LEAPS-MPS: Analysis of Initial and Boundary Value Problems

该奖项的全部或部分资金来自《2021年美国救援计划法案》(Public Law 117-2)。自从艾萨克·牛顿爵士使用微积分描述行星物体的运动以来，微分方程式一直在物理现象的研究中发挥着核心作用，并且是理解物理、工程、生物、化学和经济学的基础。微分方程式与量或系统如何随空间和时间变化有关，并作为现实世界情况的模型而推导出来。在最优投资、传染病、化学反应和海啸等应用中，所使用的微分模型通常是扩散或分散型的。扩散和色散非线性偏微分方程组的数学理论是不完整的，即使在基本设置中也是如此，例如当考虑的区域可以被建模为半直线或有界区间时，例如在处理海底或光缆时。这个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基本的物理现象。该项目将为本科生和研究生以及职业生涯初期的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的成员参与科学、技术和经济管理。这个项目的中心目标是使用均匀变换法(UTM)来研究半直线上和有界区间上的色散和扩散方程，该方法是最近发展起来的对初始和边界数据的傅立叶变换的推广。UTM允许通过复平面上的轮廓积分来构造迭代映射，但尚未成功地应用于具有很低正则性空间中的初值和边值数据的非线性薛定谔方程、Korteweg-de Vries方程和Burgers方程。这个项目将使用UTM将先前在非常低和非常高的正则性空间中获得的关于无界区域的结果扩展到有界区间，主要目标是开发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NSF的法定使命，并通过使用基金会的智力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.