Kardar-Parisi-Zhang equation in the weak noise limit: pattern formation and upper critical dimension

We extend the previously developed nonperturbative weak noise scheme, applied to the noisy Burgers equation in one dimension, to the Kardar-Parisi-Zhang equation for a growing interface in arbitrary dimensions. By means of the Cole-Hopf transformation we show that the growth morphology can be interpreted in terms of dynamically evolving textures of localized growth modes with superimposed diffusive modes. In the Cole-Hopf representation the growth modes are static solutions to the diffusion equation and the nonlinear Schrödinger equation, subsequently boosted to finite velocity by a Galilei transformation. We discuss the dynamics of the pattern formation and, briefly, the superimposed linear modes. Implementing the stochastic interpretation we discuss kinetic transitions and in particular the preliminary scaling properties pertaining to the pair mode or dipole sector. In the dipole sector we obtain the Hurst exponent H=(3-d)/(4-d) or dynamic exponent Zdip(4-d)/(3-d) for the random walk of growth modes. Below d=3 the dipole growth modes show anomalous diffusion, above d=3 the dipole growth modes freeze. Finally, applying Derrick's theorem based on constrained minimization we show that the upper critical dimension is d=4 in the sense that growth modes cease to exist above this dimension.