Isomonodromic differential equations and differential categories

Abstract

We study isomonodromicity of systems of parameterized linear differential equations and related conjugacy properties of linear differential algebraic groups by means of differential categories. We prove that isomonodromicity is equivalent to isomonodromicity with respect to each parameter separately under a filtered-linearly closed assumption on the field of functions of parameters. Our result implies that one does not need to solve any non-linear differential equations to test isomonodromicity anymore. This result cannot be further strengthened by weakening the requirement on the parameters as we show by giving a counterexample. Also, we show that isomonodromicity is equivalent to conjugacy to constants of the associated parameterized differential Galois group, extending a result of P. Cassidy and M. Singer, which we also prove categorically. We illustrate our main results by a series of examples, using, in particular, a relation between Gauss-Manin connection and parameterized differential Galois groups.

31 pages

References in corpus (9)

• Nonhomogeneous quadratic duality and curvature
• Parameterized Picard-Vessiot extensions and Atiyah extensions
• Tannakian categories, linear differential algebraic groups, and parameterized linear differential equations
• Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations
• Unipotent differential algebraic groups as parameterized differential Galois groups
• Tannakian approach to linear differential algebraic groups
• A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function
• Computing the differential Galois group of a one-parameter family of second order linear differential equations
• Scalar Extension of Abelian and Tannakian Categories

Cited by in corpus (14)

• Parameterized Picard-Vessiot extensions and Atiyah extensions
• Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations
• A density theorem for parameterized differential Galois theory
• Galois groups for integrable and projectively integrable linear difference equations
• Approche galoisienne de la transcendance différentielle
• Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence
• Difference algebraic relations among solutions of linear differential equations
• Computing the differential Galois group of a parameterized second-order linear differential equation
• Finiteness properties of affine difference algebraic groups
• Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation
• Difference integrability conditions for parameterized linear difference and differential equations
• Parallel Telescoping and Parameterized Picard--Vessiot Theory
• Triviality of differential Galois cohomologies of linear differential algebraic groups
• Calculating Galois groups of third order linear differential equations with parameters