The main theme of the proposed work is on understanding the widespread phenomenon of dynamic scaling in extended physical systems, building on stability analysis of coherent structures such as self-similar solutions, nonlinear wave pulses and vortices. The paradox of dynamic scaling laws is that typically they can be predicted by back of the envelope calculations, yet defy rigorous analysis for years. We will study scaling and stability in a variety of problems of high current interest: the trend to self-similar behavior in Smoluchowski's mean-field model of coagulation; scaling and stability of fingering patterns in viscous fluid flows; models of domain coarsening in phase transitions; existence and stability of solitary waves in infinite-dimensional Hamiltonian systems; and stability of Bose-Einstein condensates with vortices. The work will involve the development of improved methods for stability analysis in nonlinear systems, for the rigorous analysis and numerical computation of self-similar solutions and nonlinear waves, and for physical modeling of step-terrace structures on crystalline surfaces.
The general aim of this work is to improve understanding of large-scale dynamic behavior in complex systems. We aim to develop new and effective mathematical methods and concepts for the modeling and analysis of dynamic behavior in a variety of problems of high scientific interest. We study fundamental mathematical models for the large-scale dynamics of: agglomeration of small particles; mixing patterns in viscous fluid flow; microscopic structure of metallic alloys and semiconductor surfaces; waves on fluid interfaces; and quantum vortices in super-cold gases. Many issues that will be investigated are of fundamental interest in condensed-matter physics, aerosol physics, chemical engineering, and materials science.
