一阶Toda场方程的Killing场和守恒律
Killing fields and Conservation Laws for rank-1 Toda field equations
摘要：
我们提出了一个连接之间产生的循环组方法可积系统和守恒律被视为特征上同调的元素的Killing字段。我们使用这种联系来生成齐采卡方程的完整守恒律集（作为特征上同调的元素），完成了Fox和Goertsches（2011）的工作。我们直接用守恒律定义了外微分系统的积分流形的有限型概念，推广了Pinkall-Sterling（1989）的定义.这个定义适用于任何具有无穷多个守恒律的外微分系统。最后，我们证明了，对于秩为1的Toda场方程，每一个特征上同调类都有一个平移不变代表作为不可微的守恒律。因此特征上同调定义了双周期解的de Rham上同调类。
英文摘要：
We present a connection between the Killing fields that arise in the loop-group approach to integrable systems and conservation laws viewed as elements of the characteristic cohomology. We use the connection to generate the complete set of conservation laws (as elements of the characteristic cohomology) for the Tzitzeica equation, completing the work in Fox and Goertsches (2011). We define a notion of finite-type for integral manifolds of exterior differential systems directly in terms of conservation laws that generalizes the definition of Pinkall-Sterling (1989). The definition applies to any exterior differential system that has infinitely many conservation laws possessing a normal form. Finally, we show that, for the rank-one Toda field equations, every characteristic cohomology class has a translation invariant representative as an undifferentiated conservation law. Therefore the characteristic cohomology defines de Rham cohomology classes on doubly periodic solutions.
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