标量发展方程零曲率表示的李代数
Lie algebras responsible for zero-curvature representations of scalar evolution equations
摘要：
零曲率表示是可积偏微分方程理论的主要工具之一。特别地，（1+1）维偏微分方程的拉克斯对可以解释为ZCR。对于任意的（1+1）维标量发展方程$E$，我们定义了一族李代数$F(E)$，它们在以下意义下负责$E$的所有ZCR.代数$F(E)$的表示将方程$E$的所有ZCR分类到局部规范变换。为了实现这一点，我们找到一个正规形式的ZCR相对于当地的规范变换组的行动。正如我们在其他出版物中所示，使用这些代数，得到了一些必要条件的可积性所考虑的偏微分方程（其中可积性理解的意义上的孤子理论）和必要条件存在的贝克隆变换之间的两个给定的方程。证明不可积性的例子和应用，以获得不存在的结果贝克隆变换在其他出版物以及。在我们的方法中，ZCR可以取决于任意阶的偏导数，其可以高于方程$E$的阶。代数$F(E)$推广了Wahlquist-Estabrook延拓代数，它负责一类小得多的ZCR。本文描述了$F(E)$的一般性质，给出了这些代数的生成元和关系。在其他出版物中，我们研究的结构$F(E)$方程的KdV，Krichever-Novikov，Kaup-Kupershirt，泽田-Kotera类型。在所得到的代数中，人们发现某些矩阵值函数在有理和椭圆代数曲线上的无限维李代数。
英文摘要：
Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs. For any (1+1)-dimensional scalar evolution equation $E$, we define a family of Lie algebras $F(E)$ which are responsible for all ZCRs of $E$ in the following sense. Representations of the algebras $F(E)$ classify all ZCRs of the equation $E$ up to local gauge transformations. To achieve this, we find a normal form for ZCRs with respect to the action of the group of local gauge transformations. As we show in other publications, using these algebras, one obtains some necessary conditions for integrability of the considered PDEs (where integrability is understood in the sense of soliton theory) and necessary conditions for existence of a Bäcklund transformation between two given equations. Examples of proving non-integrability and applications to obtaining non-existence results for Bäcklund transformations are presented in other publications as well. In our approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation $E$. The algebras $F(E)$ generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs. In this paper we describe general properties of $F(E)$ and present generators and relations for these algebras. In other publications we study the structure of $F(E)$ for equations of KdV, Krichever-Novikov, Kaup-Kupershmidt, Sawada-Kotera types. Among the obtained algebras, one finds infinite-dimensional Lie algebras of certain matrix-valued functions on rational and elliptic algebraic curves.
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