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On a Poisson structure on the space of Stokes matrices. (English) Zbl 0939.34073

The author solves the problem of computing the Poisson structure of monodromy preserving the deformation equations (MPDEs) in the monodromy data coordinates for one particular example of the operators with one regular and one irregular singularity of Poincaré rank 1, \[ A(z)= {d\over dz}- U- {V\over z}, \] \(U\) is a diagonal matrix with pairwise distinct entries, and \(V\) is a skewsymmetric square matrix of order \(n\). The Poisson structure of MPDEs for the operator \(A\) coincides with the standard linear Poisson bracket on the Lie algebra \(so(n)\) belonging to \(V\).
The paper is very well organized beginning with a very complete description of a monodromy \(A(z)\) around two singular points. The MPDEs for this operator are clearly presented and continue with its connection as related to the somewhat analogous Fuchsian system. The Poisson structure on the space of monodromy data of the Fuchsian system is described and culminates with results between the monodromy data of the two systems. A very detailed and explicit calculation gives the value of the Poisson bracket on a six-dimensional space of the Stokes matrices of the form \[ S= \begin{pmatrix} 1 & p & q & r\\ 0 & 1 & x & y\\ 0 & 0 & 1 & z\\ 0 & 0 & 0 & 1\end{pmatrix}. \]

MSC:

34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain
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