Category of nonlinear evolution equations, algebraic structure, and r-matrix

Abstract

This paper deals with the category of nonlinear evolution equations (NLEEs) associated with the spectral problem and provides an approach for constructing their algebraic structure and $r$-matrix. First we introduce the category of NLEEs, which composes of various positive order and negative order hierarchies of NLEEs both integrable and non-integrable. The whole category of NLEEs possesses a generalized Lax representation. Next, we present two different Lie algebraic structures of the Lax operator, one of them is universal in the category,i.e. independent of the hierarchy, while the other one is nonuniversal in the hierarchy, i.e. dependent on the underlying hierarchy. Moreover, we find that two kinds of adjoint maps are $r$-matrices under the algebraic structures. In particular, the Virasoro algebraic structures without central extension of isospectral and non-isospectral Lax operators can be viewed as reductions of our algebraic structure. Finally, we give several concrete examples to illustrate our methods. Particularly, the Burgers category is linearized when the generator, which generates the category, is chosen to be independent of the potential function. Furthermore, an isospectral negative order hierarchy in the Burger's category is solved with its general solution. Additionally, in the KdV category we find an interesting fact: the Harry-Dym hierarchy is contained in this category as well as the well-known Harry-Dym equation is included in a positive order KdV hierarchy.