Vortices and Bound States

这项研究项目涉及超导和非线性光学中某些类型的行为的数学建模和分析。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，例如模拟光在光纤中传播的脉冲解，或者在流体和凝聚态物理中产生的涡旋解。这项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(某些类型的超导体的典型特征)。在第二种情况下，我们将研究超导体中涡旋是如何产生的。第三个项目将考虑用于计算非线性光学中产生的脉冲解的数值算法。这项研究将集中在应用数学中出现的三个问题上。第一个项目将研究在超导中产生的渐近大量涡旋的平均场行为。分析需要渐近处理大量的涡旋，每个涡旋都具有渐近较大的能量。为了做到这一点，将开发精确的估计来跟踪能量集中情况。在含时背景下理解涡旋成核的计划将有助于理解II型超导体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，这应该会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.