An -Matrix Approach to Nonstandard Classes of Integrable Equations

  • Boris Konopelchenko

    Russian Academy of Sciences, Novosibirsk, Russian Federation
  • Walter Oevel

    Loughborough University, UK


Three different decompositions of the algebra of pseudo-differential operators and the corresponding -matrices are considered. Three associated classes of nonlinear integrable equations in and dimensions are discussed within the framework of generalized Lax equations and Sato's approach. The -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 -dimensional equations as well as to a variety of integrable equations in dimensions. It is shown, how the multi-Hamiltonian structure of the -dimensional equations can be obtained from the underlying -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.

Cite this article

Boris Konopelchenko, Walter Oevel, An -Matrix Approach to Nonstandard Classes of Integrable Equations. Publ. Res. Inst. Math. Sci. 29 (1993), no. 4, pp. 581–666

DOI 10.2977/PRIMS/1195166743