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.

Figure 1

The radially symmetric bound states of the NLSE are depicted for $k=k_0=1$ in $d=1$, $d=2$, $d=3$, and $d=3.5$.

Figure 2

Three-dimensional plot of a 4-mode height profile with modes at $(\pm 20,\pm 20)$ and charges 4.0, $-2.5$, $-3.0$, and 1.5 (in units of $1/k_0$). The core radius is set to $ϵ=10$.