Algebraic Structures of Characteristics in Involutive Systems of Non-Linear Partial Differential Equations

An involutive system of partial differential equations is, roughly speaking, such a system that its general solution can be obtained by solving successively equations of Cauchy-Kowalevsky's type. It was E. Cartan who introduced the notion of involution for exterior differential systems. On this subject one has the Cartan-Kahler theorem and the prolongation theorem due to E. Cartan, M. Kuranishi and M. Matsuda (cf. Cartan [2, 4]; Kuranishi [11-12]; Matsuda [13-14]). M. Kuranishi constructed the process of standard prolongation, which was applied to infinite Lie groups by himself. It was also applied to the equivalence problem of G-structures by V.W. Guillemin, LM. Singer and S. Sternberg. In this course they clarified the algebraic structures of involutive systems. By their results M. Kuranishi gave a clear definition of involutive systems. His standard prolongation was improved by M. Matsuda, who combined it with the classical method of prolongation due to Lagrange and Jacobi. On the other hand, it is well-known that, in the classical and modern theory of partial differential equations, consideration of characteristics in various senses leads us to fruitful results. In our subject the two concepts of Cauchy characteristics and Monge ones will be particularly important. Effectiveness of the former was shown by E. Cartan for general systems ([2-4]). However, that of the latter seems to have been shown only for special systems (cf. Cartan [4], Chap. IV, Part III). The principal aim of this paper is to investigate algebraic structures