Linear Pfaffian Systems and Classical Solutions of Triangular Schlesinger Equations

Abstract

In this paper, by classical solutions we mean solutions to Fuchsian type meromorphic linear integrable Pfaffian systems dy = Ωy on the complex linear spaces ℂⁿ, n ≥ 1, where y(z) = (y₁(z),..., y_n(z)^T ∈ ℂⁿ is a column vector and Ω is a meromorphic matrix differential 1-form such that Ω = ∑_1≤i<j≤nJ_ij (β)(z_i − z_j)⁻¹ d(z_i − z_j), with constant matrix coefficients Jij(β) depending on complex parameters β = (β₁,..., β₁). Under some constraints on the constant matrix coefficients J_ij (β), the solution components y_i(z), 1 ≤ i ≤ n, can be expressed as integrals of products of powers of linear functions; i.e., they are generalizations of the integral representation of the classical hypergeometric function F(z, a, b, c). Moreover, under some additional constraints on the parameters β, the components of the solutions are hyperelliptic, superelliptic, or polynomial functions. We describe such constraints on the coefficients J_ij(β) of Fuchsian type systems, as well as describe constraints on the sets of matrices (B₁(z),...,B_n(z)) for which the nonlinear Schlesinger equations dB_i(z) = $-\sum{_{j=1, j\neq i}^n}$[B_i(z),B_j(z)](z_i − z_j)⁻¹ d(z_i − z_j) reduce to linear integrable Pfaffian systems of the type described above and have solutions of the indicated type.