Localized Growth Modes, Dynamic Textures, and Upper Critical Dimension for the Kardar-Parisi-Zhang Equation in the Weak-Noise Limit

A weak-noise scheme is applied to the Kardar-Parisi-Zhang equation for a growing interface in all dimensions. It is shown that the solutions can be interpreted in terms of a growth morphology of a dynamically evolving texture of localized growth modes with superimposed diffusive modes. By applying Derrick's theorem, it is conjectured that the upper critical dimension is four.