Vortices and Bound States

本研究计画系关于超导与非线性光学中某些行为之数学模型与分析。 物理学中出现的许多问题都可以用非线性偏微分方程来建模，其中一些微分方程产生了具有有趣结构的解，例如模拟光通过光纤的脉冲式解或流体和凝聚态物理中出现的涡旋解。 本研究旨在了解其中的一些结构。 在第一种情况下，将研究超导材料中大量涡旋的行为（某些类别超导体的典型特征）。 在第二种情况下，将检查超导体中如何产生涡旋。 第三个项目将考虑用于计算非线性光学中出现的脉冲式解的数值算法。本研究将集中于应用数学中出现的三个问题。 第一个项目将研究超导性中产生的渐进大量涡旋的平均场行为。 该分析需要处理渐近大量的旋涡，其中每一个都有渐近大的能量。 为此，将为追踪能源集中情况制定精确的估计数。 在时间相关的设置中理解涡旋成核的项目将有助于理解在II型超导体中向混合涡旋态的转变。 最后，关于计算束缚态的迭代方法的项目将研究为什么某些方法是无条件收敛的，这将导致更好的算法设计。

This research project is concerned with the mathematical modeling and analysis of certain types of behaviors in superconductivity and nonlinear optics. Many problems arising in physics can be modeled by nonlinear partial differential equations, and some of these differential equations generate solutions with interesting structures, such as pulse-like solutions that model light traveling through fiber optics or vortex solutions that arise in fluids and condensed matter physics. This research aims to understand some of these structures. In the first case, the behavior of large numbers of vortices in superconducting materials (a typical feature of certain classes of superconductors) will be studied. In the second case, how vortices are generated in superconductors will be examined. The third project will consider numerical algorithms that are used to compute pulse-like solutions that arise in nonlinear optics.This research will focus on three problems arising in applied mathematics. The first project will study the mean-field behavior of asymptotically large numbers of vortices arising in superconductivity. The analysis requires handling asymptotically large numbers of vortices, each of which has asymptotically large energy. To do so, refined estimates will be developed for tracking concentrations of energy. The project on the understanding of vortex nucleation in the time-dependent setting should help to understand the transition to the mixed-vortex state in Type-II superconductors. Finally, the project on iterative methods for computing bound states will study why certain methods are unconditionally convergent, which should lead to better algorithm design.