Geometric Criteria for (In)Stability

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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.