Category of Nonlinear Evolution Equations, Algebraic Structure, and r-Matrix

Authors

Zhijun Qiao, The University of Texas Rio Grande ValleyFollow
Cewen Cao, Zhengzhou University
Walter Strampp, Universitat-GH Kassel

Document Type

Article

Publication Date

1-2003

Abstract

In this paper we deal with the category of nonlinear evolution equations ~NLEEs! associated with the spectral problem and provide an approach for constructing their algebraic structure and r-matrix. First we introduce the category of NLEEs, which is composed of various positive order and negative order hierarchies of NLEEs both integrable and nonintegrable. 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 a central extension of isospectral and nonisospectral 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 Burgers’ 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.

Comments

© 2003 American Institute of Physics. Original published version available at https://doi.org/10.1063/1.1532769

Recommended Citation

Qiao, Zhijun, Cewen Cao, and Walter Strampp. 2003. “Category of Nonlinear Evolution Equations, Algebraic Structure, and r-Matrix.” Journal of Mathematical Physics 44 (2): 701–22. https://doi.org/10.1063/1.1532769.

First Page

701

Last Page

722

Publication Title

Journal of Mathematical Physics

DOI

10.1063/1.1532769