Localized patterns in partial differential equations

The general area of my research concerns the study of localized patterns in partial differential equations. Such patterns arise in many different applications such as cellular aggregation, morphogenesis, non-linear laser optics, chemical reactions and flame propagation. Specific examples include animal skin patterns, vegetation patches in deserts, spiral waves in certain chemical reactions, pulses in optical cavities and hot spots in flame propagation problems.The ongoing program is to elucidate the common underlying mechanisms that initiate and sustain pattern formation. This involves the classification of different types systems and patterns; analysis of transitions from one pattern state to another; identifying pattern locations in space; oscillating patterns; and temporal evolution of patterns.In the past decade, much progress has been made in studying localized patterns in one spatial dimension. By now, many of the one-dimensional patterns are well understood. On the other hand, two and higher dimensional patterns are more difficult to analyse, often requiring novel approaches. Moreover, two dimensions allow for a much richer collection of patterns, and their classification is an ongoing project. It is my intention to concentrate on the methods applicable to two and higher dimensional settings.