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Morphological change of self-affine surfaces to paraboloids with cusps during growing processes.

A generalization of the continuum model (Kardar-Parisi-Zhang equation) for growth of self-affine surfaces is considered to take nonlocal effect into account in the simplest way, which leads to unstable growth. A dynamic renormalization-group calculation reveals crossover phenomena with characteristic lengths and times, below both of which the power law remains meaningful. The numerical solutions for (1+1)-dimensional systems show that the morphology of profiles changes to global paraboloid segments joined together by singular cusps after the crossover time.