Global Instability, Limiting-State Selection and Nonlinear Pattern Formation

Abstract

Interfacial pattern formation in phase transition and crystal growth and material science is one of the most important subjects in the broad field of nonlinear science. This subject involves the concepts and issues, include the basic states, global stability, limiting-state selection, quantization conditions of eigenvalues, scaling law and free boundary problems of dynamic system far away from the equilibrium state. This talk attempts to explore these issues through the two prototype problems: (1). dendritic growth from melt; (2). disc-like crystal growth from melt. These problems are highly challenging fundamental problems in condensed matter physics and material science, which have preoccupied many investigators from various areas of science, including applied mathematics for a long period of time. We shall summarize the major results achieved in terms of a unified systematic asymptotic approach during the last decade. For the case of dendritic growth, these results described the wave-characteristics of interface evolution and led to the so-called interfacial wave (IFW) theory.