On moment maps associated to a twisted Heisenberg double

We review the concept of the (anomalous) Poisson-Lie symmetry in a way that emphasises the notion of Poisson-Lie Hamiltonian. The language that we develop turns out to be very useful for several applications: we prove that the left and the right actions of a group $G$ on its twisted Heisenberg double $(D,\kappa)$ realize the (anomalous) Poisson-Lie symmetries and we explain in a very transparent way the concept of the Poisson-Lie subsymmetry and that of Poisson-Lie symplectic reduction. Under some additional conditions, we construct also a non-anomalous moment map corresponding to a sort of quasi-adjoint action of $G$ on $(D,\kappa)$. The absence of the anomaly of this "quasi-adjoint" moment map permits to perform the gauging of deformed WZW models.Comment: 52 pages, LaTeX, introduction substantially enlarged, several explanatory remarks added, final published versio

The Atiyah--Hitchin bracket and the open Toda lattice

The dynamics of finite nonperiodic Toda lattice is an isospectral deformation of the finite three--diagonal Jacobi matrix. It is known since the work of Stieltjes that such matrices are in one--to--one correspondence with their Weyl functions. These are rational functions mapping the upper half--plane into itself. We consider representations of the Weyl functions as a quotient of two polynomials and exponential representation. We establish a connection between these representations and recently developed algebraic--geometrical approach to the inverse problem for Jacobi matrix. The space of rational functions has natural Poisson structure discovered by Atiyah and Hitchin. We show that an invariance of the AH structure under linear--fractional transformations leads to two systems of canonical coordinates and two families of commuting Hamiltonians. We establish a relation of one of these systems with Jacobi elliptic coordinates.Comment: 26 pages, 2 figure

Reductions of the Volterra and Toda chains

The Volterra and Toda chains equations are considered. A class of special reductions for these equations are derived.Comment: LaTeX, 6 page

Invariant critical sets of conserved quantities

For a dynamical system we will construct various invariant sets starting from its conserved quantities. We will give conditions under which certain solutions of a nonlinear system are also solutions for a simpler dynamical system, for example when they are solutions for a linear dynamical system. We will apply these results to the example of Toda lattice

The Poisson geometry of SU(1,1)

We study the natural Poisson structure on the Lie group SU(1,1) and related questions. In particular, we give an explicit description of the Ginzburg-Weinstein isomorphism for the sets of admissible elements. We also establish an analogue of Thompson's conjecture for this group.Comment: 11 pages, minor correction

Integrability of V. Adler's discretization of the Neumann system

We prove the integrability of the discretization of the Neumann system recently proposed by V. Adler.Comment: 9 pp., LaTe

The twistor theory of Whitham hierarchy

We have established a 1-1 correspondence between a solution of the universal Whitham hierarchy and a twistor space. The twistor space consists of a complex surface and a family of complex curves together with a meromorphic 2-form. The solution of the Whitham hierarchy is given by deforming the curve in the surface. By treating the family of algebraic curves in $CP^1 X CP^1$ as a twistor space, we were able to express the deformations of the isomonodromic spectral curve in terms of the deformations generated by the Whitham hierarchy.Comment: 27 page

Transformations of ordinary differential equations via Darboux transformation technique

A new approach for obtaining the transformations of solutions of nonlinear ordinary differential equations representable as the compatibility condition of the overdetermined linear systems is proposed. The corresponding transformations of the solutions of the overdetermined linear systems are derived in the frameworks of the Darboux transformation technique.Comment: 7 pages, LaTeX2