54 research outputs found
A Note on the Rotationally Symmetric SO(4) Euler Rigid Body
We consider an SO(4) Euler rigid body with two 'inertia momenta' coinciding.
We study it from the point of view of bihamiltonian geometry. We show how to
algebraically integrate it by means of the method of separation of variables.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
Bihamiltonian Geometry, Darboux Coverings, and Linearization of the KP Hierarchy
We use ideas of the geometry of bihamiltonian manifolds, developed by
Gel'fand and Zakharevich, to study the KP equations. In this approach they have
the form of local conservation laws, and can be traded for a system of ordinary
differential equations of Riccati type, which we call the Central System. We
show that the latter can be linearized by means of a Darboux covering, and we
use this procedure as an alternative technique to construct rational solutions
of the KP equations.Comment: Latex, 27 pages. To appear in Commun. Math. Phy
Krichever Maps, Faa' di Bruno Polynomials, and Cohomology in KP Theory
We study the geometrical meaning of the Faa' di Bruno polynomials in the
context of KP theory. They provide a basis in a subspace W of the universal
Grassmannian associated to the KP hierarchy. When W comes from geometrical data
via the Krichever map, the Faa' di Bruno recursion relation turns out to be the
cocycle condition for (the Welters hypercohomology group describing) the
deformations of the dynamical line bundle on the spectral curve together with
the meromorphic sections which give rise to the Krichever map. Starting from
this, one sees that the whole KP hierarchy has a similar cohomological meaning.Comment: 16 pages, LaTex using amssymb.sty. To be published in Lett. Math.
Phy
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