lax representation for the map,
to be used to establish its integrability
Geometric Poisson brackets on Grassmannians and conformal spheres
We relate the geometric Poisson brackets on the 2-Grassmannian in4 and on the (2, 2) Mbius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the Mbius sphere does not restrict to the space of differential invariants of Schwarzian type. But when the concept of conformal natural frame is transported from the conformal sphere into the Grassmannian, and the Poisson bracket is written in terms of the Grassmannian natural frame, it restricts and results in either a decoupled system or a complexly coupled system of Korteweg-de Vries (KdV) equations, depending on the character of the invariants. We also show that the bi-Hamiltonian Grassmannian geometric brackets are equivalent to the non-commutative KdV bi-Hamiltonian structure. Both integrable systems and Hamiltonian structure can be brought back to the conformal sphere
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