On the Miura and Bäcklund Transformations Associated With the Supersymmetric Gelfand–dickey Bracket

Abstract

The supersymmetric version of the Miura and Bäcklund transformations associated with the supersymmetric Gelfand-Dickey bracket are investigated from the point of view of the Kupershmidt-Wilson theorem.Soon after the work of Zamolodchikov, 1 much attention has been paid to the Walgebras in two-dimensional conformal field theory and integrable systems. 2 Now it is well-known that the classical version of the W n -algebras arises naturally as the second Gelfand-Dickey (GD) bracket of the nth generalized Korteweg de Vries (KdV) hierarchy 3,4 defined by the Lax operator L = ∂ n +u n−1 ∂ n−1 +· · ·+u 0 . In such connection, the Kupershmidt-Wilson (KW) theorem 5,6 plays an important role in such a way that under the factorization L = (∂−φ n )(∂−φ n−1 ) · · · (∂−φ 1 ), the second GD bracket is transformed into a much simple one in terms of the Miura fields φ i . Thus the factorization not only provides the Miura transformation which maps the nth KdV hierarchy to the corresponding modified hierarchies but also gives a free-field realization of the associated classical W n -algebras. 7 On the other hand, as observed by Adler in Ref. 8, the cyclic permutation of the Miura fields φ i in the factorized Lax operator generates the Bäcklund transformation of the nth KdV hierarchy. Hence from the point of view of the KW theorem, one has a unified description of the Miura and Bäcklund transformations associated with the second GD structure for the nth KdV hierarchy. In general, the above scheme is encoded in a particular form of the Lax operator and its associated Poisson structure. Several integrable systems have been studied based on this scheme. For example, the KW theorem and Adler's work have been generalized to the constrained KP hierarchy 9-14 *

On the Bäcklund transformation for the Moyal Korteweg-de Vries hierarchy

“…The approach is intimately related to the Kupershnidt-Wilson (KW) theorem [4] (see also [5,6]) for the second Gelfand-Dickey (GD) bracket [2,3], the Hamiltonian structure of the KdV, in which the Miura variables play an important role. Based upon which, some generalizations including Toda [1], Drinfeld-Sokolov [7], KP [8][9][10], supersymmetric KdV [11] and q-deformed KdV [12] hierarchies have been discussed in a similar way.…”