TY - JOUR T1 - Canonical structure and symmetries of the Schlesinger equations JF - Comm. Math. Phys. 271 (2007) 289-373 Y1 - 2007 A1 - Boris Dubrovin A1 - Marta Mazzocco AB - The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n+1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m×m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation ofthe general Schlesinger equations S (n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates. UR - http://hdl.handle.net/1963/1997 U1 - 2199 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - On the reductions and classical solutions of the Schlesinger equations JF - Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys. 9 (2007) 157-187 Y1 - 2007 A1 - Boris Dubrovin A1 - Marta Mazzocco AB - The Schlesinger equations S(n,m) describe monodromy preserving\\r\\ndeformations of order m Fuchsian systems with n+1 poles. They\\r\\ncan be considered as a family of commuting time-dependent Hamiltonian\\r\\nsystems on the direct product of n copies of m×m matrix algebras\\r\\nequipped with the standard linear Poisson bracket. In this paper we address\\r\\nthe problem of reduction of particular solutions of “more complicated”\\r\\nSchlesinger equations S(n,m) to “simpler” S(n′,m′) having n′ < n\\r\\nor m′ < m. PB - SISSA UR - http://hdl.handle.net/1963/6472 U1 - 6418 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Picard and Chazy solutions to the Painlevé VI equation JF - Math. Ann. 321 (2001) 157-195 Y1 - 2001 A1 - Marta Mazzocco AB -

I study the solutions of a particular family of Painlevé VI equations with the parameters $\beta=\gamma=0, \delta=1/2$ and $2\alpha=(2\mu-1)^2$, for $2\mu\in\mathbb{Z}$. I show that the case of half-integer $\mu$ is integrable and that the solutions are of two types: the so-called Picard solutions and the so-called Chazy solutions. I give explicit formulae for them and completely determine their asymptotic behaviour near the singular points $0,1,\infty$ and their nonlinear monodromy. I study the structure of analytic continuation of the solutions to the PVI$\mu$ equation for any $\mu$ such that $2\mu\in\mathbb{Z}$. As an application, I classify all the algebraic solutions. For $\mu$ half-integer, I show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For $\mu$ integer, I show that all algebraic solutions belong to a one-parameter family of rational solutions.

PB - Springer UR - http://hdl.handle.net/1963/3118 U1 - 1215 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Monodromy of certain Painlevé-VI transcendents and reflection groups JF - Invent. Math. 141 (2000) 55-147 Y1 - 2000 A1 - Boris Dubrovin A1 - Marta Mazzocco AB - We study the global analytic properties of the solutions of a particular family of Painleve\\\' VI equations with the parameters $\\\\beta=\\\\gamma=0$, $\\\\delta={1\\\\over2}$ and $\\\\alpha$ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of our Painleve\\\' VI equation. PB - Springer UR - http://hdl.handle.net/1963/2882 U1 - 1818 U2 - Mathematics U3 - Mathematical Physics ER - TY - THES T1 - Algebraic Solutions to the Painlevé-VI Equation and Reflection Groups Y1 - 1998 A1 - Marta Mazzocco KW - Painlevé VI equation PB - SISSA UR - http://hdl.handle.net/1963/5574 U1 - 5402 U2 - Mathematics U3 - Mathematical Physics U4 - -1 ER - TY - JOUR T1 - Kam theorem for generic analytic perturbations of the Guler system JF - Z. Angew. Math. Phys. 48 (1997), no. 2, 193-219 Y1 - 1997 A1 - Marta Mazzocco AB - We apply here KAM theory to the fast rotations of a rigid body with a fixed point, subject to a purely positional potential. The problem is equivalent to a small perturbation of the Euler system. The difficulty is that the unperturbed system is properly degenerate, namely the unperturbed Hamiltonian depends only on two actions. Following the scheme used by Arnol\\\'d for the N-body problem, we use part of the perturbation to remove the degeneracy: precisely, we construct Birkhoff normal form up to a suitable finite order, thus eliminating the two fast angles; the resulting system is nearly integrable and (generically) no more degenerate, so KAM theorem applies. The resulting description of the motion is that, if the initial kinetic energy is sufficiently large, then for most initial data the angular momentum has nearly constant module, and moves slowly in the space, practically following the level curves of the initial potential averaged on the two fast angles; on the same time the body precesses around the instantaneous direction of the angular momentum, essentially as in the Euler-Poinsot motion. We also provide two simple physical examples, where the procedure does apply. PB - Springer UR - http://hdl.handle.net/1963/1038 U1 - 2818 U2 - Mathematics U3 - Mathematical Physics ER -