This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
Fronts are traveling waves that asymptotically connect two equilibrium states of a system. This research project is concerned with the analysis of the stability of fronts. The mechanism of the instability of fronts is defined by the competition between the rate of growth of perturbations and the rate of their transport. In the convective regime perturbations to the front are transported away faster than they grow and thus decay pointwise. Critical information about nonlinear stability of the front is contained in the spectrum of the linearization of the system about the wave, but in many cases the spectral information is not definitive. That happens, for example, when the continuous spectrum of the linearized operator crosses the imaginary axis. Standard bifurcation theory techniques then typically fail. Reducing the domain for the system to some weighted spaces often works on the linear level, but there are serious issues related to the proof of nonlinear stability in the weighted spaces. One of the goals of this project is to develop general criteria for the convective nature of instability for classes of applied problems. An instability caused by the continuous spectrum can also manifest itself in the appearance of new local or global structures. Another goal of this project is to investigate whether the instability caused by the continuous spectrum may be the key point in the explanation of phenomena that are characterized by a sudden transition from one coherent structure to another.
Fronts arise in a variety of applied problems from different fields: optical communication, combustion theory, biomathematics (calcium waves in tissue, nerve conduction, population dynamics), chemistry, ecology, to name a few, therefore their stability is of a great interest. For many models the stability of a front in a full nonlinear equation cannot be simply inferred from the properties of its linear approximation. This project is focused on finding criteria for the convective (or transient) nature of the instability in such cases and investigating the mechanism of the transition between drastically different regimes within the same system, such as a sudden transition from a slow process to a much faster one. Capturing this phenomenon analytically will assist in predicting when the transition happens and exploring ways to control it. Progress in this direction will be of importance for applications in combustion theory, ecology, and biomathematics. The techniques of the analysis will be based on the relation between the geometric structure of the wave and its stability.
