Research will be conducted on microscopically realistic models of interfacial patterns in non-eqilibrium condensed matter systems. These models should be sufficiently accurate so as to allow for quantitative tests of current theoretical concepts. These concepts include the "solvability" theory for ordered structures, bifurcation theoretic analysis of secondary instabilities in cellular patterns, mean field methods for "ensemble-averaged" Diffusion Limited Aggregation and ideas regarding the emergence of braod-band spectra in non-linear systems via self-organized criticality." These ideas must be combined with specific and detailed models if they are to yield quantitative predictions regarding pattern forming behavior. In the course of this work, studies will be made of the solidification of monolayers, where the reduced dimensionality enables one to perform exact shape calculations and compare to general schemes for bifurcation of cellular states; a new mathematical model will be developed for the recent stream instability experiment which exhibits a transition to broad-band dynamics; solvability theory for free dendritic growth will be extended to the realistic case of non- axisymmetric crystals with cubic surface tension anisotropy so as to be able to directly compare with experiment; and, mean field Diffusion Limited Aggregation calculations will be done on a variety of geometries and dimensions so as to ascertain the generality and usefulness of this new approach. %%% A variety of problems in the exciting new field of nonlinear science which impact condensed matter physics and materials research will be studied. The problems focus on models of the nonlinear dynamics of interfacial patterns which are realistic and can, thus, be compared with experimental results.
