2 research outputs found
An integrable discretization of the rational su(2) Gaudin model and related systems
The first part of the present paper is devoted to a systematic construction
of continuous-time finite-dimensional integrable systems arising from the
rational su(2) Gaudin model through certain contraction procedures. In the
second part, we derive an explicit integrable Poisson map discretizing a
particular Hamiltonian flow of the rational su(2) Gaudin model. Then, the
contraction procedures enable us to construct explicit integrable
discretizations of the continuous systems derived in the first part of the
paper.Comment: 26 pages, 5 figure
Bruhat Order in Full Symmetric Toda System
In this paper we discuss some geometrical and topological properties of the full symmetric Toda system. We show by a direct inspection that the phase transition diagram for the full symmetric Toda system in dimensions coincides with the Hasse diagram of the Bruhat order of symmetric groups and . The method we use is based on the existence of a vast collection of invariant subvarieties of the Toda flow in orthogonal groups. We show how one can extend it to the case of general . The resulting theorem identifies the set of singular points of Toda flow with the elements of the permutation group , so that points will be connected by a trajectory, if and only if the corresponding elements are Bruhat comparable. We also show that the dimension of the submanifolds, spanned by the trajectories connecting two singular points, is equal to the length of the corresponding segment in the Hasse diagramm. This is equivalent to the fact, that the full symmetric Toda system is in fact a Morse-Smale system