标题： 光滑形状的演化与可积系统 显示英文标题

标题： Evolution of smooth shapes and integrable systems

摘要： 我们考虑从单位圆开始的光滑形状空间中的同伦演化。 基于Loewner-Kufarev方程，我们给出了这种演化的哈密顿形式，并提供了守恒定律。 演化的对称性由Virasoro代数给出。 “正”Virasoro生成元跨越了单位圆盘到复平面上共形嵌入空间上的复化向量丛的全纯部分，并在边界上光滑。 在协变形式中，它们沿哈密顿流被守恒。 “负”Virasoro生成元可以通过利用规范泊松结构的迭代方法恢复。 我们研究了Loewner-Kufarev轨迹到Segal-Wilson格拉斯曼流形的嵌入，构造了tau函数、Baker-Akhiezer函数，并最终给出了KP方程的一类解。 显示英文摘要

摘要： We consider a homotopic evolution in the space of smooth shapes starting from the unit circle. Based on the Loewner-Kufarev equation we give a Hamiltonian formulation of this evolution and provide conservation laws. The symmetries of the evolution are given by the Virasoro algebra. The `positive' Virasoro generators span the holomorphic part of the complexified vector bundle over the space of conformal embeddings of the unit disk into the complex plane and smooth on the boundary. In the covariant formulation they are conserved along the Hamiltonian flow. The `negative' Virasoro generators can be recovered by an iterative method making use of the canonical Poisson structure. We study an embedding of the Loewner-Kufarev trajectories into the Segal-Wilson Grassmannian, construct the tau-function, the Baker-Akhiezer function, and finally, give a class of solutions to the KP equation.