Existence and Energetic Stability of Traveling Waves in the Presence of Symmetry

对称性下行波的存在性和能量稳定性

• 批准号: 1812436
• 负责人: Samuel Walsh
• 金额: $ 18.02万
• 依托单位: University of Missouri-Columbia
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-01 至 2022-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812436&HistoricalAwards=false
• 关键词: Existence; Energetic; Stability; Traveling; Waves

Traveling waves are solutions to time-dependent systems that propagate without changing their shape. They are observed throughout the natural world and include surface and internal waves in the ocean, ignition fronts in combustion theory, and even stripe patterns in animal fur. Through their ubiquity, traveling waves command the interest of researchers in nearly every corner of the physical and biological sciences. This project aims to develop novel tools for constructing large-amplitude waves, especially for the important case of pulses, which are localized disturbances that travel through unbounded domains (such as tsunami waves or rough waves). In a related direction, the project will apply and extend a new theoretical framework for diagnosing stability of traveling waves solutions, i.e., capacity of the waves to persist when subjected to a disturbance. This project will advance the mathematical understanding of traveling waves; the results will have implications to oceanography, fluid mechanics, and combustion theory. More generally, the techniques developed by this study are intended to provide a framework with considerable potential for future applications to a broad array of disciplines. Graduate students will be trained and actively involved in this research. This project aims to advance the mathematical understanding of traveling waves along two parallel tracks: existence theory and stability theory. The first set of activities concern the existence of large-amplitude traveling waves on unbounded domains. While many tools currently exist for constructing small-amplitude waves in a neighborhood of known explicit solutions, the non-perturbative regime is far less well understood. This is particularly true for problems set on unbounded domains, for which issues of compactness seriously frustrate what tools are available. This project will develop a global bifurcation theoretic machinery designed to overcome these obstructions using symmetry and monotonicity properties. This new framework will then be used to address important open problems in a variety of systems. Specifically, these applications include (i) construct large-amplitude bore solutions to a two-phase fluid system in a channel; (ii) prove the existence of large-amplitude traveling waves evolving according to general non-compact symmetry groups, for example scroll ring solutions to reaction-diffusion equations; and (iii) extend beyond the appearance of internal stagnation points a family of large-amplitude solitary stratified water waves constructed in earlier work. A second set of projects will develop new systematic tools for diagnosing the orbital stability or instability of traveling solutions to abstract Hamiltonian systems that possess symmetries. This work aims to relax key hypotheses in existing theory to allow more direct application to highly nonlinear systems like those governing water waves. Using this machinery, it is planned to (i) prove the instability of internal waves in a two-phase system confined to a channel and (ii) give a new systematic proof of the stability and instability of Korteweg?de Vries solitons. Moreover, a local well-posedness theory in the Hadamard sense for the water wave problem with a point vortex will be provided.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.

行进波是时间依赖性系统的解决方案，它们在不改变形状的情况下传播。它们在整个自然世界中都被观察到，包括海洋中的表面和内部波，燃烧理论中的点火锋，甚至是动物毛皮中的条纹模式。通过无处不在，旅行浪使研究人员在物理和生物科学的几乎每个角落都有利益。该项目旨在开发用于构建大振幅波的新颖工具，尤其是对于重要的脉冲情况，脉冲是局部的干扰，这些干扰穿过无界的域（例如海啸波或粗波波）。在相关的方向上，该项目将应用并扩展一个新的理论框架，以诊断行驶波解决方案的稳定性，即在受到干扰时持续存在的波的能力。该项目将提高对行进浪潮的数学理解。结果将对海洋学，流体力学和燃烧理论有影响。更一般而言，这项研究开发的技术旨在为将来在各种各样的学科中应用提供巨大潜力的框架。研究生将接受培训并积极参与这项研究。 该项目旨在沿着两个平行的轨道：存在理论和稳定理论来推进对行进波的数学理解。第一组活动涉及在无限域中存在大振幅行进波。 尽管目前存在许多用于在已知的显式解决方案附近构建小振幅波的工具，但非扰动制度的理解程度不高。 对于在无限域中设定的问题尤其如此，紧凑性问题严重挫败了哪些工具可用。 该项目将开发全球分叉理论机制，旨在使用对称性和单调性能克服这些障碍物。 然后，该新框架将用于解决各种系统中的重要开放问题。 具体而言，这些应用包括（i）在通道中的两相流体系统中构建大振幅孔解决方案； （ii）证明存在根据一般的非紧密对称基团的大振幅行进波的存在，例如滚动环溶液到反应扩散方程； （iii）延伸超出了内部停滞的出现点，一个在早期工作中构建的大振幅孤立的水波家族。第二组项目将开发新的系统工具，用于诊断出具有对称性的术语途径的轨道稳定性或不稳定性。这项工作旨在放宽现有理论中的关键假设，以使在高度非线性系统（如管理水波）中更直接应用。 使用这种机械，计划（i）证明局限于通道的两相系统中内部波的不稳定性，（ii）给出了Korteweg？de vries solitons的稳定性和不稳定的新系统证明。此外，将提供与Hadamard意义上的当地体系良好的理论有关水波问题，并提供点涡流。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响审查标准，被认为值得通过评估来获得支持。

项目成果

On the Stability of Solitary Water Waves with a Point Vortex

具有点涡的孤立水波的稳定性

• DOI: 10.1002/cpa.21891
• 发表时间: 2020
• 期刊: Communications on Pure and Applied Mathematics
• 影响因子: 3
• 作者: Varholm, Kristoffer;Wahlén, Erik;Walsh, Samuel
• 通讯作者: Walsh, Samuel

On the Existence and Instability of Solitary Water Waves with a Finite Dipole

有限偶极子孤立水波的存在性和不稳定性

• DOI: 10.1137/18m1231638
• 发表时间: 2019
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Le, Hung

Orbital Stability of Internal Waves

内波的轨道稳定性

• DOI: 10.1007/s00220-022-04332-x
• 发表时间: 2022
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Chen, Robin Ming;Walsh, Samuel
• 通讯作者: Walsh, Samuel

Large-Amplitude Solitary Waves in Two-Layer Density Stratified Water

两层密度分层水中的大振幅孤立波

• DOI: 10.1137/20m1383537
• 发表时间: 2021
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Sinambela, Daniel

Broadening Global Families of Anti-Plane Shear Equilibria

• DOI: 10.1137/21m1392838
• 发表时间: 2021-01
• 期刊: SIAM J. Math. Anal.
• 作者: T. Hogancamp