Dynamics and scattering of vortices and vortex rings

这个项目集中在一个经典的主题上，由于最近的实验结果，这个主题已经找到了新的相关性。包括Helmholtz和Kirchhoff在内的19世纪的研究人员写下了一组方程，描述了点涡旋-无穷小的漩涡--如何在流体中诱导运动，进而影响涡旋的运动。还有一个类似的系统来描述更复杂的涡环问题。玻色-爱因斯坦凝聚体(BEC)，一种在20世纪90年代首次实验产生的奇异物质相，以及包括液氦在内的超流体，都可以支持点涡旋和涡环。最近的实验工作已经成像了BEC中相互作用的点涡，显示出与数学预测的强烈一致。这个项目将从数学上研究点涡旋相互作用中的一些现象，以及与之密切相关的相互作用涡环问题。特别是，有一种运动被称为跳跃轨道，在这种运动中，一对涡环沿着一条线传播，在彼此移动的同时周期性地膨胀和收缩(这是众所周知的雪茄吸烟者用烟圈表演的一个技巧)。这在两个涡环的情况下是很好理解的，计算机模拟已经将这种现象推广到三个或更多个涡环，但还没有进行全面的数学研究。这个项目将对这个广义问题进行第一次详细的数学研究。很可能不可能为这些解写下精确的公式，所以数值技术将是必不可少的，特别是数值连续和分支方法，以及最近发展起来的称为拉格朗日描述符的方法，它使我们能够看到在其他不透明的混沌系统中的结构。第二个问题将研究一种称为点涡的混沌散射现象：可以使一对涡沿直线以恒定的速度传播。这种对之间的碰撞可以导致极其复杂的动力学，这将用动力系统的方法来研究。这个项目将把动力系统的技术应用到两类经典的相互作用点涡旋或涡环动力学的问题上。第一种是对跳跃涡问题的几种推广，即两对涡沿着一条直线运动，同时反复周期性地加宽，然后变窄，其中一对穿过另一对，而前导发生变化。对于点涡旋，这已经得到了很好的研究，但对于涡环，这一点的研究就不那么好了。这项研究旨在将这一系统推广到由三对或更多对涡旋(或三个或更多涡环)组成的系统，并跨越其他几何形状的轨道，如球面上的轨道。虽然对这类轨道的数值研究有限，但它们仍然没有得到数学上的探索。哈密顿约化是我们理解四涡旋系统的核心技术，导致两个两自由度的哈密顿量，这是服从于进一步的分析简化。对于较大的涡集，哈密顿约化不会产生足够小的系统来提供更多的洞察力，因此需要其他方法，包括数值延拓和分叉方法。该项目还将利用新开发的拉格朗日描述符方法，尽管系统的规模意味着需要就如何在初始条件空间中找到表面做出选择，这将使人们能够深入了解动力学。第二类问题是碰撞点涡对的混沌散射。该系统与早期NSF资助的研究中所研究的混沌散射有一些共同的特征：一个由一个分界线耦合到一个快动力系统的慢动力系统组成的两自由度系统。然而，这种情况下的动力学要复杂得多：没有小参数。相反，时间尺度的分离是由于动态而产生的，但可能不会一直保持下去。在以前的系统中，一个分离线描述了解决方案如何逃脱到无穷远，而这里的逃脱机制仍然未知。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project focuses on a classical subject that has found new relevance due to recent experimental results. Nineteenth century researchers including Helmholtz and Kirchhoff wrote down a system of equations describing how point vortices---infinitesimal whirlpools---induce motion in a fluid, which in turn affects the motion of the vortices. A similar system exists that describes the more complex problem of vortex rings. Both Bose-Einstein Condensates (BEC), an exotic phase of matter first produced experimentally in the 1990s, and superfluids including liquid helium, can support point vortices and vortex rings. Recent experimental work has imaged interacting point vortices in a BEC, showing strong agreement with mathematical predictions. This project will mathematically investigate a number of phenomena in the interactions of point vortices, and in the closely-related problem of interacting vortex rings. In particular, there is a motion called the leapfrogging orbit in which a pair of vortex rings propagate along a line, periodically expanding and contracting while moving through each other (this is a well-known trick performed by cigar smokers with smoke rings). This is well understood in the case of two vortex rings, and computer simulations have generalized this phenomenon to three or more rings, but have not performed a full mathematical study. This project will perform the first detailed mathematical study of the generalized problem. It is likely impossible to write down exact formulas for these solutions, so numerical techniques will be essential, especially numerical continuation and bifurcation methods and a recently-developed method called Lagrangian descriptors, which allows us to see structure in otherwise opaque chaotic systems. A second problem will study a phenomenon called chaotic scattering of point vortices: A pair of vortices can be made to propagate at a constant speed along a straight line. Collisions between such pairs can lead to extremely complex dynamics which will be investigated using dynamical systems methods.This project will apply techniques from dynamical systems to two classes of problems in the classical subject of the dynamics of interacting point vortices or vortex rings. The first is several generalizations of the problem of leapfrogging vortices, in which two pairs of vortices travel along a straight line while repeatedly and periodically widening, then narrowing, with one pair passing through the other, and the lead changing. This has been very well-studied for point vortices, but less well for vortex rings. The study aims to generalize this system to systems of three or more pairs of vortices (or three or more vortex rings), and to leapfrogging orbits in other geometries such as on the surface of a sphere. While limited numerical studies exist of such orbits, they remain unexplored mathematically. Hamiltonian reduction is the technique at the center of our understanding of the four-vortex system, leading two a two-degree-of-freedom Hamiltonian that is amenable to further analytical simplification. With larger sets of vortices, Hamiltonian reduction does not lead to systems of small enough dimension for the method to provide much insight, and other methods are required, including numerical continuation and bifurcation methods. The project will also make use of the newly-developed method of Lagrangian descriptors, although the size of the system means that choices need to be made about how to find surfaces in initial-condition space that will lead allow insight into the dynamics. The second class of problems is the chaotic scattering of colliding point vortex pairs. This system has some features in common with chaotic scattering studied in earlier NSF-sponsored research: a two degree-of-freedom system consisting of a slow dynamical system with a separatrix coupled to a fast dynamical system. However, the dynamics in this case are far more complex: there is no small parameter. Instead, the separation of time scales arises due to the dynamics but may not hold for all time. In the previous system, a separatrix describes how solutions escape to infinity, while the mechanism of escape here is still unknown.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.