R^n上的几何Airy曲线流
The Geometric Airy Curve Flow on R^n
摘要：
Langer和Perline证明了，如果x是R^n上的几何Airy曲线流的解，则存在沿x（.，t），使得对应的主曲率满足（n-1）个分量修正KdV（vmKdV_n）。构造了高阶曲线流，其主曲率是vmKdV_n孤子族中高阶曲线流的解。本文给出了R^n中弧长参数化曲线空间上的Poisson结构，证明了几何Airy曲线流是Hamilton的，给出了一列交换Hamilton算子，构造了Backlund变换和显式孤子解.
英文摘要：
Langer and Perline proved that if x is a solution of the geometric Airy curve flow on R^n then there exists a parallel normal frame along x(. ,t) for each t such that the corresponding principal curvatures satisfy the (n-1) component modified KdV (vmKdV_n). They also constructed higher order curve flows whose principal curvatures are solutions of the higher order flows in the vmKdV_n soliton hierarchy. In this paper, we write down a Poisson structure on the space of curves in R^n parametrized by the arc-length, show that the geometric Airy curve flow is Hamiltonian, write down a sequence of commuting Hamiltonians, and construct Backlund transformations and explicit soliton solutions.
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