This is a proposal to study the formation and stability properties of spatially localized states in one and two spatial dimensions. Such states can be viewed in terms of homoclinic or heteroclinic connections in phase space, with one spatial coordinate playing the role of time. In many cases the presence of such states is related to a phemenon called homoclinic snaking. In this proposal we shall investigate different mechanisms leading to the formation of spatially localized structures and their stability with respect to both one and two-dimensional perturbations. The study will employ the techniques of spatial dynamics for reversible systems coupled with detailed numerics, and will focus on several fourth order partial differential equations of importance in the physical sciences. The theory will be extended to systems in which reversibility is weakly broken.
This proposal focuses on the origin and properties of spatially localized structures in partial differential equations of importance in the physical sciences. Such structures are observed in experiments in optics, ferrofluids in a uniform magnetic field and in convection, and take the form of isolated bumps in an otherwise uniform background, or a front connecting two different uniform states. Spatially localized oscillations, called oscillons, are observed in vibrating granular media and in chemical reactions. Techniques will be developed to explain the origin of these states, and their stability properties. The theory will be extended to study the formation of related states in systems placed on a slight incline.
