Connection Problem for Painlevé tau Functions
thesisposted on 16.10.2019 by Andrei Prokhorov
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We derive the differential identities for isomonodromic tau functions, describing their monodromy dependence.
For Painlev\'e equations we obtain them from the relation of tau function to classical action which is a consequence of quasihomogeneity of corresponding Hamiltonians.
We use these identities to solve the connection problem for generic solution of Painlev\'e-III(D8) equation, and homogeneous Painlev\'e-II equation.
We formulate conjectures on Hamiltonian and symplectic structure of general iso\-mo\-no\-dro\-mic deformations we obtained during our studies and check them for Painlev\'e equations.
Development of methods of spectral analysis, scattering theory and integrable systems in modern problems of mathematical physics.
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