Dynamic scaling behavior in models of aggregation and clustering. Numerous complex systems exhibit clumping behaviors that are observed to scale in well-defined ways that are rather poorly explained by current theory. The plan is to study dynamic scaling limits in coagulation models that (a) are fundamentally related to branching processes and genealogy, (b) describe phase separation, mixing continuous growth and clustering, (c) model interstellar smoke, and (d) relate to repair mechanisms in chromosomes. A key goal is to develop renormalization-group ideas that strengthen methods previously used to study scaling-symmetric systems. Following the remarkable recent discovery of Menon concerning complete integrability for random shock clustering in scalar conservation laws, work will try to identify and understand important families of solutions, and extend the theory to handle more general models of ballistic aggregation.
Existence and stability theory for solitary waves in fluids and lattices. Solitary waves are one of the earliest and most prominent kinds of coherent structures to be studied in nonlinear science. Recent work has provided a proof of linear stability for solitary water waves of small amplitude, and has opened up new avenues for investigation. Stability is explained using spatially weighted norms that measure the scattering of infinitesimal perturbations away from the nonlinear wave. The proposed work is directed toward issues for which the use of weighted norms promises to produce progress, especially: i) existence of solitary wave fronts in periodic media, including multi-dimensional particle lattices and localized water waves with periodic bathymetry; and ii) stability of multidimensional soliton fronts for the Kadomtsev-Petviashvili model of three-dimensional water waves.
