Gauge Transformations and Reciprocal Links in 2 + 1

Generalized Lax equations are considered in the spirit of Sato theory. Three decompositions of an underlying algebra of pseudo-diierential operators lead, in turn, to three diierent classes of integrable nonlinear hierarchies. These are associated with Kadomtsev-Petviashvili, modiied Kadomtsev-Petviashvili and Dym hierarchies in 2+1 dimensions. Miura-and auto-BB acklund transformations are shown to originate naturally from gauge transformations of the Lax operators. General statements on reciprocal links between these hierarchies are established, which, in particular , give rise to novel reciprocal auto-BB acklund transformations for the Dym hierarchy. These links are formulated as Darboux theorems for the associated Lax operators. 1. Introduction 2. Pseudo-diierential operators and integrable nonlinear hierarchies in 2+1 dimensions 2.1 The Kadomtsev-Petviashvili hierarchy 2.2 The modiied Kadomtsev-Petviashvili hierarchy 2.3 The 2+1-dimensional Dym hierarchy 3. Gauge transformations and reciprocal links 3.1 Gauge transformations 3.2 Reciprocal links 4. Results for 2+1-dimensional integrable hierarchies 4.1 The auto-BB acklund transformation for the KP hierarchy 4.2 The Miura link between the KP and modiied KP hierarchies 4.3 Auto-BB acklund transformations for the modiied KP hierarchy 4.4 The link between the modiied KP and Dym hierarchies 4.5 Auto-BB acklund transformations for the Dym hierarchy 5. Remarks on the commutativity of the Bianchi diagrams 6. Perspectives References