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) Backlund and auto-Backlund transformations for large classes of integrable equations covered by the KP, the modified KP, and the Dym hierarchies.
Cite this article
Boris Konopelchenko, Walter Oevel, An <i>r</i>-Matrix Approach to Nonstandard Classes of Integrable Equations. Publ. Res. Inst. Math. Sci. 29 (1993), no. 4, pp. 581–666DOI 10.2977/PRIMS/1195166743