Spontaneous formation of singularities through critical collapse

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This project is devoted to the study of mathematical nature of phenomena of collapse that arise in a variety of biological and physical systems. For example, collapse may occur when a powerful laser beam enters a transparent medium, such as usual glass. Interaction of the medium with the laser beam results in self-focusing when the intense light creates an effective lens inside the medium and amplifies itself during propagation until the singularity of the light amplitude is reached. Mathematically a singularity means that the corresponding mathematical solution approaches infinity. Qualitatively similar collapse phenomena occur in hydrodynamics as well as in the bacterial growth (for example, for E. Coli). In the latter case, communications between bacteria (through a chemical substance called chemoattractant) cause spontaneous aggregation of the bacterial colony to a very high (almost singular) bacterial density. Solutions of the corresponding nonlinear systems of partial differential equations that model phenomena with collapse experience spontaneous formation of singularities in finite time (blow-up). Blow-up is often accompanied by a dramatic contraction of the spatial extent of a solution, hence the term "collapse." Near the singularity point there is a qualitative change in underlying nonlinear phenomena; the initial mathematical models lose their applicability and other mechanisms become more important, such as optical breakdown and formation of plasma in optical media, or bacterial crowding and formation of multicellular organisms from the bacterial colony.This research will focus on phenomena of collapse in the Nonlinear Schrödinger equation (NLSE), Keller-Segel equation (KSE) and Davey-Stewartson equation (DSE) which are archetypal equations to study finite time singularities in the critical dimension two. These equations are among most universal and widespread equations in nonlinear science with numerous applications in nonlinear optics, hydrodynamics and biology. The need in understanding collapse became especially pressing since the advent of lasers in early 60s. More than 50 years of research produced well-established collapse theories for NLSE and KSE. However, until recently it remained a puzzle why these theoretical results were never confirmed by either direct simulations or in experiments. The explanation is that the existing theories required unrealistically large amplitudes for their applicability. This project will develop a detailed theory of collapse scaling and collapse regularization in NLSE and KSE for the realistic amplitudes. The principal investigator will also develop a DSE collapse scaling theory, which has been lacking for many years. The effort will be made to make the theory a practical tool for applications. Both perturbative and nonperturbative approaches will be used, as well as matched asymptotic techniques and extensive supercomputing. Diverse applications of NLSE, KSE and DSE will promote cross-fertilization of ideas across the fields of nonlinear optics, hydrodynamics, Bose-Einstein condensation and biology.