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

Refs:

Hans C. Fogedby
"Localized growth modes, dynamic textures,
and upper critical dimension for the
Kardar-Parisi-Zhang equation in the weak noise limit"
Phys. Rev. Lett. 94, 195702 (2005)

"Kardar-Parisi-Zhang equation in the
weak noise limit: Pattern formation and upper critical dimension"
Phys. Rev. E 73, 031104 (2006)