CAREER: Wyoming Applied Analysis & Computing Group: Behavior of Solutions of Nonlinear Partial Differential Equations

该奖项支持一系列协调的研究和教育活动，目的是阐明某些非线性偏微分方程解的行为，并刺激怀俄明大学在分析、微分方程和相关计算方面的研究和学习环境。这项活动以怀俄明州应用分析与计算小组为中心，该小组由本科生和研究生、怀俄明大学教职员工以及怀俄明州社区学院网络的教职员工组成。非线性偏微分方程常出现在物理现象的数学模型中。然而，对它们的性质的广泛理解仍然是纯数学和应用数学中的一个基本挑战。作为本研究重点的微分方程具物理动机，其特征(非线性、多空间维度、多尺度)代表了此类方程分析中的一些中心挑战。研究议程包括调查最近关于聚焦非线性薛定谔方程中剧烈振荡形成的一个猜想；这个猜想是一个新出现的信念的一部分，即这些振荡的形成可能具有普遍的特征。这种普遍行为在微分方程式中的出现得到证实，必将推动非线性现象分析的一波新发展。在数学层面上，攻击计划基于广泛的分析技术，包括埃文斯函数、奇异摄动、动力系统、线性算子的谱理论、复数分析和数值分析。如果实现了预期的结果，可能会对数学的应用和其他领域产生影响。其中一个研究项目旨在扩展在可积偏微分方程分析中出现的Riemann-Hilbert问题的渐近分析；这种扩展有望在正交多项式和随机矩阵的看似无关的分析中有用。研究议程所需的广泛数学工具确保怀俄明州应用分析与计算小组将成为下一代研究人员和学生在应用分析和计算方面的理想培训基地。除了研究议程上的集体工作外，该组织还将为选定的社区学院教职员工举办为期两周的年度沉浸研究活动。此外，该小组的活动还将包括参加一年一度的“衔接会议”，这些会议旨在将怀俄明大学的数学和科学系与怀俄明州七所社区学院的数学和科学系联系起来。这些活动将加强全州的数学网络。该小组的招募和培训战略部分是基于开发一门新的微积分课程，该课程将数值方法视为微积分的内在组成部分。该课程为学生在其职业生涯的早期阶段参与该小组做好准备。

This award supports a coordinated set of research and educational activities with the goals of elucidating the behavior of solutions of certain nonlinear partial differential equations and stimulating the environment for research and study in analysis, differential equations, and related computation at the University of Wyoming. This activity centers on the Wyoming Applied Analysis & Computing Group, a team consisting of undergraduate and graduate students, University of Wyoming faculty, and faculty from Wyoming's network of community colleges. Nonlinear partial differential equations arise generically in mathematical models of physical phenomena. However, a broad understanding of their properties remains a fundamental challenge in pure and applied mathematics. The differential equations that are the focus of this research are physically motivated, and their features (nonlinearity, multiple space dimensions, multiple scales) represent some of the central challenges in the analysis of such equations. The research agenda includes investigation of a recent conjecture about the formation of violent oscillations in the focusing nonlinear Schrodinger equation; this conjecture is part of an emerging belief that the formation of these oscillations may have a universal character. The validation of the appearance of such universal behavior in differential equations is certain to drive a wave of new developments in the analysis of nonlinear phenomena. On a mathematical level, the plan of attack is based on a broad array of analytical techniques, including Evans functions, singular perturbations, dynamical systems, spectral theory for linear operators, complex analysis, and numerical analysis. The anticipated results, if achieved, have the potential to make an impact on applications and other areas of mathematics. One of the research projects aims to extend the asymptotic analysis of Riemann-Hilbert problems arising in the analysis of integrable partial differential equations; such extensions are expected to be useful in the seemingly unrelated analysis of orthogonal polynomials and random matrices.The breadth of mathematical tools required by the research agenda ensures that the Wyoming Applied Analysis & Computing Group will serve as an ideal training ground in applied analysis and computation for the next generation of researchers and students. In addition to collective work on the research agenda, the group will host an annual two-week immersion into research activities for select community-college faculty. Additionally, the group's activities will include participation in the annual "articulation conferences" that are held to connect the math and science departments at the University of Wyoming with those at Wyoming's seven community colleges. These activities will strengthen the mathematical network across the state. The group's recruitment and training strategy is partially based on the development of a novel calculus course that treats numerical methods as an intrinsic part of calculus. The course prepares students to participate in the group at an early stage of their careers.