Bäcklund变换与圈群作用
Bäcklund Transformations and Loop Group Actions
摘要：
我们构造了一个局部行动的合理的映射组$S^2$ $GL(n,C)$上的流的局部解的$sl(n,C)$ -层次。我们证明了简单元素的作用（线性分式变换）给出了局部Bäcklund变换，并且我们从二次元的不同因式分解导出了一个置换公式。我们证明了简单元在真空上的作用既可以给出整体光滑解，也可以给出奇异解。然而，满足U（n）-现实条件$g(\barł)^*g(ł)=I$的有理映射的子群在$u(n)$ -层次流的整体快速衰减解空间上的作用是整体的，并且一个简单元素的作用给出了一个整体的Bäcklund变换.有理环群中某些元素对真空的作用产生了显式的时间周期多孤子（多呼吸子）。我们表明，这一理论推广了经典的Bäcklund理论的sine-Gordon方程。各种层次的Bäcklund变换的群结构由它们的现实条件决定。我们确定了$sl(n,R)$，$u(k,n-k)$，KdV，Kupershmidt-Wilson和Gel'fand-Dikii层次的现实条件（群结构）。满足一个现实条件的线性分式变换的作用，模有理映射群的中心，给出了由现实条件定义的层次的Bäcklund和Darboux变换。由于分解不能总是在这些现实条件下进行，作用又是局部的，并且Bäcklund变换仅为这些层次生成局部解，除非允许奇异解。
英文摘要：
We construct a local action of the group of rational maps from $S^2$ to $GL(n,C)$ on local solutions of flows of the ZS-AKNS $sl(n,C)$-hierarchy. We show that the actions of simple elements (linear fractional transformations) give local Bäcklund transformations, and we derive a permutability formula from different factorizations of a quadratic element. We prove that the action of simple elements on the vacuum may give either global smooth solutions or solutions with singularities. However, the action of the subgroup of the rational maps that satisfy the U(n)-reality condition $g(\barł)^*g(ł)=I$ on the space of global rapidly decaying solutions of the flows in the $u(n)$-hierarchy is global, and the action of a simple element gives a global Bäcklund transformation. The actions of certain elements in the rational loop group on the vacuum give rise to explicit time periodic multi-solitons (multi-breathers). We show that this theory generalizes the classical Bäcklund theory of the sine-Gordon equation. The group structures of Bäcklund transformations for various hierarchies are determined by their reality conditions. We identify the reality conditions (the group structures) for the $sl(n,R)$, $u(k,n-k)$, KdV, Kupershmidt-Wilson, and Gel'fand-Dikii hierarchies. The actions of linear fractional transformations that satisfies a reality condition, modulo the center of the group of rational maps, gives Bäcklund and Darboux transformations for the hierarchy defined by the reality condition. Since the factorization cannot always be carried out under these reality condition, the action is again local, and Bäcklund transformations only generate local solutions for these hierarchies unless singular solutions are allowed.
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