Spatially localized structures in dissipative systems: open problems

Stationary spatially localized structures, sometimes called dissipative solitons, arise in many interesting and important applications, including buckling of slender structures under compression, nonlinear optics, fluid flow, surface catalysis, neurobiology and many more. The recent resurgence in interest in these structures has led to significant advances in our understanding of the origin and properties of these states, and these in turn suggest new questions, both general and system-specific. This paper surveys these results focusing on open problems, both mathematical and computational, as well as on new applications.