Isomonodromic differential equations and differential categories

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• 作者：Sergey Gorchinskiy^a ; ¹ ; ^{gorchins@mi.ras.ru" class="auth_mail} ; Alexey Ovchinnikov^b ; ^c ; ² ; ^{aovchinnikov@qc.cuny.edu" class="auth_mail}
• 关键词：primary ; 12H05 ; secondary ; 12H20 ; 13N10 ; 20G05 ; 20H20 ; 34M15 ; 34M56 ; 37K20
• 刊名：Journal des Mathematiques Pures et Appliquees
• 出版年：July 2014
• 年：2014
• 卷：102
• 期：1
• 页码：48-78
• 全文大小：448 K

文摘

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.