Dynamics of Partial Differential Equations: Topological Implications for Stability and Analysis in Higher Spatial Dimensions

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Mathematical equations known as dynamical systems and partial differential equations are used to model a wide variety of processes, including ecological, biological, and physical applications. Mathematically predicting solution behavior provides a mechanism for predicting real-world system behavior, and also for predicting how such behavior will change as system parameters are varied. Stable states are attracting, and thus stable states are those that are mostly likely to be observed. Topological properties related to stability are those that are robust under sufficiently nice deformations, and can therefore be used to make predictions about large classes of systems, possibly without reference to the finer details of the equation under study. Although much is mathematically understood about models with only one space dimension, there are many interesting open questions about higher spatial dimensions, which is particularly relevant for applications. This project is focused on developing theoretical tools for predicting the long-time behavior of solutions to these types of equations. This project will also support efforts to train women PhD students and conduct outreach work, such as a week-long summer math camp for high school students. This project is focused on developing theoretical tools for analyzing the dynamics associated with a large class of partial differential equations (PDEs), including reaction-diffusion equations and fourth-order systems such as the Swift-Hohenberg equation, in both one and higher spatial dimensions. Understanding such systems requires predicting not only what types of solutions will exist, but also their stability, which has direct consequences for their observability in the real-world. There are two primary goals: (i) Investigating topological implications for stability in higher-order PDEs and systems of PDEs; and (ii) Analyzing solutions to PDEs in spatial dimensions greater than one. The first goal is connected with the results of classical Sturm-Liouville theory for scalar, second order eigenvalues problems. However, for fundamental reasons that theory cannot be directly extended to higher-order systems, and so truly new methods must be developed. The second goal will be achieved by developing a useful spatial dynamics, which refers to treating a distinguished spatial variable as a time-like evolution variable, for systems in multiple spatial dimensions. This will allow one to apply the tools of dynamical systems theory to understand the associated solutions.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.