齐性空间中的不变曲线流和对偶可积系统

The object is to study the relationship between Camassa-Holm type integrable systems and invariant curve flows in homogenoues space. Based on the equivariant moving frame metod, the differential invariants and Serret-Frenet equation for curves in homogeneous space can be obtained. Under certain conditions(such as the arc-length preserving condition), the Cartan structure equation of the curve flows induces the evolution equations for the curvatures or other geometric quantities of the curves. These equations includes the calssical integrable systems which have smooth soliton solutions or their dual Camassa-Holm type integable systems (known or new) which have non-smooth soliton solutions. In this way, some (new) dual integrable systems can be geometric realized(constructed). Furthermore, the integrability such as Lax pair, conservation law, recursion operator, nonlocal symmetries can be studied with aid of the geometric structure of the corresponding curve flows. The (Bi-)Hamiltonian structure of the dual integrable systems can also be obtained by using the Beffa's geometric Poisson reduction method. These study is helpful to investigate the relationship between differntial geometry and intreble systems in deep, as long as the properties of the dual integrable systems.

本项目拟研究齐性空间中不变曲线流和Camassa-Holm型可积系统之间的关系。利用等变活动标架法,可以建立齐性空间中曲线的微分不变量和Serret-Frenet公式。当曲线流满足一定条件（如保弧长条件）时，曲线流的Cartan结构方程给出曲线曲率或其他几何量所满足的方程，这包括经典具有光滑孤子解的可积系统和与其对偶的具有非光滑孤子解的Camassa-Holm型可积系统（已知的或新的），这样我们可以几何实现（或构造）一些新的对偶可积系统。利用曲线流的几何结构可以研究这些方程的可积性质，如Lax对，守恒律，递推算子，（非局部）对称，或几何变换等。利用Beffa的几何Poisson约化技巧，还可以得到这些可积系统的（双）哈密顿结构。这些研究有助于我们深入了解微分几何与可积系统间的关系及一些对偶可积系统的性质。