Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Matrix Szeg? biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szeg? polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szeg? matrix and the associated Szeg? recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson‐type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szeg? polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non‐Fuchsian type, are considered. For both cases, a new matrix version of the discrete Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed.     

On the heritability of the property <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">π</Emphasis></Subscript> by subgroups

We consider the so-called Jordan-Pochhammer systems, a special class of linear Pfaffian systems of Fuchsian type on complex linear (or projective) spaces. These systems appeared as systems of differential equations for hypergeometric type integrals in which the integrand is a product of powers of linear functions. These systems also arise in some reductions of the Knizhnik-Zamolodchikov equations. The main advantage of these systems is the possibility of presenting a basis in the solution space of such systems in an explicit integral form and, as a consequence, of describing their monodromy representation. The main focus in the paper is placed on the applications of Jordan-Pochhammer systems. We describe the relationship of Jordan-Pochhammer systems to isomonodromic deformations of Fuchsian systems that are described by the Schlesinger equations, as well as to the linearization of the dynamical system of bending spatial polygons. We also describe the application of Jordan-Pochhammer systems to constructing Kohno systems on the Manin-Schechtman configuration spaces.     

Generation and removal of apparent singularities in linear ordinary differential equations with polynomial coefficients

We discuss several examples of generating apparent singular points as a result of differentiating particular homogeneous linear ordinary differential equations with polynomial coefficients and formulate two general conjectures on the generation and removal of apparent singularities in arbitrary Fuchsian differential equations with polynomial coefficients. We consider a model problem in polymer physics.     

Kohno Systems on Manin–Schechtman Configuration Spaces

We consider multidimensional Fuchsian linear differential equations on the Manin–Schechtman configuration spaces, obtained by the Kohno construction. We study integrability of these systems in the sense of Frobenius and some of their reductions.     

Construction of a system of linear differential equations from a scalar equation

As is well known, given a Fuchsian differential equation, one can construct a Fuchsian system with the same singular points and monodromy. In the present paper, this fact is extended to the case of linear differential equations with irregular singularities.     

On the Galois group of some Fuchsian systems

In this paper, we give a new result ofn the differential Galois theory of linear ordinary differential equations. In particular, we compute the differential Galois group for a special type of nonresonant Fuchsian system.     

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)     

Constructible sets of linear differential equations and effective rational approximations of polylogarithmic functions

The aim of this paper is to investigate rational approximations to solutions of some linear Fuchsian differential equations from the perspective of moduli of linear differential equations with fixed monodromy group. One of the main arithmetic applications concerns the study of linear forms involving polylogarithmic functions. In particular, we give an explanation of the well-poised hypergeometric origin of Rivoal’s construction on linear forms involving odd zeta values.     

On the Cauchy problem for a class of iterated Fuchsian partial differential equations

In this paper, we consider the Cauchy problem with ramified data for a class of iterated Fuchsian partial differential equations. We give an explicit representation of the solution in terms of Gauss hypergeometric functions. Our results are illustrated through some examples.     

On the extendability of Noether’s integrals for orbifolds of constant negative curvature

This paper is concerned with the problem of the integrable behavior of geodesics on homogeneous factors of the Lobachevsky plane with respect to Fuchsian groups (orbifolds). Locally the geodesic equations admit three independent Noether integrals linear in velocities (energy is a quadratic form of these integrals). However, when passing along closed cycles the Noether integrals undergo a linear substitution. Thus, the problem of integrability reduces to the search for functions that are invariant under these substitutions. If a Fuchsian group is Abelian, then there is a first integral linear in the velocity (and independent of the energy integral). Conversely, if a Fuchsian group contains noncommuting hyperbolic or parabolic elements, then the geodesic flow does not admit additional integrals in the form of a rational function of Noether integrals. We stress that this result holds also for noncompact orbifolds, when there is no ergodicity of the geodesic flow (since nonrecurrent geodesics can form a set of positive measure).     

General linear problem of the isomonodromic deformation of Fuchsian systems

In contrast to nonresonance systems whose continuous deformations are always Schlesinger deformations, systems with resonances provide great possibilities for deformations. In this case, the number of continuous parameters of deformation, in addition to the location of the poles of the system, includes the data describing the Levelt structure of the system, or, in other words, the distribution of resonance directions in the space of solutions. The question of classifying the form and structure of deformations according to these parameters arises. In the present paper, we consider continuous isomonodromic deformations of Fuchsian systems, including those with respect to additional parameters, describe the corresponding linear problem, and present the Pfaff form of the linear problem of general continuous isomonodromic deformation of Fuchsian systems.     

Solvability of a periodic boundary-value problem for systems of functional differential equations with cyclic matrices

We consider first-order systems of linear functional differential equations with regular operators. For families of systems of two equations we obtain the general necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem. For families of systems of n linear functional differential equations with cyclic matrices we obtain effective necessary and sufficient conditions for the unique solvability of a periodic boundary-value problem.     

Fourier Method for Riemann-Hilbert-Poincaré Problem of Analytic Function

Spectrum problem with Riemann-Hilbert-Poincaré boundary condition is studied. This problem will lead to inhomogeneous Fuchsian differential equations with its right hand side depending on some constants to be determined simultaneously. We find out that the multiplicities of eigenfunctions for different eigenvalues are not necessary the same, which are in sharp contrast to the known results of Riemann-Hilbert problem for analytic functions.     

Integral representation of solutions to Fuchsian system and Heun's equation

We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlevé equation, and to Heun's equation.     

On the solution of boundary value problems with nonseparated multipoint and integral conditions

We suggest a numerical method for solving systems of linear nonautonomous ordinary differential equations with nonseparated multipoint and integral conditions. By using this method, which is based on the operation of convolution of integral conditions into local ones, one can reduce the solution of the original problem to the solution of a Cauchy problem for systems of ordinary differential equations and linear algebraic equations. We establish bounded linear growth of the error of the suggested numerical schemes. Numerical experiments were carried out for specially constructed test problems.     

Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations

A numerical method is suggested for solving systems of nonautonomous loaded linear ordinary differential equations with nonseparated multipoint and integral conditions. The method is based on the convolution of integral conditions into local ones. As a result, the original problem is reduced to an initial value (Cauchy) problem for systems of ordinary differential equations and linear algebraic equations. The approach proposed is used in combination with the linearization method to solve systems of loaded nonlinear ordinary differential equations with nonlocal conditions. An example of a loaded parabolic equation with nonlocal initial and boundary conditions is used to show that the approach can be applied to partial differential equations. Numerous numerical experiments on test problems were performed with the use of the numerical formulas and schemes proposed.     

Confluence of fuchsian second-order differential equations

Ordinary linear homogeneous second-order differential equations with polynomial coefficients including one in front of the second derivative are studied. Fundamental definitions for these equations: of s-rank of the singularity (different from Poincaré rank), of s-multisymbol of the equation, and of s-homotopic transformations are proposed. The generalization of Fuchs' theorem for confluent Fuchsian equations is proved. The tree structure of types of equations is shown, and the generalized confluence theorem is proved.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 2, pp. 233–247, August, 1995.     

On globally nilpotent differential equations

In a previous work of the authors, a middle convolution operation on the category of Fuchsian differential systems was introduced. In this note we show that the middle convolution of Fuchsian systems preserves the property of global nilpotence.This leads to a globally nilpotent Fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems.     

Differential systems with Fuchsian linear part: Correction and linearization, normal forms and multiple orthogonal polynomials

Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable) and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable.More generally, normal forms are found.The corrections and the normal forms are found constructively. Expansions in multiple orthogonal polynomials and their generalization to matrix-valued polynomials are instrumental to these constructions.     

Generalized normal forms for two-dimensional systems of ordinary differential equations with linear and quadratic unperturbed parts

Various definitions of normal forms for systems of ordinary differential equations are discussed. The notion of a generalized normal form and the problem of formal equivalency of systems of differential equations in terms of resonant equations are considered. The method of resonant equations is applied to two-dimensional systems whose unperturbed parts are linear in the first equation and quadratic in the second one.