Traveling waves are solutions of partial differential equations that preserve their shape while moving in a preferred direction. Traveling waves are basic coherent structures in partial differential equations and they often serve as building blocks for complex patterns. This project is about traveling waves in parabolic and partly parabolic systems (in partly parabolic systems some quantities diffuse and others do not). More precisely, this project is focused on the investigation of three phenomenologically different manifestations of instability of traveling waves: (1) If a system supports waves that move with different speeds, what kind of instability can cause a transition from one wave to another? In such systems of coupled equations, will the preferred wave speed be defined by linear or nonlinear dynamics? (2) How to analytically prove or predict that instability is convective? For a convectively unstable wave small perturbations are moving away from the interface of the wave faster than they grow. (3) An instability sometimes manifests itself in the appearance of new structures. These can be patterns in a neighborhood of an asymptotic rest state of the wave, or new global structures such as modulated waves which are composite waves that consist of a wave and periodic patterns. Modulated waves are known to exist in parabolic systems. A component of this project is devoted to the study of modulated waves in partly parabolic systems.
Traveling waves are abundant in nature and human activities. They arise in applied problems from different fields: optical communication, combustion theory, biomathematics, chemistry, population dynamics, to name a few. Information about whether traveling waves exist, how many of them are there in the system, and their resilience under perturbations assists in understanding of complex phenomena. Proposed work will help to identify situations when multiple waves exist and when the transition between them can be anticipated and controlled. Traveling waves that are observed in real life are usually stable (resilient under small perturbations). Stable waves withstand inhomogeneity of the medium that carries them and perturbations that may be caused by a variety of factors. Scientists and engineers who study traveling waves are interested to know in what parameter regimes waves are stable, but it is equally important to know what to expect from an unstable wave. The proposed work will allow detailed and rigorous conclusions to be drawn about how different perturbations to a wave behave and what the outcome of an instability is.
