Generalized needle solutions, interfacial instabilities, and pattern formation

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Abstract
During the past several years, the study of interfacial instability and pattern formation phenomena has preoccupied many researchers in the broad area of nonlinear science. These phenomena occur in a variety of dynamical systems, far from equilibrium, especially in some practically very important physical systems, always displaying some fascinating patterns at the interface between solid and liquid or liquid and another liquid. A prototype of these phenomena is dendrite growth in solidification. It is now well recognized that this phenomenon is induced by some global interfacial instabilities involved in the systems. In the present article, we shall consider the generalized needle crystal solution and its stability properties. In terms of the unified asymptotic approach developed in the previous papers, two different types of global instability mechanisms have been identified: (1) the global trapped wave instability, and (2) the instability caused by perturbations with zero frequency that we call the null-f instability. It connects the so-called microscopic solvability condition theory. On the basis of these results, a solution to the selection problem for pattern formation is clarified. © 1996 The American Physical Society.