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J. Math. Pures Appl. (9), 2014, Volume 102, Issue 1, Pages 48–78 (Mi jmpa1)  

Isomonodromic differential equations and differential categories

S. Gorchinskiya, A. Ovchinnikovbc

a Steklov Mathematical Institute, Gubkina str. 8, Moscow, 119991, Russia
b CUNY Queens College, Department of Mathematics, 65-30 Kissena Blvd, Queens, NY 11367, USA
c CUNY Graduate Center, Department of Mathematics, 365 Fifth Avenue, New York, NY 10016, USA

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 the Gauss–Manin connection and parameterized differential Galois groups.

Funding Agency Grant Number
Russian Foundation for Basic Research 11-01-00145
12-01-31506
12-01-33024
13-01-12420
Ministry of Education and Science of the Russian Federation 11.G34.31.0023
Dynasty Foundation
National Science Foundation CCF-0952591
PSC-CUNY 60001-40 41
S. Gorchinskiy was supported by the grants RFBR 11-01-00145, 12-01-31506, 12-01-3302, 13-01-12420, AG Laboratory NRU HSE, RF government grant, ag.11.G34.31.0023, and by Dmitry Zimin’s Dynasty Foundation. A. Ovchinnikov was supported by the grants: NSF CCF-0952591 and PSC-CUNY No. 60001-40 41.


DOI: https://doi.org/10.1016/j.matpur.2013.11.001


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Document Type: Article
Received: 19.09.2012
Language: English

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