Another Look on Recursion Operators

Recursion operators of partial diierential equations are identiied with BB acklund auto-transformations of linearized diieties. Relations to the classical concept and its recent Guthrie's generalization are discussed. Traditionally,a recursion operator of a PDE is a linear operator L acting on symmetries: if f is a symmetry then so is Lf. This provides a convenient way to generate innnite families of symmetries (Olver 13]). Standard reference is 14]. The presence of a recursion operator has been soon recognized as one of the attributes of the complete integrability. An exact and rich theory has been developed within the class of evolution equations (see Fokas 2] and references therein). Recursion operators are, as a rule, nonlocal and so may turn out to be the symmetries they generate (giving a powerful source of nonlocal symmetries 10, 5]). A remarkable problem of inverting a recursion operator motivated Guthrie 3] to a generalization, with consequences for generation of nonlocal symmetries 4]. His nonlocalities are no longer limited to mere inverses D ?1 of total derivatives. It seems to be an interesting problem to give an interpretation of recursion operators in terms of the Vinogradov 17] category of diieties. Recently, Krasilshchik and Kersten 6, 8] established a fundamental relation between recursion operators and deformations of diiety structures. In this paper (Section 3) we propose another answer, according to which recur-sion operators are BB acklund auto-transformations of so called linearized diiety (Section 2). Classical recursion operators are discussed in Section 4. The Guthrie operators turn out to be equivalent to what we call nitary linear operators, in the case of two independent variables (Section 5). To cover known examples in multi-dimension (e.g., the KP equation in 2]) nitary constructions are manifestly insuf-cient, but a theory of innnitary coverings still awaits a necessary development. In