Geometric Criteria for (In)Stability

许多科学现象在某种程度上涉及诸如波浪、图案或稳定的流体流动等永久结构。这些结构出现在整个科学领域，特别是在非线性光学、海洋和大气流体流动以及生物学等关键技术领域。因此，这类结构的稳定性评估已成为应用数学研究的一个关键部分。目前缺乏确定维度大于1的空间中结构稳定性的技术，例如，包括我们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，该奖项下的研究者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该状态的内在特征的系统特定状态的稳定性标准。研究者设想的方法依赖于这种联系的悠久历史，包括解的节点性质或测地线的共轭点与其莫尔斯指数之间的直接关系。潜在的思想植根于动力系统，因此几乎完全在一个空间维度。本工作的目标是为多维领域建立一个可行的理论。该方法通过引入域扫描技术，通过收缩域的边界参数化域，将多维问题置于动态系统环境中。马斯洛夫指数在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，无限维的公式是必要的，这将进一步发展和适应具体问题的条件。特别关注玻色-爱因斯坦凝聚和流体力学的问题。所寻求的结果是将底层结构的信息与动态方程的线性化谱联系起来。应用于玻色-爱因斯坦凝聚体的一个目标将是了解局部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，拉格朗日动力学的构型与由点谱引起的不稳定性之间的关系是预期的。

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.