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On Gelfand-Dickey and Drinfeld-Sokolov systems. (English) Zbl 0818.35104
It is well-known that a solution \((\varphi_ 1, \dots, \varphi_ n)\) of the Drinfeld-Sokolov (DS) equations yields \(n\) solutions \((q_{0,k}, \dots, q_{n-2,k}),\) \(1 \leq k \leq n\) of the corresponding Gelfand-Dickey (GD) equations. For a given a solution \((q_{0,1}, \dots, q_{n-2, 1})\) of the GD equations a solution \((\varphi_ 1, \dots, \varphi_ n)\) of the corresponding DS equations and \(n - 1\) further solutions \((q_{0,k}, \dots, q_{n - 2,k})\), \(2 \leq k \leq n\) of the GD equations related to each other by generalized Miura-type transformations associated with factorizations of \(n\)-th order linear differential expressions are constructed in this paper.
The auto-Bäcklund transformations for the GD hierarchy are obtained as a by-product. These results are obtained in the case of general matrix- valued coefficients with entries in a commutative algebra over an arbitrary field.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
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