（1+1）维偏微分方程的高阶喷流延拓李代数和Backlund变换
Higher jet prolongation Lie algebras and Backlund transformations for (1+1)-dimensional PDEs
摘要：
对于任意的（1+1）维（多分量）演化PDE，定义了一个李代数序列$F^p$，$p=0,1,2,3,...$，它们负责该PDE的所有拉克斯对和零曲率表示（ZCR）.在我们的结构中，允许任意顺序的射流。在低阶喷注情形下，代数$F^p$推广了Wahlquist-Estabrook延拓代数.为了实现这一点，我们找到一个规范形式的（非线性）ZCR相对于组的规范变换的行动。一个证明了任何ZCR都是局部规范等价于由代数$F^p$的矢量场表示产生的ZCR，其中$p$是ZCR的$x$ -部分中包含的喷注的阶。更准确地说，我们定义了一个李代数$F^p$为每个非负整数$p$和每个点$a$的无限延长$E$的发展PDE。所以代数的完整符号是$F^p(E,a)$。利用这些代数，人们得到了一个必要条件，两个给定的发展偏微分方程被连接的一个Backlund变换。本文计算了KdV型偏微分方程的代数$F^p(E,a)$.在另一篇文章中，G. Manno，我们计算了Golubchik和Sokolov的多组分Landau-Lifshitz系统的$F^p(E,a)$。在所得到的李代数中，人们遇到了某些矩阵值函数在某些代数曲线上的无限维代数。此外，在$F^p(E,a)$的描述中还出现了一些可解理想和半单李代数。应用分类KdV和Krichever-Novikov型方程的Backlund变换也简要讨论。
英文摘要：
For any (1+1)-dimensional (multicomponent) evolution PDE, we define a sequence of Lie algebras $F^p$, $p=0,1,2,3,...$, which are responsible for all Lax pairs and zero-curvature representations (ZCRs) of this PDE. In our construction, jets of arbitrary order are allowed. In the case of lower order jets, the algebras $F^p$ generalize Wahlquist-Estabrook prolongation algebras. To achieve this, we find a normal form for (nonlinear) ZCRs with respect to the action of the group of gauge transformations. One shows that any ZCR is locally gauge equivalent to the ZCR arising from a vector field representation of the algebra $F^p$, where $p$ is the order of jets involved in the $x$-part of the ZCR. More precisely, we define a Lie algebra $F^p$ for each nonnegative integer $p$ and each point $a$ of the infinite prolongation $E$ of the evolution PDE. So the full notation for the algebra is $F^p(E,a)$. Using these algebras, one obtains a necessary condition for two given evolution PDEs to be connected by a Backlund transformation. In this paper, the algebras $F^p(E,a)$ are computed for some PDEs of KdV type. In a different paper with G. Manno, we compute $F^p(E,a)$ for multicomponent Landau-Lifshitz systems of Golubchik and Sokolov. Among the obtained Lie algebras, one encounters infinite-dimensional algebras of certain matrix-valued functions on some algebraic curves. Besides, some solvable ideals and semisimple Lie algebras appear in the description of $F^p(E,a)$. Applications to classification of KdV and Krichever-Novikov type equations with respect to Backlund transformations are also briefly discussed.
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