This book deals with the method of characteristics and the envelope method for solving the Cauchy problem for first order fully nonlinear partial differential equations (PDE). Second order hyperbolic PDE of Monge-Ampere type and the Cauchy problem for them are considered too.
The solutions of these nonlinear PDE can be interpreted as smooth surfaces (integral surfaces) developing specific singularities. Local existence results based on geometrical ideas are well known (Cauchy, Lagrange-Charpit, Goursat, Darboux and many others). We give the corresponding proofs and moreover, in several cases we construct solutions with maximal domain of definition, multivalued solutions and find their smooth single valued branches.
An Appendix with some recent results on the generic singularities of ruled and developable surfaces is given at the end of the book. Possible applications are to the equations of Clairaut, eikonal and Monge-Ampere types.The book contains different applications to geometry and mechanics, namely: geodesics, reconstruction of surfaces by their Gauss curvatures, canonical transformations, integration of the Hamilton-Jacobi systems, characteristics in the hodograph plane of two dimensional steady, isentropic, irrotational flow (the latter turning out to be epicycloids) and others. Anomalous singularities of the solutions of semilinear non-strictly hyperbolic systems with one or two space variables are studied too.
The first four Chapters can be used by (graduate) students in Math., Physics, Engineerings as a manual on nonlinear PDE equipped with more than 50 exercises. The rest part of the book could be of interest to PhD students and researchers in the domain of Analysis and Geometry.