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Darboux transformations, covariance theorems and integrable systems. (English) Zbl 0963.37066

Semenov-Tian-Shansky, Michael (ed.), L. D. Faddeev’s seminar on mathematical physics. Providence, RI: American Mathematical Society (AMS). Transl., Ser. 2, Am. Math. Soc. 201, 179-209 (2000).
In this paper a survey is provided of the results obtained by extending the idea originally introduced in 1882 by Gaston Darboux in order to manufacture ODEs of Sturm-Liouville type whose solutions could be exhibited in explicit form. The extension covers linear and nonlinear ODEs and PDEs, as well as discretized versions (including difference, and difference-differential, equations, and equations with generalized shift operators). The material covered constitutes an important chapter of the modern theory of integrable systems. Several of the results surveyed were originally obtained by the author and/or by his collaborators.
The paper is complemented by 61 references (the ordering criterion with which these references are numbered is puzzling for this reviewer). This paper is well-written and clear, hence it can be recommended as a good introduction to this topic.
In the KP-I equ. (2.22) there is a trivial misprint (the expression written should be equated to zero).
For the entire collection see [Zbl 0947.00010].

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35R30 Inverse problems for PDEs
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