Stability of coherent structures in evolutionary partial differential equations: a geometric approach

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Most continuously evolving physical processes can be described by differential equations. Examples include water waves, biological populations, interacting atoms and molecules, and the constituents of a chemical reaction. In any such system a distinguished role is played by steady statessolutions that do not change in time because all external forces are in perfect equilibrium. It is important to know whether such states are stable, in the sense that small perturbations to the initial condition or external forces will eventually fade away, or unstable, meaning the perturbations will be amplified exponentially and lead to radically different behaviour in the long run. For this reason it is typically only the stable states that can be observed in nature, or physically realized in a laboratory setting, so it is important to identify them and understand what properties lead to their stability.The ultimate goal is to predict a state's stability from its general shape and structure, and to determine what properties are indicative of instability. A classical problem describes a signal propagating in one dimension (such as light traveling along an optical fibre, or an electrical impulse in a neuron). In this case it is known that a pulse solution (which looks like a small bump moving along the fibre) is unstable, whereas a front (which is shaped like a cliff or a step) is stable. The difference between the two is that the pulse has a local maximum while the front does not, and this is enough to distinguish stability from instability.When the problem involves multiple spatial dimension (as all real physical systems do), it is much more difficult, and results from the one-dimensional case no longer apply. The proposed research addresses this shortcoming by simultaneously developing two different tools for higher-dimensional problems: 1) the Maslov index; and 2) the Evans function. Both methods are well understood in the one-dimensional context, but have only recently begun to receive attention in a more general setting. Thus the proposed research is likely to have a strong impact on both the mathematical and physical sciences, with theoretical advancements allowing for new applications to problems in fluid dynamics, materials science and nonlinear optics, to name just a few examples.These new theoretical tools will be advanced through the consideration of a large, robust family of physical applications. Student researchers will have the opportunity to communicate with scientists in related disciplines to determine the most important applications of these methods, and guide their efforts accordingly. As a result, the work outlined in the proposal will effectively train these researchers not just as mathematicians, but as active, productive members of the general scientific community, and as such will promote the development of innovative new scientific methods in Canada.