On nonlinear differential Galois theory. (English) Zbl 1009.12005

Malgrange, B.

Chin. Ann. Math., Ser. B 23, No. 2, 219-226 (2002).

The author continues his paper [see Le groupoide de Galois d’un feuilletage, Monogr. Enseign. Math. 38, 465-501 (2001; Zbl 1033.32020)] discussing a new Galois theory of nonlinear differential equations. Let \(X\) denote a (smooth) complex analytic manifold, and let \(\operatorname{Aut}(X)\) be the space of germs of invertible maps \((X,a)\to (X,b)\), \(a,b\in X\). Some subgroupoids of \(\operatorname{Aut} (X)\) (Lie groupoids) defined by a system of partial differential equations are discussed in the paper. The author attaches such a groupoid to a foliation with singularities on \(X\) and calls it “the Galois groupoid of the foliation”. In this paper he considers several new examples for the theory.
Other approaches for building nonlinear differential Galois theory were provided by H. Umemura [algebraic case, see Nagoya Math. J. 144, 59-135 (1996; Zbl 0878.12002)] and J. F. Pommaret [analytic case, see Differential Galois theory, New York, Gordon & Bread (1983; Zbl 0539.12013)].

Reviewer: N.V.Grigorenko (Kyïv)

MSC:

12H05	Differential algebra
34M35	Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
35A99	General topics in partial differential equations

Keywords:

Galois theory of nonlinear differential equations; foliation with singularities; Galois groupoid

Citations:

Zbl 1033.32020; Zbl 0878.12002; Zbl 0539.12013