Compatibility, multi-brackets and integrability of systems of PDEs, prepr. Univ. Tromsø 2006-49; ArXive: math.DG/0610930

• Main Authors: Boris Kruglikov, Valentin Lychagin
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Subjects: Tromsø; Lagrange
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.234.9570; http://arxiv.org/pdf/math/0610930v1.pdf

We establish an efficient compatibility criterion for a system of generalized complete intersection type in terms of certain multi-brackets of differential operators. These multi-brackets generalize the higher Jacobi-Mayer brackets, important in the study of evolutionary equations and the integrability problem. We also calculate Spencer δ-cohomology of generalized complete intersections and evaluate the formal functional dimension of the solutions space. The results are applied to establish new integration methods and solve several differential-geometric problems. 1 Introduction and main results In this paper we introduce multi-brackets of non-linear vector differential operators. In the case of bi-brackets they coincide with the well-known Jacobi bracket, which is a generalization of the classical Lagrange-Jacobi bracket important in the theory of 1st order differential equations. These latter brackets