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

行波是时变系统的解，它在不改变其形状的情况下传播。它们在自然界中随处可见，包括海洋的表面和内部波浪，燃烧理论中的点火锋，甚至动物皮毛上的条纹图案。由于行波无处不在，它引起了物理和生物科学各个领域研究人员的兴趣。该项目旨在开发用于构建大振幅波的新工具，特别是对于脉冲的重要情况，脉冲是通过无界域（如海啸波或巨浪）传播的局部扰动。在一个相关的方向上，该项目将应用和扩展一个新的理论框架来诊断行波解的稳定性，即波在受到干扰时持续存在的能力。这个项目将推进对行波的数学理解；研究结果将对海洋学、流体力学和燃烧理论产生影响。更一般地说，本研究开发的技术旨在为未来广泛学科的应用提供一个具有相当大潜力的框架。研究生将接受培训并积极参与这项研究。该项目旨在推进沿两条平行轨道的行波的数学理解：存在理论和稳定性理论。第一组活动涉及无界域上大振幅行波的存在性。虽然目前存在许多工具用于在已知显式解的邻域中构造小振幅波，但对非摄动状态的理解远远不够。对于设置在无界域上的问题尤其如此，对于无界域，紧凑性问题严重阻碍了可用的工具。该项目将开发一个全局分岔理论机制，旨在利用对称性和单调性来克服这些障碍。然后，这个新框架将用于解决各种系统中的重要开放问题。具体来说，这些应用包括：(i)构建通道中两相流体系统的大振幅孔解；（ii）证明了根据一般非紧对称群演化的大振幅行波的存在性，例如反应扩散方程的涡旋环解；并且（iii）在早期工作中构建的一系列大振幅孤立分层水波延伸到内部停滞点的外观之外。第二组项目将开发新的系统工具，用于诊断具有对称性的抽象哈密顿系统的行进解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，以便更直接地应用于高度非线性系统，如控制水波的系统。使用这种机器，计划(i)证明限制在通道内的两相系统中的内波的不稳定性，（ii）给出Korteweg？德弗里斯孤子。此外，还给出了具有点涡的水波问题的Hadamard意义上的局部适定性理论。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。

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.