44 research outputs found
On the multiplicity of the hyperelliptic integrals
Let be an Abelian integral, where
is a hyperelliptic polynomial of Morse type, a
horizontal family of cycles in the curves , and a polynomial
1-form in the variables and . We provide an upper bound on the
multiplicity of , away from the critical values of . Namely: $ord\
I(t) \leq n-1+\frac{n(n-1)}{2}\deg \omega <\deg H=n+1\delta(t)nHHI(t)\gamma(t)\textbf C^ n\gamma(t)\omegaHI(t)\{H=t\}
\subseteq \textbf C^2\omega\gamma(t)\textbf C^{n+1}ord I(t)\deg \omega$.Comment: 18 page
Cubic and quartic transformations of the sixth Painleve equation in terms of Riemann-Hilbert correspondence
A starting point of this paper is a classification of quadratic polynomial
transformations of the monodromy manifold for the 2x2 isomonodromic Fuchsian
systems associated to the Painleve VI equation. Up to birational automorphisms
of the monodromy manifold, we find three transformations. Two of them are
identified as the action of known quadratic or quartic transformations of the
Painleve VI equation. The third transformation of the monodromy manifold gives
a new transformation of degree 3 of Picard's solutions of Painleve VI.Comment: Added: classification of quadratic transformations of the Monodromy
manifold; new cubic (and quartic) transformations for Picard's case. 26
Pages, 3 figure
Non-Schlesinger Deformations of Ordinary Differential Equations with Rational Coefficients
We consider deformations of and matrix linear ODEs with
rational coefficients with respect to singular points of Fuchsian type which
don't satisfy the well-known system of Schlesinger equations (or its natural
generalization). Some general statements concerning reducibility of such
deformations for ODEs are proved. An explicit example of the general
non-Schlesinger deformation of -matrix ODE of the Fuchsian type with
4 singular points is constructed and application of such deformations to the
construction of special solutions of the corresponding Schlesinger systems is
discussed. Some examples of isomonodromy and non-isomonodromy deformations of
matrix ODEs are considered. The latter arise as the compatibility
conditions with linear ODEs with non-singlevalued coefficients.Comment: 15 pages, to appear in J. Phys.
Deformations of Fuchsian Systems of Linear Differential Equations and the Schlesinger System
We consider holomorphic deformations of Fuchsian systems parameterized by the
pole loci. It is well known that, in the case when the residue matrices are
non-resonant, such a deformation is isomonodromic if and only if the residue
matrices satisfy the Schlesinger system with respect to the parameter. Without
the non-resonance condition this result fails: there exist non-Schlesinger
isomonodromic deformations. In the present article we introduce the class of
the so-called isoprincipal deformations of Fuchsian systems. Every isoprincipal
deformation is also an isomonodromic one. In general, the class of the
isomonodromic deformations is much richer than the class of the isoprincipal
deformations, but in the non-resonant case these classes coincide. We prove
that a deformation is isoprincipal if and only if the residue matrices satisfy
the Schlesinger system. This theorem holds in the general case, without any
assumptions on the spectra of the residue matrices of the deformation. An
explicit example illustrating isomonodromic deformations, which are neither
isoprincipal nor meromorphic with respect to the parameter, is also given
Riemann-Hilbert problem for Hurwitz Frobenius manifolds: regular singularities
In this paper we study the Fuchsian Riemann-Hilbert (inverse monodromy)
problem corresponding to Frobenius structures on Hurwitz spaces. We find a
solution to this Riemann-Hilbert problem in terms of integrals of certain
meromorphic differentials over a basis of an appropriate relative homology
space, study the corresponding monodromy group and compute the monodromy
matrices explicitly for various special cases.Comment: final versio
Shear coordinate description of the quantised versal unfolding of D_4 singularity
In this paper by using Teichmuller theory of a sphere with four
holes/orbifold points, we obtain a system of flat coordinates on the general
affine cubic surface having a D_4 singularity at the origin. We show that the
Goldman bracket on the geodesic functions on the four-holed/orbifold sphere
coincides with the Etingof-Ginzburg Poisson bracket on the affine D_4 cubic. We
prove that this bracket is the image under the Riemann-Hilbert map of the
Poisson Lie bracket on the direct sum of three copies of sl_2. We realise the
action of the mapping class group by the action of the braid group on the
geodesic functions . This action coincides with the procedure of analytic
continuation of solutions of the sixth Painlev\'e equation. Finally, we produce
the explicit quantisation of the Goldman bracket on the geodesic functions on
the four-holed/orbifold sphere and of the braid group action.Comment: 14 pages, 2 picture
On the Geometry of Isomonodromic Deformations
This note examines the geometry behind the Hamiltonian structure of
isomonodromy deformations of connections on vector bundles over Riemann
surfaces. The main point is that one should think of an open set of the moduli
of pairs of vector bundles and connections as being obtained by
"twists" supported over points of a fixed vector bundle with a fixed
connection ; this gives two deformations, one, isomonodromic, of
, and another induced from the isomonodromic deformation of
. The difference between the two will be Hamiltonian.Comment: 20 page
An Isomonodromy Cluster of Two Regular Singularities
We consider a linear matrix ODE with two coalescing regular
singularities. This coalescence is restricted with an isomonodromy condition
with respect to the distance between the merging singularities in a way
consistent with the ODE. In particular, a zero-distance limit for the ODE
exists. The monodromy group of the limiting ODE is calculated in terms of the
original one. This coalescing process generates a limit for the corresponding
nonlinear systems of isomonodromy deformations. In our main example the latter
limit reads as , where is the -th Painlev\'e equation. We
also discuss some general problems which arise while studying the
above-mentioned limits for the Painlev\'e equations.Comment: 44 pages, 8 figure