LEAPS-MPS: Analysis of Initial and Boundary Value Problems

该奖项全部或部分由2021年美国救援计划法案（公法117-2）资助。自艾萨克·牛顿爵士用微分法描述行星物体的运动以来，微分方程在物理现象的研究中发挥了核心作用，是理解物理学、工程学、生物学、化学和经济学的基础。微分方程与数量或系统如何随空间和时间变化有关，是作为现实世界情况的模型推导出来的。在最优投资、传染病、化学反应和海啸等应用中，使用的微分模型通常是弥散型或弥散型。扩散和色散非线性偏微分方程的数学理论是不完整的，即使在基本的设置中，例如当考虑的区域可以被建模为半线或有界区间时，例如处理海底或光纤电缆的情况。该项目旨在进一步对这些模型进行数学研究，以帮助科学家理解这些潜在的物理现象。该项目将为本科生和研究生以及早期职业研究人员提供研究和培训机会，重点是促进代表性不足群体成员参与STEM。该项目的中心目标是使用统一变换方法（UTM）研究半线上和有界区间上的色散和扩散方程，该方法最近被发展为初始和边界数据的傅里叶变换的扩展。UTM允许在复平面上通过轮廓积分构建迭代映射，但尚未成功应用于非线性Schrödinger、Korteweg-de Vries和Burgers方程，这些方程具有非常低规则性空间中的初始值和边值数据。本项目将使用UTM将之前在非常低和非常高正则性空间中的无界域的结果扩展到有界区间，其主要目标是发展研究解的适定性和渐近行为所需的技术和理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2)Differential equations have played a central role in the study of physical phenomena since Sir Isaac Newton described the motion of planetary objects using differential calculus, and are fundamental to the understanding of physics, engineering, biology, chemistry, and economics. Differential equations relate how quantities or systems change over space and time and are derived as models for real world situations. In applications such as optimal investing, infectious diseases, chemical reactions, and tsunamis, the differential models used are often of diffusive or dispersive type. The mathematical theory for diffusive and dispersive nonlinear partial differential equations is incomplete, even in basic setups such as when the region considered can be model as the half line or a bounded interval, which for example is the case when dealing with the ocean floor or a fiber optic cable. This project intends to further the mathematical study of these models, to aid scientists in their understanding of the underlining physical phenomena. The project will provide research and training opportunities for undergraduate and graduate students and for early career researchers, with a focus in promoting the participation of members of underrepresented groups in STEM. The central aim of this project is to study dispersive and diffusive equations on the half line and on bounded intervals using the Uniform Transform Method (UTM), which has recently been developed as an extension to the Fourier transform for initial and boundary data. The UTM allows to construct iterative maps via contour integrals in the complex plane but has not yet been successfully applied to the Nonlinear Schrödinger, Korteweg-de Vries, and Burgers equations, with initial and boundary value data in spaces of very low regularity. This project will use the UTM to extend to bounded intervals previous results obtained for unbounded domains in spaces of both very low and very high regularity, with the main goal of developing the techniques and theory necessary to study well-posedness and asymptotic behavior of the solutions.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.