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

TL;DR

This paper extends the weak noise approach to the KPZ equation in arbitrary dimensions, revealing pattern formation mechanisms, growth mode dynamics, and identifying the upper critical dimension as four.

Contribution

It introduces a generalized weak noise scheme for KPZ in higher dimensions and analyzes the pattern formation and critical dimension using the Cole-Hopf transformation.

Findings

Growth modes are static solutions in the Cole-Hopf representation.

The Hurst exponent for growth mode random walk is (3-d)/(4-d).

The upper critical dimension for the KPZ equation is d=4.

Abstract

We extend the previously developed weak noise scheme, applied to the noisy Burgers equation in 1D, 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 Schroedinger 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 properties in the pair mode or dipole sector. We find the Hurst exponent H=(3-d)/(4-d) for the random walk of growth modes in the dipole sector. 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.