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Introduction to the geometry of nonlinear differential equations. (Vvedenie v geometriyu nelinejnykh differentsial’nykh uravnenij). (Russian) Zbl 0592.35002

Moskva: “Nauka”. Glavnaya Redaktsiya Fiziko-Matematicheskoj Literatury. 336 p. R. 2.40 (1986).
A formal approach to the theory of nonlinear differential equations, based on the geometry of jet spaces, is developed in this book. In the frame of the accepted formalism, for example, an ordinary differential equation is a surface in the jet-space and its solution is a manifold which lies on this surface and which is tangent to the standard Cartan distribution.
Systematic exploitation of the jet-language allows to give formal analogues of the basic notions of the theory of ordinary and partial differential equations and to pose such problems as the problem of symmetry existence, solvability and others.
The book consists of 9 chapters. Chapter 0. Introduction. Basic notions are introduced here together with some nonformal interpretations and ”translations” into the usual mathematical language. Chapter 1. Linear differential operator in commutative algebras. Basic categories and functors of the theory of differential equations are discussed here. Chapter 2. Nonlinear differential operators and U-geometry on the jet- manifolds. Chapter 3. Evolutions and linearizations. Here the space \(J^{\infty}(\pi)\) of infinite jets of a fiber bundle \(\pi\) is introduced.
Chapter 4. The geometry of Cartan distribution. The theory of Lie- Bäcklund transformations is also under discussion. Chapter 5. Nonlinear differential equations, their solutions, symbols and symmetries. It is shown here that the notions of generalized solution, singularities, formal integrability are connected with the Spencer \(\delta\)-cohomologies of the symbol of the differential equation.
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. The problem of invariance under transformations mixing up dependent and independent variables is investigated. The category of nonlinear differential equations is introduced. Chapter 8. Some applications of the theory of symmetries of partial differential equations. Several examples are considered, such as the KdV equation, the Burgers equation, the Khokhlov-Zabolotskaya equation and others.
Reviewer: I.Ya.Dorfman

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A30 Geometric theory, characteristics, transformations in context of PDEs
58B25 Group structures and generalizations on infinite-dimensional manifolds
58A20 Jets in global analysis