Dynamics in pattern-forming systems

We study the time evolution of the initial instability toward the final steady state for two pattern-forming systems: explosive crystallization and the viscous-fingering problem. We show analytically that a scaling solution exists for the power spectrum of the interface shape in the case of explosive crystallization. The scaling exponents are computed exactly. For the viscous-fingering problem, a similar scaling solution is obtained, and the scaling exponents are determined numerically. We also study the robustness of the scaling solutions against external perturbations. We find that local perturbation on the surface tension does not alter the scaling exponents, although it has dramatic influence on the late-stage steady-state patterns.