Category of nonlinear evolution equations, algebraic structure, and r-matrix
Category of nonlinear evolution equations, algebraic structure, and r-matrix
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通讯作者：
Zhijun Qiao;Cewen Cao;W. Strampp
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作者：
Zhijun Qiao;Cewen Cao;W. Strampp
本文讨论了一类非线性发展方程~NLEES！与谱问题有关，并提供了一种构造其代数结构和r-矩阵的方法。首先，我们引入了NLEE范畴，它由各种正阶和负阶的NLEE可积和不可积的族组成。整个NLEE范畴都有一个广义Lax表示。接下来，我们给出了Lax算子的两种不同的李代数结构：一种是范畴中通用的，即独立于层次的；另一种是层次中的非通用的，即依赖于底层层次。此外，我们还fi证明了两类伴随映射在代数结构下都是r-矩阵。特别地，没有等谱和非等谱Lax算子的中心扩张的Virasoro代数结构可以看作是我们的代数结构的约化。最后，我们给出了几个具体的例子来说明我们的方法。具体地说，当生成该类别的生成器被选择为独立于势函数时，Burgers的类别被线性化。在此基础上，给出了Burgers范畴中一个等谱负序族的通解。此外，在KdV范畴中，我们fi发现了一个有趣的事实：Harry-Dym谱包含在这个范畴中，众所周知的Harry-Dym方程包含在正序KdV谱中。煤气瓶2003
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. © 2003