During the past several decades, the study to interfacial instability and pattern formation phenomena has preoccupied many researchers in the board field of nonlinear science. These phenomena occur in a variety of dynamical systems far from equilibrium, especially, in some practically very important physical systems. They always display some fascinating patterns at the interface between solid and liquid, or liquid and another kind liquid. Two prototypes of these phenomena are dendrite growth in solidification and viscous fingering in a Hele-Shaw cell. In the late 1980’s and early 1990’s, two important instability mechanisms were consecutively discovered. (1) The instability caused by perturbations with zero frequency that we call the null-f instability. It connects the Microscopic Solvability Condition (MSC) theory. (2) The global trapped wave (GTW) instability, which connects the Interfacial Wave (IW) Theory. The MSC theory and the interfacial wave theory adopted different approaches and derived different results. Since they appeared, a profound controversy around the selection problem and instability has arisen in the science community. In the present article, it will be shown that these two theories, which in some way appeared to conflict to each other, can be reconciled. Dendrite growth system with anisotropy of surface tension are subject to both the null-f instability and the GTW instability. These two mechanisms can be derived by using the unified approach developed in the interfacial wave theory; when the anisotropy tends to zero, the null-f instability disappears, but the GTW mechanism remains.
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