Geometric construction and properties of some families of solutions of nonlinear partial differential equations. I

This series of papers deals with ‘‘The 19th Century Theory of Partial Differential Equations from an Advanced Standpoint.’’ In the treatises of Darboux, Goursat, and Forsythe one finds methods for classifying nonlinear differential equations according to the geometric properties of families of solutions. This work was used by Elie Cartan in his theory of exterior differential systems, and is involved in an indirect way in today’s work on ‘‘nonlinear physics.’’ I plan to present several major themes of the classical work (e.g., ‘‘general,’’ ‘‘singular,’’ and ‘‘complete’’ solutions, ‘‘intermediate integral’’) using geometric methods developed by Cartan, Vessiot, Ehresmann, and Spencer. My aim is to develop this material from a point of view that is both fundamental and directed toward its ultimate application. Another possible utilization is in the development of the symbolic computation computer systems such as MACSYMA and REDUCE. This first paper concentrates on a description of what is meant by a ‘‘gener...