Discrete Schlesinger Transformations, their Hamiltonian Formulation, and Difference Painlev\'e Equations
| Autor: | Dzhamay, Anton, Sakai, Hidetaka, Takenawa, Tomoyuki |
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| Rok vydání: | 2013 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Schlesinger transformations are algebraic transformations of a Fuchsian system that preserve its monodromy representation and act on the characteristic indices of the system by integral shifts. One of the important reasons to study such transformations is the relationship between Schlesinger transformations and discrete Painlev\'e equations; this is also the main theme behind our work. We derive \emph{discrete Schlesinger evolution equations} describing discrete dynamical systems generated by elementary Schlesinger transformations and give their discrete Hamiltonian description w.r.t.~the standard symplectic structure on the space of Fuchsian systems. As an application, we compute explicitly two examples of reduction from Schlesinger transformations to difference Painlev\'e equations. The first example, d-$P\big(D_{4}^{(1)}\big)$ (or difference Painlev\'e V), corresponds to B\"acklund transformations for continuous $P_{\text{VI}}$. The second example, d-$P\big(A_{2}^{(1)*}\big)$ (with the symmetry group $E_{6}^{(1)}$), is purely discrete. We also describe the role played by the geometry of the Okamoto space of initial conditions in comparing different equations of the same type. Comment: 29 pages, 17 figures. Changes: this is a significant rewrite of the previous version. We now first derive explicit evolution equations for elementary Schlesinger transformations and then use those equations to obtain discrete Hamiltonian functions. We also simplified examples of reductions to difference Painlev\'e equations, and corrected some typos and inaccuracies |
| Databáze: | arXiv |
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