标题： 稳定的打结字符串 显示英文标题

标题： Stable Knotted Strings

摘要： 我们解决了闵可夫斯基空间$M^{3+1}$中相对论闭弦的柯西问题，包括初始数据具有纽结拓扑结构的情况。 我们给出了闭纽结弦的世界面成为时间周期曲面的一般条件。 在初始弦速度为零的特殊情况下，世界面的周期与初始弦长度（$\ell$）的一半成正比，并且对于$t=\ell/4$来说，纽结弦总是会坍缩为一个链环。 相对论闭弦是在时空中的动态演化或脉动结构，无论是否打结，这些结构在时间上都保持稳定。 从具有非零速度的初始简单链环构型开始，可以生成任意$n$重纽结。 显示英文摘要

摘要： We solve the Cauchy problem for the relativistic closed string in Minkowski space $M^{3+1}$, including the cases where the initial data has a knot like topology. We give the general conditions for the world sheet of a closed knotted string to be a time periodic surface. In the particular case of zero initial string velocity the period of the world sheet is proportional to half the length ($\ell$) of the initial string and a knotted string always collapses to a link for $t=\ell/4$. Relativistic closed strings are dynamically evolving or pulsating structures in spacetime, and knotted or unknotted like structures remain stable over time. The generation of arbitrary $n$-fold knots, starting with an initial simple link configuration with non zero velocity is possible.