An r-Matrix Approach to Nonstandard Classes of Integrable Equations

Three different decompositions of the algebra of pseudo-differential operators and the corresponding r -matrices are considered. Three associated classes of nonlinear integrable equations in 1 +1 and 2 + 1 dimensions are discussed within the framework of generalized Lax equations and Sato's approach. The 2 + 1 -dimensional hierarchies are associated with the Kadomtsev–Petviashvili (KP) equation, the modified KP equation and a Dym equation, respectively. Reductions of the general hierarchies lead to other known integrable 2 + 1 -dimensional equations as well as to a variety of integrable equations in 1 +1 dimensions. It is shown, how the multi-Hamiltonian structure of the 1 +1 -dimensional equations can be obtained from the underlying r -matrices. Further, intimate relations between the equations associated with the three different r-matrices are revealed. The three classes are related by Darboux theorems originating from gauge transformations and reciprocal links of the Lax operators. These connections are discussed on a general level, leading to a unified picture on (reciprocal) Bäcklund and auto-Bäcklund transformations for large classes of integrable equations covered by the KP, the modified KP, and the Dym hierarchies.