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Transformation groups applied to mathematical physics. Transl. from the Russian. (English) Zbl 0558.53040
The investigation of the qualitative behavior of solutions of nonlinear partial differential equations by means of invariance and symmetry and by the transformation of the equation into forms which are easier to tackle are methods with a long and rich history. The role group theoretic procedures play in the treatment of differential equations of mathematical physics can hardly be overestimated. A systematic and thorough compilation of such methods therefore is at any rate valuable. Concerning the width of the field a volume of 394 pages (in the English translation) cannot be encyclopedic, but can, as is done here, show some of its most fruitful aspects.
The term ”transformation group” is used here in some different meanings, namely: Lie group, smoothly operating on differentiable manifolds, one- parameter-groups of local diffeomorphisms and their prolongations (better called continuations) to jet-bundles, one-parameter-groups of tangent transformations of infinite order (Lie-Bäcklund-transformations).
The book takes the local point of view and stresses the calculating aspects more than the abstract and conceptual ones. So the geometrical significance of the constructions sometimes gets a little bit more veiled. On the other hand this manner of presentation will be highly appreciated by the physicist. The historical context of the problems is shown in a short but sufficient and stimulating way. For the detailed table of contents, which - even in the opinion of the author - replaces a survey [cf. the author, Transformation groups in mathematical physics (1983; Zbl 0529.53014)].
The volume consists of two parts: I. Point transformations, II. Tangent transformations. An introductory chapter contains besides necessary foundations concerning local transformation groups and differential equations some material of the theory of Lie algebras never used later. Chapter 1 (Motions in Riemannian spaces) treats generally the continuation of point transformations to Riemannian and pseudo-Riemannian metrics, specially conformal and isometric transformations. Chapter 2 is concerned with the Huyghens principle and its relation to conformal transformations. The second part of the book is devoted to the Lie- Bäcklund transformations. Chapter 3 introduces into the theory of Lie- Bäcklund groups, beginning with a nice description of the historical development of the ideas. The Lie-Bäcklund transformations are treated by formal one-parameter-groups and their generators. Chapter 4 shows equations with infinite Lie-Bäcklund groups, the heat equation, the Korteweg-de Vries equation and the wave equation in particular. The last chapter is concerned with conservation laws.
The volume contains an index and a list of the relevant literature with 219 items. The English translation is type-written. One could imagine a better lay-out. There are a lot of misprints, but not very grave ones.
Reviewer: K.Horneffer

MSC:
53B50 Applications of local differential geometry to the sciences
22-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups
35-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations
35A22 Transform methods (e.g., integral transforms) applied to PDEs
22E70 Applications of Lie groups to the sciences; explicit representations
22E05 Local Lie groups
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties