Deformations of Fuchsian Systems of Linear Differential Equations and the Schlesinger System

Abstract

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.

References in corpus (2)

• Symplectic Manifolds and Isomonodromic Deformations
• Right and Left Joint System Representation of a Rational Matrix Function in General Position (System Representation Theory for Dummies)

Cited by in corpus (5)

• Isomonodromy Deformations at an Irregular Singularity with Coalescing Eigenvalues
• Canonical decomposition of linear differential operators with selected differential Galois groups
• Notes on Non-Generic Isomonodromy Deformations
• Families of Painleve VI equations having a common solution
• Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution