Vortices and Bound States

涡旋和束缚态

• 批准号: 1516565
• 负责人: Daniel Spirn
• 金额: $ 20.82万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1516565&HistoricalAwards=false
• 关键词: Vortices; Bound; States

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.

该研究项目与超导性和非线性光学的某些类型行为的数学建模和分析有关。 物理学出现的许多问题可以通过非线性部分差分方程进行建模，其中一些微分方程生成具有有趣结构的溶液，例如类似脉冲的溶液，这些溶液模拟了在流体和凝结物理学中出现的光纤或涡旋溶液中传播的光。 这项研究旨在了解其中一些结构。 在第一种情况下，将研究超导材料中大量涡流的行为（某些类别超导体的典型特征）。 在第二种情况下，将检查如何在超导体中产生涡旋。 第三个项目将考虑用于计算非线性光学中出现的类似脉冲溶液的数值算法。这项研究将重点放在应用数学中出现的三个问题上。 第一个项目将研究超导性产生的渐近涡流的平均场行为。 该分析需要渐近地处理大量的涡旋，每个涡流都具有渐近的能量。 为此，将开发出精制的估计来跟踪能量浓度。 在时间依赖性设置中理解涡旋成核的项目应有助于理解II型超导体中混合涡流状态的过渡。 最后，关于计算结合状态的迭代方法的项目将研究为什么某些方法无条件收敛，这应该导致更好的算法设计。