Geometry of jet spaces and nonlinear partial differential equations. Transl. from the Russian by A. B. Sosinskij. (English) Zbl 0722.35001

A formal approach to the theory of nonlinear differential equations based on the geometry of jet spaces, is developed in this book. In the framework of the accepted formalism a differential equation is a surface in the jet-space and the solution is a manifold tangent to the standard Cartan distribution. Systematic use of the jet-language allows to give formal interpretation of the basic notions of the theory of partial differential equations. Such problems as the problem of symmetry existence, solvability and others, are considered.
Chapter 0. Introduction. Basic notions are introduced together with “translations” of jet-language into the usual mathematical language.
Chapter 1. Linear differential operators in commutative algebras. Basic categories and functors of the theory of differential equations are discussed here.
Chapter 2. Nonlinear differential operators and the geometry of jet- manifolds.
Chapter 3. Evolutions and linearizations. Here the space of infinite jets on a fiber bundle is introduced.
Chapter 4. The geometry of Cartan distribution. Lie Bäcklund transformations are also under discussion.
Chapter 5. Nonlinear differential equations, their solutions, symbols and symmetries.
Chapter 6. The geometry of Cartan distribution on the space of infinite jets.
Chapter 7. The projective point of view and geometry of infinitely prolonged equations. Invariance under transformations mixing up dependent and independent variables is under discussion.
Chapter 8. Some applications of the theory of symmetries of partial differential equations. Such examples as the KdV equation, the Burgers equation, the Khokhlov-Zabolotskaya equation and other are considered.


35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
35G20 Nonlinear higher-order PDEs
58A20 Jets in global analysis
58B25 Group structures and generalizations on infinite-dimensional manifolds
35Q53 KdV equations (Korteweg-de Vries equations)