1,667 research outputs found
On the Reductions and Classical Solutions of the Schlesinger equations
The Schlesinger equations describe monodromy preserving
deformations of order Fuchsian systems with poles. They can be
considered as a family of commuting time-dependent Hamiltonian systems on the
direct product of copies of 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
to ``simpler'' having or .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 -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 -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 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
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