关于（1+1）维标量发展偏微分方程零曲率表示和Backlund变换的李代数
On Lie algebras responsible for zero-curvature representations and Backlund transformations of (1+1)-dimensional scalar evolution PDEs
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PDE的拉克斯对可以被解释为ZCR。在[arXiv：1303.3575]中，对于任何$(1+1)$维标量发展方程$E$，我们定义了一族李代数$F(E)$，它们在以下意义上负责$E$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。此外，使用这些代数，一个获得的必要条件存在的Backlund变换之间的两个给定的方程。代数$F(E)$在[arXiv：1303.3575]中根据生成元和关系定义。在该方法中，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。在本预印本中，我们证明了[arXiv：1303.3575]中宣布的关于$F(E)$的一些结果。我们目前的应用程序$F(E)$理论的Backlund变换更详细地描述了显式结构（非本质幂零理想）的代数$F(E)$的订单$3$和$5$的一些方程。
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. In [arXiv:1303.3575], for any $(1+1)$-dimensional scalar evolution equation $E$, we defined 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. Also, using these algebras, one obtains necessary conditions for existence of a Backlund transformation between two given equations. The algebras $F(E)$ are defined in [arXiv:1303.3575] in terms of generators and relations. In this 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 preprint we prove a number of results on $F(E)$ which were announced in [arXiv:1303.3575]. We present applications of $F(E)$ to the theory of Backlund transformations in more detail and describe the explicit structure (up to non-essential nilpotent ideals) of the algebras $F(E)$ for a number of equations of orders $3$ and $5$.
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