On the Reductions and Classical Solutions of the Schlesinger equations

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 $m\times m$ matrix algebras equipped with the standard linear Poisson bracket. In this paper we address the problem of reduction of particular solutions of ``more complicated'' Schlesinger equations $S_{(n,m)}$ to ``simpler'' $S_{(n',m')}$ having $n'< n$ or $m' < m$.Comment: 32 pages. To the memory of our friend Andrei Bolibruc

On the Genus Two Free Energies for Semisimple Frobenius Manifolds

We represent the genus two free energy of an arbitrary semisimple Frobenius manifold as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so-called "genus two G-function". Conjecturally the genus two G-function vanishes for a series of important examples of Frobenius manifolds associated with simple singularities as well as for ${\bf P}^1$-orbifolds with positive Euler characteristics. We explain the reasons for such Conjecture and prove it in certain particular cases.Comment: 37 pages, 3 figures, V2: the published versio

The Extended Bigraded Toda hierarchy

We generalize the Toda lattice hierarchy by considering N+M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are $\epsilon$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy. Using R-matrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups of the $A$ series.Comment: 32 pages, corrected typo

Stokes matrices for the quantum differential equations of some Fano varieties

The classical Stokes matrices for the quantum differential equation of projective n-space are computed, using multisummation and the so-called monodromy identity. Thus, we recover the results of D. Guzzetti that confirm Dubrovin's conjecture for projective spaces. The same method yields explicit formulas for the Stokes matrices of the quantum differential equations of smooth Fano hypersurfaces in projective n-space and for weighted projective spaces.Comment: 20 pages. Introduction has been changed. Small corrections in the tex

On the numerical evaluation of algebro-geometric solutions to integrable equations

Physically meaningful periodic solutions to certain integrable partial differential equations are given in terms of multi-dimensional theta functions associated to real Riemann surfaces. Typical analytical problems in the numerical evaluation of these solutions are studied. In the case of hyperelliptic surfaces efficient algorithms exist even for almost degenerate surfaces. This allows the numerical study of solitonic limits. For general real Riemann surfaces, the choice of a homology basis adapted to the anti-holomorphic involution is important for a convenient formulation of the solutions and smoothness conditions. Since existing algorithms for algebraic curves produce a homology basis not related to automorphisms of the curve, we study symplectic transformations to an adapted basis and give explicit formulae for M-curves. As examples we discuss solutions of the Davey-Stewartson and the multi-component nonlinear Schr\"odinger equations.Comment: 29 pages, 20 figure