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Infinite-dimensional Lie groups and algebraic geometry in soliton theory. (English) Zbl 0582.35103

We study several methods of describing ’explicit’ solutions to equations of Korteweg-de Vries type: (i) the method of algebraic geometry [I. M. Krichever, Usp. Mat. Nauk 32, No.6(198), 183-208 (1977; Zbl 0372.35002)]; (ii) the Grassmannian formalism of the Kyoto school; (iii) acting on the trivial solution by the ’group of dressing transformations’ [V. E. Zakharov and A. B. Shabat, Funkts. Anal. Prilozh. 13, No.3, 13-22 (1979; Zbl 0448.35090)]. I show that the three methods are more or less equivalent, and in particular that the ’\(\tau\)-functions’ of methods (ii) arise very naturally in the context of method (iii).

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
35A30 Geometric theory, characteristics, transformations in context of PDEs
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
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