Fuchsian Systems Research Articles

On Orders of Movable Poles of the Schlesinger Equation

We treat here movable singular points of the Schlesinger equation of isomonodromic deformations as the points of jumping of the splitting type of the vector bundle constructed from the initial Fuchsian equation. We show that solutions to the Schlesinger equation have at these points poles of order two and we also illustrate the results mentioned above by means of some Fuchsian system of two differential equations with four singular points.

On the Deligne-Simpson problem

Consider the Deligne-Simpson problem: give necessary and sufficient conditions for the choice of the conjugacy classes C j ∃( n, C) (or C j λ( n, C) so that there exist irreducible (p + 1) -tuples of matrices M j ε C j (or A j ε c j) satisfying the equality m 1 … M p+1 = I (or A 1 + … + A p+1 = 0) . We solve the problem for generic eigenvalues; their set is defined by explicit algebraic inequalities. There exist no reducible ( p + l)-tuples of matrices for generic eigenvalues. The matrices M 1 and A j are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann's sphere.

Control of evolutionary processes and deformation theory

A general approach to the study of control of general evolutionary processes is discussed. Such a process is described by a vector field with polynomial, analytic or smooth coefficients in phase space. The corresponding multidimensional systems are investigated by means of perturbation analysis of the associated bifurcation sets and the discriminants. The approach is based essentially on the modem deformation theory, the theory of discriminants and on the theory of integrable connexions associated to deformations of multidimensional systems with singularities. Such a connexion is presented as a Fuchsian system whose coefficients have logarithmic poles along the discriminant of a deformation.

On the Algebro-Geometric Integration¶of the Schlesinger Equations

A new approach to the construction of isomonodromy deformations of 2× 2 Fuchsian systems is presented. The method is based on a combination of the algebro-geometric scheme and Riemann–Hilbert approach of the theory of integrable systems. For a given number 2g+ 1, g≥ 1, of finite (regular) singularities, the method produces a 2g-parameter submanifold of the Fuchsian monodromy data for which the relevant Riemann–Hilbert problem can be solved in closed form via the Baker–Akhiezer function technique. This in turn leads to a 2g-parameter family of solutions of the corresponding Schlesinger equations, explicitly described in terms of Riemann theta functions of genus g. In the case g= 1 the solution found coincides with the general elliptic solution of the particular case of the Painleve VI equation first obtained by N. J. Hitchin [H1].

Quantum States of Monodromy Groups

Any irreducible finitely generated matrix group (with generators M1,…,Mp+1 satisfying the only relation M1…Mp+1 e I) is the monodromy group of some fuchsian linear system on Riemann's sphere. The eigenvalues of the matrices Mj define λk,j, the eigenvalues of the matrices-residues of the system only up to integers. There are always infinitely many possible choices of λk,j, a priori they must satisfy the only condition that their sum is 0. However, not always all a priori possible choices can be made. Some of them can be impossible due to the positions of the poles. Consider the a priori possible choices when the eigenvalues of only one matrix-residuum change (we presume that its pole is at 0). We show that infinitely many a priori possible choices are impossible if and only if the fuchsian system is obtained from another fuchsian system with a smaller number of poles and with a pole at 0 by the change of time t\mapsto t^k / (p_kt^k+p_{k-1}t^{k-1}+\ldots + p_0), p_i \in {\Bbb C}, p_0 \neq 0, k > 1. The result is applied to the Riemann–Hilbert problem.

On Reducible Monodromies Realized by Reducible Fuchsian Systems

It is shown that under the conditions of Theorems 1 and 2 in l2r, the Fuchsian systems realizing the reducible monodromy are in fact reducible systems. On the other hand, when the reducible monodromy is realized by a Fuchsian system, sufficient conditions on the monodromy matrices under which the final Fuchsian system can be chosen reducible are given.

La classification des systèmes aux q--différences singuliers réguliers par la matrice de connexion de Birkhoff

G.D. Birkhoff extended the Riemann-Hilbert problem for fuchsian linear q-difference systems with rational coefficients. He solved it in the generic (semi-simple) case: the classifying object is made up of the connection matrix P, together with the exponents at 0 and ∞. We follow his method in the case of a general regular singular system, but treat symetrically 0 and ∞ and use neither wildly growing nor multivalued functions; the matrix P then has elliptic coefficients.

Problème de Riemann—Hilbert pour une représentation de monodromie triangulaire supérieure de dimension 5

We show that any upper-triangular representation of dimension 5 can be realized as the monodromy representation of some upper-triangular Fuchsian system.

On isomonodromic deformations of Fuchsian systems

On isomonodromic deformations of Fuchsian systems

Monodromy Groups of Systems of Total Differential Equations of Two Variables

Fuchsian systems of ordinary differential equations that are irreducible and free from accessory parameters are classified into eight classes. Canonical systems of these classes were determined by Haraoka [SIAM J. Math. Anal., 25 (1994), pp. 1203--1226]. Among them the canonical systems of exactly four classes are extended to the completely integrable systems of total differential equations of two variables that are introduced by Yokoyama [Funkcial. Ekvac., 35 (1992), pp. 65--93]. This paper presents monodromy groups of the four systems of total differential equations. The theory for computing monodromy groups of the systems developed by Yokoyama is applied.

On Isomonodromic Deformations of Fuchsian Systems

Isomonodromic deformations of Fuchsian systems are considered. A description of all possible forms of such isomonodromic deformations is presented.

Regular systems whose monodromies differ from monodromies of Fuchsian systems

Regular systems whose monodromies differ from monodromies of Fuchsian systems

On the existence of monodromy groups of fuchsian systems on Riemann's sphere with unipotent genrators

In this paper we consider the following problem: For what choice of the (p+1)-tuple of conjugacy classesC 1,…,C p+1 in GL(n,ℂ),p≥2, do there exist irreducible (p+1)-tuples of matricesM j ∈C j such that the productM 1,…,M p+1 equals identity? We present the necessary and sufficient conditions for the existence of such tuples in the case whereM j are unipotent.

Vector bundles associated with monodromies and asymptotics of Fuchsian systems

Connections between the splitting type of a holomorphic vector bundle and the asymptotics of solutions to the corresponding Fuchsian system onP1 (ℂ) are presented. As an application, the exact number of apparent singularities arising as a result of the attempt to construct a scalar Fuchsian equation with a given monodromy is obtained.

Algorithmic methods for Fuchsian systems of linear partial differential equations

Algorithmic methods for Fuchsian systems of linear partial differential equations

Monodromy Representations of Systems of Differential Equations Free from Accessory Parameters

In a paper by Haraoka [SIAM J. Math. Anal., 25(1994), pp. 1203–1226], by following Okubo’s theory [K. Okubo, Seminar reports of Tokyo Metropolitan University, 1987], the canonical forms of all generic classes of Fuchsian systems of differential equations on ${\bf P}^1 ({\bf C})$ free from accessory parameters are obtained. Among them the explicit forms of six classes are new. In the present paper monodromy representations of the systems in the six classes are calculated. The technique employed is similar to one used to obtain the canonical forms. Hermitian forms invariant under those monodromy groups are also calculated. It turns out that the space of the invariant Hermitian forms for each system is real one-dimensional.

Holonomic quantum field theory of bosons in the Poincaré disk and the zero curvature limit

We formulate a holonomic quantum field theory of bosons on a Poincaré disk of radius R ( D R) based on a monodromy preserving deformation of the laplacian operator defined on D R . First the isomonodromy problem for the laplacian is converted to an isomonodromy problem for a fuchsian system using the hyperbolic Laplace transform defined on D R . The deformation equations for the fuchsian system (Schlesinger equations) and an associated closed one-form ω are then discussed. Locally ω = d log τ defines a τ-function which is then identified with an n-point function of bosons defined on D R . At every step we discuss the limit R → ∞ where the problem reduces to one on the euclidean plane. This facilitates a detailed comparison with the original results of Sato, Miwa and Jimbo on the euclidean plane. The two-point function is discussed in some detail and it is shown that it can be expressed in terms of a Painlevé transcendent of the sixth kind.

Fuchsian Systems Associated with the $P^2 (\mathbb{F}_2 )$-Arrangement

A family of Fuchsian systems of differential equations in two variables that interpolate Appell’s $F_4$, and $F_1 $ are constructed. Their geometric properties are also studied.MSC codes33A35KeywordsFuchsian systemAppell’s hypergeometric equationsorbifolduniformizationWeyl group

Monodramy preserving deformation theory of the Klein-Gordon equation in the hyperbolic plane

We study the complex linear space of multivalued solutions to y 2(ϖ x 2ƒ + ϖ y 2ƒ) = s(s − 1)ƒ, s > 1, on H 2 − {a 1,…,a n} , H 2 = hyperbolic plane , satisfying (1) ƒ→ exp(2π il v)ƒ is continued about a small loop enclosing a v ( v = 1,…, n), 0 l(inv)1, and (2) ƒ H 2 ( dx λ d y)/y 2 (y 2 ϖƒ ϖƒ + 1 4s (s − 1)ƒƒ) > ∞ . The solution space is of dimension n; and the extended system of equations are transformed, using the hyperbolic Laplace transform, to a Fuchsian system.

PFAFFIAN SYSTEMS OF FUCHS TYPE ON A COMPLEX ANALYTIC MANIFOLD

In this paper we give necessary and sufficient conditions for a completely integrable Pfaffian system with regular singular points on to be a Fuchsian system, where is a divisor with normal crossings in a compact Kahler manifold . We prove that the condition of being a Fuchsian system is equivalent to the solvability of some first Cousin problem on . This condition appears particularly simple when is complex projective space.Bibliography: 12 titles.