Geometric Criteria for (In)Stability

许多科学现象在某种程度上涉及永久性结构，如波、模式或稳定的流体流动。这些结构存在于整个科学领域，特别是在诸如非线性光学、海洋和大气流体流动以及生物学等关键技术领域。因此，对这类结构稳定性的评估已成为应用数学研究的关键部分。缺乏用于确定维度大于1的空间中的结构稳定性的技术，例如，包括我们所生活的物理三维空间。这种方法论的缺乏严重阻碍了数学技术在许多应用领域的使用，该奖项下的研究人员和合作者的工作将解决这一重要问题。该奖项下的工作重点将是基于系统特定状态的内在特征的稳定性标准。研究人员设想的方法依赖于这种联系的长期历史，包括解的节点性质或测地线的共轭点与它们的莫尔斯指数之间的直接关系。潜在的想法植根于动力系统，因此几乎只存在于一个空间维度。这项工作的目标是为多维领域建立一个可行的理论。该方法将通过引入域扫描技术将多维问题置于动态系统的背景下，该技术用收缩域的边界来参数化域。Maslov指数在区域和解几何以及线性化算子的谱之间起着关键的桥梁作用。在这种多维的背景下，无限维的表述是必要的，这将得到进一步的发展和适应具体问题的条件。特别关注的是玻色-爱因斯坦凝聚体和流体力学的问题。将寻求将关于底层结构的信息与动力学方程的线性化的谱相关联的结果。应用于玻色-爱因斯坦凝聚体的一个目标将是理解局域涡旋和背景光学晶格的相互作用如何导致不稳定性。在二维流体流动中，拉格朗日动力学构型与点谱引起的不稳定性之间存在着一定的关系。

Many scientific phenomena involve, in some way, permanent structures such as waves, patterns or steady fluid flow. These structures occur throughout science, in particular in such key technological areas as nonlinear optics, oceanic and atmospheric fluid flow as well as biology. Assessments of the stability of such structures have thus become a key part of applied mathematical investigations. There is a dearth of techniques for determining the stability of structures in spaces of dimension greater than one, including, for instance, the physical three-dimensional space in which we live. This lack of methodology has severely hampered the use of mathematical techniques in many application areas and the work of the investigator and collaborators under this award will address this important issue.The focus of the work under this award will be on criteria for the stability of particular states of a system that are based on intrinsic features of that state. The approach envisioned by the investigator rests on a long history of such connections, including the direct relationships between nodal properties of solutions, or conjugate points of geodesics, and their Morse indices. The underlying ideas are rooted in dynamical systems and thus have been almost exclusively in one space dimension. The goal of this work is to build a viable theory for multi-dimensional domains. The approach will place the multi-dimensional problem in a dynamical systems context by introducing a domain sweeping technique that parametrizes the domain with the boundaries of shrinking domains. A key bridging role between the domain and solution geometry, on the one hand, and the spectrum of the linearized operator on the other, is played by the Maslov index. In this multi- dimensional context aninfinite-dimensional formulation is necessary and this will be further developed and adapted to the conditions of the specific problems. A particular focus will be on problems from Bose-Einstein condensates and fluid mechanics. Sought will be results that relate information about the underlying structure to the spectrum of the linearization of the dynamic equation. A goal in the application to Bose-Einstein condensates will be to understand how the interaction of localized vortices and background optical lattices leads to instabilities. In the 2D fluid flows, the relation is anticipated to be between the configuration of the Lagrangian dynamics and the instabilities due to point spectrum.