Connection Problem for Painlevé tau Functions

2019-10-16T16:19:40Z (GMT) by Andrei Prokhorov
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.