LEAPS-MPS: Long-time behavior for nonlinear dispersive equations

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。非线性色散偏微分方程出现在许多物理环境中，其特征是不同频率的波以不同速度传播的趋势。在这种模型中，色散效应和非线性效应之间经常存在竞争，从而产生丰富的可能的解行为集。这些包括衰减和散射，被称为孤立波的相干结构的存在，甚至波塌缩（或爆炸）。该项目包括考虑与非线性色散方程解的长期行为有关的问题，包括孤立波的稳定性，低正则解的全局衰减估计，以及处于或接近某些尖锐散射阈值的解的行为。该项目侧重于几个具体的模型，这些模型在物理上有意义，但仍然足够简单，可以进行深入分析。这样的选择使我们能够对色散方程领域中一些重要问题背后的基本数学困难进行提炼。通过这种方式，拟议的研究有可能为未来的进展铺平道路，甚至超出了本项目所考虑的具体问题。该项目包含适合本科生、硕士和博士学生参与的问题。该项目包括几项活动，通过向公立学校推广、组织会议和指导本科生研究项目，鼓励代表性不足的学生或农村学生参与STEM。该项目将首先解决一维非线性薛定谔方程（NLS）在吸引delta势存在下的解的渐近稳定性，这是非线性光学中出现的一个简单模型。一些主要目标包括建立整个稳定孤立波族的渐近稳定性，以及不稳定状态下稳定流形的构造。接下来，该项目将解决完全可积模型（包括1d立方NLS）的低正则解的全局时空估计问题。该项目旨在发展适应于新的微观守恒定律的维里和莫拉韦茨型估计，这些守恒定律最近在此类方程的低正则性适定性理论中发挥了关键作用。第三，该项目将解决与破对称NLS模型解的阈值行为相关的几个问题，包括非齐次NLS和具有外部电位的NLS。除了在尖锐散射阈值处对可能的解动力学进行分类外，该项目还将涉及为缺乏非线性基态的模型构建具有行波行为的解。最后，该项目旨在通过从高中阶段开始培养学生对STEM学科的兴趣，以及为本科和研究生阶段的数学学生建立一个支持性社区，来增加代表性不足群体对数学的参与。实现这一目标的具体步骤包括向公立高中推广，为数学专业的本科生组织定期会议和演讲，监督本科生的研究，以及继续组织研究研讨会和邀请研究访问者。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). Nonlinear dispersive partial differential equations arise in many physical settings and are characterized by the tendency of waves of different frequencies to travel at different velocities. In such models there is often a competition between dispersive and nonlinear effects, resulting in a rich set of possible solution behaviors. These include decay and scattering, the presence of coherent structures known as solitary waves, or even wave collapse (or blowup). This project includes consideration of problems related to the long-time behavior of solutions to nonlinear dispersive equations, including the stability properties of solitary waves, global decay estimates for low regularity solutions, and the behavior of solutions living at or near certain sharp scattering thresholds. The project focuses on several specific models that are physically meaningful but still simple enough to admit deep analysis. Such choices allow for the distillation of the essential mathematical difficulties underlying some important problems in the field of dispersive equations. In this way, the proposed research has the potential to pave the way for future progress even beyond the specific problems under consideration in this project. The project contains problems that are suitable for the involvement of students at the undergraduate, Masters, and PhD levels. The project includes several activities to encourage participation of underrepresented or rural students in STEM via outreach to public schools, organization of meetings and mentoring of undergraduate research projects.The project will first address asymptotic stability properties for solutions to the one-dimensional nonlinear Schrodinger equation (NLS) in the presence of an attractive delta potential, a simple model arising in nonlinear optics. Some of the main goals include establishing asymptotic stability for the entire family of stable solitary waves, as well as the construction of stable manifolds in the unstable regime. Next, the project will address the problem of global space-time estimates for low regularity solutions to completely integrable models, including the 1d cubic NLS. The project seeks to develop virial and Morawetz-type estimates adapted to the novel microscopic conservation laws that have recently played a key role in the low-regularity well-posedness theory for such equations. Third, the project will address several problems related to threshold behaviors for solutions to NLS models with broken symmetries, including the inhomogeneous NLS and the NLS with external potentials. In addition to classifying the possible solution dynamics at the sharp scattering threshold, the project will involve the construction of solutions with traveling wave behavior for models that lack a nonlinear ground state. Finally, the project seeks to increase participation from underrepresented groups in mathematics by fostering student interest in STEM subjects, beginning at the high school level, as well as developing a supportive community for mathematics students at both the undergraduate and graduate level. Specific steps towards this goal include outreach to public high schools, the organization of regular meetings and presentations for undergraduate math majors, the supervision of undergraduate research, and the continued organization of research seminars and invitation of research visitors.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.