Foliations and separation of variables. (English) Zbl 0553.58037

Sacksteder, Richard

Structure transverse des feuilletages, Toulouse 1982, Astérisque 116, 214-221 (1984).

[For the entire collection see Zbl 0534.00014.]
Let L be a linear partial differential operator, U and V be differential ideals defining a pair of complementary foliations, and \(F_ U\) and \(F_ V\) be the spaces of functions constant on the leaves of the two foliations. Then U and V are said to split L if there is a positive function R and partial differential operators \(M: F_ U\to F_ U\) and \(N: F_ V\to F_ V\) such that for all \(u\in F_ U\) and \(v\in F_ V\) \(L(uv)=R(vM(u)+uN(v)).\) The author gives necessary and sufficient conditions for a given pair of foliations to split L, and shows by examples how to apply the conditions to find all splittings of a given operator. He also discusses the simultaneous splitting of a 1-parameter family of operators.

Reviewer: B.Reinhart

MSC:

58J72	Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
37C85	Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
57R30	Foliations in differential topology; geometric theory

Keywords:

separation of variables; linear partial differential operator; differential ideals; complementary foliations; simultaneous splitting

Citations:

Zbl 0534.00014