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

LEAPS-MPS：非线性色散方程的长时间行为

• 批准号: 2350225
• 负责人: Jason Murphy
• 金额: $ 16.97万
• 依托单位: University of Oregon Eugene
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-10-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2350225&HistoricalAwards=false
• 关键词: LEAPS; MPS; Long; time; behavior

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.

该奖项是根据2021年《美国救援计划法》（公法117-2）全部或部分资助的。 非线性分散偏微分方程在许多物理环境中都会出现，其特征是不同频率的波趋于以不同速度行驶。 在这样的模型中，分散效应和非线性效应之间通常会有竞争，从而产生了丰富的解决方案行为。 这些包括腐烂和散射，存在称为孤立波甚至波浪塌陷（或爆炸）的相干结构的存在。 该项目包括考虑与非线性分散方程解决方案的长期行为相关的问题，包括孤立波的稳定性，低规律性解决方案的全球衰减估计以及生活在某些急剧散射阈值或附近的解决方案的行为。 该项目着重于几种物理上有意义但仍然足够简单的特定模型，可以接受深入分析。 这种选择允许在分散方程领域中蒸馏出一些重要问题的基本数学困难。 通过这种方式，拟议的研究有可能为未来的进步铺平道路，甚至超出了该项目中所考虑的特定问题。 该项目包含适合在本科，硕士和博士学位水平的学生参与的问题。 该项目包括几项活动，以鼓励通过向公立学校推出，会议的组织和指导本科研究项目的范围内代表性不足或农村学生参与。 一些主要目标包括为整个稳定的孤立波浪建立渐近稳定性，以及在不稳定政权中构建稳定的歧管。 接下来，该项目将解决针对完全可以集成的模型（包括1D立方NLS）的全球时空估计问题。 该项目旨在开发病毒和摩拉维型型估计，该估计适合于新型的微观保护定律，该法律最近在低调性良好的方程式中发挥了关键作用。第三，该项目将解决与具有断裂对称性的NLS模型解决方案的阈值行为有关的几个问题，包括不均匀的NLS和具有外部电势的NLS。 除了对急剧散射阈值处可能的解决方案动力学进行分类之外，该项目还将涉及用于缺乏非线性基态模型的波动行为的解决方案。 最后，该项目试图通过培养学生对STEM学科的兴趣，从高中级开始，并在本科和研究生层面上为数学学生开发支持社区，从而增加数学中代表性不足的人的参与。实现这一目标的具体步骤包括向公立高中的宣传，定期会议的组织和本科数学专业的演讲，对本科研究的监督，以及持续的研究研讨会的组织以及研究访问者的邀请。这奖反映了NSF的法定任务，并通过该基金会的知识绩效和广泛的影响来通过评估来进行评估，并通过评估值得进行评估。