R-Matrix Approach to Nonstandard Classes of Integrable Equations

摘要

Three different decompositions of the algebra of pseudodifferential operators andthe 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-Pietviashvili (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-Hamiltoninan structure of the 1 + 1 dimensional equations can be obtained from the underlying r matrices. Furthermore, 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) Baecklund and auto-Baecklund transformations for large classes of integrable equations covered by the KP, the modified KP, and the Dym hierarchies.