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Topological and algebraic structure of linear problems associated with completely integrable systems. (English) Zbl 0536.35067

Group theoretical methods in physics, Proc. XIth int. Colloq., Istanbul 1982, Lect. Notes Phys. 180, 65-90 (1983).
[For the entire collection see Zbl 0527.00003.]
The Bäcklund transformations represent themselves as one of the most important tools in the study of completely integrable systems. Various geometric and analytical interpretations of Bäcklund transformations as differential correspondences were examined in recent years especially in connection with the isospectral deformation equations. The author reviews the applications of the Bäcklund transformations to study of completely integrable systems. The review presented is organized as follows. In Ch. I the author develops the concept of Bäcklund transformation as an isomonodromy deformation of solutions of the Riemann boundary value problem consisting in creation of apparent singularities and addition of integers to local multiplicities. This development is based on a striking analogy with the Padé approximation theory. In particular, in Ch. I the topological interpretation of Bäcklund transformations based on the category theory is discussed also. And, finally, the Bäcklund transformations realized as Darboux ones are applied to three-dimensional completely integrable systems. In Ch. II the author reviews briefly the generation of two-dimensional isospectral deformation equations on a basis of an arbitrary linear differential operator with matrix coefficients. In particular, the author demonstrates a way by which the solutions of all matrix two-dimensional systems are naturally imbedded into solutions of scalar three-dimensonal systems of the Kadomtsev- Petviashvili type. That construction is of importance in order to incorporate all matrix two-dimensional isospectral deformation systems into a single universal ”difference” relation.
Reviewer: E.Kryachko

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35A30 Geometric theory, characteristics, transformations in context of PDEs
41A21 Padé approximation
18A05 Definitions and generalizations in theory of categories

Citations:

Zbl 0527.00003