Scaling properties of generalized two-dimensional Kuramoto-Sivashinsky equations

TL;DR

This study investigates the scaling behavior of two-dimensional generalized Kuramoto-Sivashinsky equations, revealing power-law surface spectra and scale-free variations, which enhance understanding of complex pattern formation in nonlinear systems.

Contribution

It provides numerical analysis of the 2D generalized KSE, highlighting the scale-invariant properties and spectral exponents influenced by system parameters.

Findings

Surface roughness exhibits power-law dependence on system size.

Surface patterns show both cellular and scale-free long-range variations.

Spectral exponent varies with the equation parameter, indicating different scaling regimes.

Abstract

This paper presents numerical results for the two-dimensional isotropic Kuramoto-Sivashinsky equation (KSE) with an additional nonlinear term and a single independent parameter. Surfaces generated by this equation exhibit a certain dependence of the average saturated roughness on the system size that indicates power-law shape of the surface spectrum for small wave numbers. This leads to a conclusion that although cellular surface patterns of definite scale dominate in the range of short distances, there are also scale-free long-range height variations present in the large systems. The dependence of the spectral exponent on the equation parameter gives some insight into the scaling behavior for large systems.