An $r$-Matrix Approach to Nonstandard Classes of Integrable Equations

Abstract

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.