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

行波是在不改变形状的情况下传播的依赖时间的系统的解。它们在整个自然界都可以观察到，包括海洋中的表面波和内波，燃烧理论中的点火前沿，甚至动物毛皮中的条纹图案。由于行波无处不在，几乎引起了物理和生物科学各个角落的研究人员的兴趣。该项目旨在开发新的工具来构造大幅度波，特别是对于脉冲这一重要情况，脉冲是通过无界区域(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，即行波在受到扰动时的持续能力。该项目将促进对行波的数学理解，其结果将对海洋学、流体力学和燃烧理论产生影响。更广泛地说，这项研究开发的技术旨在提供一个具有相当大潜力的框架，将来应用于广泛的学科。研究生将接受培训，并积极参与这项研究。本项目旨在促进对沿两条平行轨道行波的数学理解：存在理论和稳定性理论。第一组活动涉及无界区域上大振幅行波的存在。虽然目前有许多工具可以在已知显式解的邻域内构造小幅度波，但对非微扰机制的了解要少得多。对于设置在无界域上的问题尤其如此，对于这些问题，紧凑性问题严重阻碍了可用的工具。这个项目将开发一种全局分叉理论机制，旨在利用对称性和单调性来克服这些障碍。然后，这个新的框架将被用来解决各种系统中的重要公开问题。具体地说，这些应用包括：(I)构造渠道中两相流体系统的大振幅钻孔解；(Ii)证明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；以及(Iii)将早期工作中构造的一族大振幅孤立分层水波推广到内部停滞点之外。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旅行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的系统，如那些控制水波的系统。利用这一机制，计划(I)证明受限于通道的两相系统中内波的不稳定性，以及(Ii)给出新的系统地证明Korteweg？de Vries孤子的稳定性和不稳定性。此外，对于点涡流的水波问题，将提供Hadamard意义下的局部适定性理论。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.