Fuchsian Systems Research Articles

Finitely smooth solutions of nonlinear singular partial differential equations

AbstractWe solve a Fuchsian system of singular nonlinear partial differential equations with resonances. These equations have no smooth solutions in general. We show the solvability in a class of finitely smooth functions. Typical examples are a homology equation for a vector field and a degenerate Monge–Ampère equation. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

Generalized Fuchsian Systems of Differential Equations

In the paper, the notion of generalized Fuchsian systems of differential equations with logarithmic singularities along a divisor D on a complex manifold is discussed. It is proved that such a system is characterized by the property of being regular singular along its singular locus in the classical sense. The proof is based on the main properties of logarithmic differential forms and vector fields; it does not use the traditional technique of resolution of singularities by means of which this problem is usually reduced to the study of divisors with normal crossings. In the case where the system in question has singularities along a free Saito divisor, a purely algebraic method of computing the integrability condition in terms of the commutation relations on its coefficient matrices is described.

Classification rationnelle et confluence des systèmes aux différences singuliers réguliers

By using meromorphic “characters” and “logarithms” built up from Euler’s Gamma function, and by using convergent factorial series, we will give, in a first part, a “normal form” to the solutions of a regular singular difference system. It will enable us to define a connection matrix for a regular singular system. Following one of Birkhoff’s idea, we will then study its link with the problem of rational classification of systems. In a second part, we will be interested in the confluence of fuchsian difference systems to differential systems. We will show more particularly how we can get, under some natural hypotheses, the local monodromies of a limit differential system from the connection matrices of the deformation that we consider. The use of factorial series (which can diverge as power series) distinguish regular singular difference systems from their differential and q-difference analogues and make their study more difficult.

Construction of rigid local systems and integral representations of their sections

We give a method for constructing all rigid local systems of semi-simple type, which is different from the Katz–Dettweiler–Reiter algorithm. Our method follows from the construction of Fuchsian systems of differential equations with monodromy representations corresponding to such local systems, which give an explicit solution of the Riemann–Hilbert problem. Moreover, we show that every section of such local systems has an integral representation. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Picard–Fuchs Equations and Gauss–Manin Systems with a View Towards the Riemann–Hilbert Problem

We study Abelian integrals associated with a tame polynomial function and their Picard---Fuchs equations using the theory of algebraic Gauss---Manin systems. Especially, we look for a basis of the Petrov module, in which the Picard---Fuchs equations become as simple as possible. As an application, we discuss the related Riemann---Hilbert problem and prove that it has a positive answer under some conditions. In this case, we compute the Jordan normal form of the residue matrices of the corresponding Fuchsian system in terms of local data.

Isomonodromic Deformations of sl(2) Fuchsian Systems on the Riemann Sphere

Isomonodromic Deformations of sl(2) Fuchsian Systems on the Riemann Sphere

Confluence q-différence vers différence pour un système Fuchsien

In a previous article we showed how one can see the classical result on existence and uniqueness of a fundamental set of factorial series solutions of a regular di.erence system as the limit, when q tends to 1, of analogous results for a q-di.erence system obtained by some kind of deformation. We extend here this property to singular-regular (or Fuchsian) systems.

On Schrödinger Equations of Okubo Type

We discuss some aspects of a well-known connection between Schrodinger equations and Fuchsian systems of Okubo type. Main attention is paid to the numerical in variants of holomorphic vector bundles on the Riemann sphere defined by a Fuchsian system of Okubo type. In particular, we compute the Chern class of the canonical extension and in some cases find its splitting type.

Discrete Symmetries of Systems of Isomonodromic Deformations of Second-Order Fuchsian Differential Equations

We compute the discrete affine group of Schlesinger transformations for isomonodromic deformations of a Fuchsian system of second-order differential equations. These transformations are treated as isomorphisms between the moduli spaces of logarithmic sl(2)-connections with given eigenvalues of the residues on ℙ1. The discrete structure is computed with the use of the modification technique for bundles with connections. The result generalizes the well-known classical computations of symmetries of the hypergeometric equation, the Heun equation, and the sixth Painleve equation.

Two-dimensional Fuchsian systems and the Chebyshev property

Let ( x( t), y( t)) ⊤ be a solution of a Fuchsian system of order two with three singular points. The vector space of functions of the form P( t) x( t)+ Q( t) y( t), where P, Q are real polynomials, has a natural filtration of vector spaces, according to the asymptotic behavior of the functions at infinity. We describe a two-parameter class of Fuchsian systems, for which the corresponding vector spaces obey the Chebyshev property (the maximal number of isolated zeros of each function is less than the dimension of the vector space). Up to now, only a few particular systems were known to possess such a non-oscillation property. It is remarkable that most of these systems are of the type studied in the present paper. We apply our results in estimating the number of limit cycles that appear after small polynomial perturbations of several quadratic or cubic Hamiltonian systems in the plane.

Some explicit solutions of the additive Deligne–Simpson problem and their applications

In this paper we construct three infinite series and two extra triples ( E 8 and E ̂ 8 ) of complex matrices B, C, and A= B+ C of special spectral types associated to Simpson's classification in Amer. Math. Soc. Proc. 1 (1992) 157 and Magyar et al. classification in Adv. Math. 141 (1999) 97. This enables us to construct Fuchsian systems of differential equations which generalize the hypergeometric equation of Gauss–Riemann. In a sense, they are the closest relatives of the famous equation, because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions. In all the cases except for E 8, we also explicitly construct scalar products such that A, B, and C are self-adjoint with respect to them. In the context of Fuchsian systems, these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions. When the eigenvalues of A, B, and C are real, the matrices and the scalar products become real as well. We find inequalities on the eigenvalues of A, B, and C which make the scalar products positive-definite. As proved by Klyachko, spectra of three hermitian (or real symmetric) matrices B, C, and A= B+ C form a polyhedral convex cone in the space of triple spectra. He also gave a recursive algorithm to generate inequalities describing the cone. The inequalities we obtain describe non-recursively some faces of the Klyachko cone.

Structure of solutions to Fuchsian systems of partial differential equations

AbstractTo a certain Volevič system of singular partial differential equations, called a Fuchsian system, all the solutions of the homogeneous equation in a complex domain are constructed and parametrized in a good way, without any assumption on the characteristic exponents.

Monodromy of Rational KZ Equations and Some Related Topics

A special class of integrable Fuchsian systems on Cn related to KZ equations is considered. We survey the construction of such systems and the list of the structural properties their monodromy representations. The relation of the Fuchsian systems obtained by the Veselov construction assosiated with a deformation of the An−1-type root system and the Gauss–Manin connection of the natural projection Cn→Cn−1 is described. In this case, we prove that the monodromy representation is equivalent to the Burau representation of the Artin braid group. For a deformations of the other root system, we introduce generalized Burau representations. We conjecture that the integrable Fuchsian systems related to essential new finite sets of the vectors described by Veselov and Chalykh are the result of the Klares–Schlesinger isomonodromic deformations (or transformation) of the integrable Fuchsian system related to the Coxeter root systems.

Quasialgebraicity of Picard—Vessiot Fields

We prove that under certain spectral assumptions on the monodromy group, solutions of Fuchsian systems of linear equations on the Riemann sphere admit explicit global bounds on the number of their isolated zeros.

The geometry of the classical solutions of the Garnier systems

Our aim is to find a general approach to the theory of classical solutions of the Garnier system in n-variables, G_n, based on the Riemann-Hilbert problem and on the geometry of the space of isomonodromy deformations. Our approach consists in determining the monodromy data of the corresponding Fuchsian system that guarantee to have a classical solution of the Garnier system G_n. This leads to the idea of the reductions of the Garnier systems. We prove that if a solution of the Garnier system G_n is such that the associated Fuchsian system has l monodromy matrices equal to ±1, then it can be reduced classically to a solution of a the Garnier system with n – 1 variables G_{n – 1}. When n monodromy matrices are equal to ±1, we have classical solutions of G_n. We give also another mechanism to produce classical solutions: we show that the solutions of the Garnier systems having reducible monodromy groups can be reduced to the classical solutions found by Okamoto and Kimura in terms of Lauricella hypergeometric functions. In the case of the Garnier system in 1-variables, i.e. for the Painleve VI equation, we prove that all classical non-algebraic solutions have either reducible monodromy groups or at least one monodromy matrix equal to ±1.

Rigidity Theorems for Multiparametric Deformations of the Universal Enveloping Algebras

The well-known one-dimensional Riemann–Hilbert problem on C concerns the proof of the existence of the Fuchsian system on C with a given monodromy, that is, with a given representation of a fundamental group, which is a braid group. In higher dimensions, the classical statement of the problem is modified so that instead of proving the existence of a system of differential equations of Fuchsian type, the problem of characterization of the representations of a fundamental group that can be a monodromy representation of the differential system of Fuchsian type is stated. It is a purely algebraic problem, but since it is closely connected with the question concerning the recovery of the functions of several complex variables if their ramification is given, its solution is based on analytical methods connected with properties of the power series on the parameters of the differential equation. In the context of the Riemann–Hilbert problem, to these methods should be added some algebraic ones concerning the algebraic structures that can be defined on algebras of such series. Some of these methods are considered in this paper, in particular, the method of multi-parametric deformations of some algebras, more exactly, the quantum enveloping algebras. In the one-parametric case, the quantum enveloping algebras are the algebras over the ring of the formal power series C[[h]]. The last object carries the topological structure defined by the powers of an ideal generated by a (quantum) parameter h. This topological structure naturally induces some structures on algebras and on modules over algebras. All methods of proving the different statements about these structures naturally are based on induction on the powers of h. Obviously, such a technique can be extended to the formal power series of several variables C[[h1, . . . , hn]]. Since many of the proofs of the theorems are easily carried over from the one-parametric case to the multi-parametric one, we have hopes that a variety of facts from the classical theory of quantum groups are valid for multi-parametric deformations. This paper is devoted to problems connected with multi-parametric generalizations of methods that are important for the multi-parametric analogue of the Drinfeld–Kohno theory. The well-known Drinfeld–Kohno theorem is a statement about the conditions of solvability of the Riemann–Hilbert problem in a class of the Knizhnik–Zamolodchikov equations [2, 7], which are particular cases of the multidimensional Fuchsian systems. Their proof and some of their generalizations can be found in [2,3,5,7]. In this paper, we first introduce some definitions that are used in the extensions of the Drinfeld– Kohno theory: the definition of the multi-parametric Drinfeld–Jimbo algebra and definitions of the braided multi-parametric quantum enveloping algebra and the braided multi-parametric quasi-bialgebra. We introduce the isomorphism of the quasi-bialgebras up to the twisting. Further, the uniqueness theorems for the homomorphisms of the trivial multi-parametric deformation of the enveloping algebra if these homomorphisms coincide on the universal enveloping algebras are proved. Then we prove some theorems on the uniqueness for the quantum enveloping algebras and for the Drinfeld–Jimbo algebra. We also prove the theorem on the isomorphism (up to the twisting) of the braided quasi-bialgebras having different associators. All these results are the necessary components in a proof of the multi-parametric analogue of the Drinfeld theorem on the isomorphism of the structure of a braided quasi-bialgebra (on

Sur le système de Gauss–Manin d'un polynôme modéré

Given a cohomologically tame polynomial we compute the determinant of a period matrix and, when the Gauss–Manin connexion is described by a fuchsian system, we give a formula for the trace of the residue matrices.

Systèmes fuchsiens à monodromie réductible

Let χ a reducible monodromy representation be realized by a Fuchsian system. It is shown that the representation χ can be realized by a Fuchsian system whose matrices of coefficients related to different poles are simultaneously reducible.

On the Riemann–Hilbert Problem in Dimension 4

We consider the problem of existence of a Fuchsian system with a prescribed 4-dimensional monodromy. We give a classification of all cases of negative solution of this problem in terms of reducibility pattern of the representation, its local structure (which is described by a modification of Jordan form), and restrictions on asymptotics of solutions to Fuchsian systems in lower dimensions. We also show that realization of a reducible 4-dimensional representation by a Fucshian system, if it exists, can be chosen in a block upper-triangular form (though not necessarily with the same reducibility pattern). At the end of the paper, we present new counterexamples to the Riemann–Hilbert problem in dimension 4.