Proof of scale invariant solutions in the Kardar-Parisi-Zhang and Kuramoto-Sivashinsky equations in 1+1 dimensions: analytical and numerical results

Abstract

Under the assumption that the Kardar-Parisi-Zhang model (KPZ) possesses scale invariant solutions, there exists an exact calculation of the dynamic scaling exponent z=3/2. The authors prove that both KPZ and the related Kuramoto-Sivashinsky model indeed possess scale invariant solutions in 1+1 dimensions which are in fact the same for both models. The proof entails an examination of the higher order diagrams in the perturbation theory in terms of the dressed Green function and the correlator. Although each higher order diagram contains logarithmic divergences, endangering the existence of the scale invariant solution, the authors show that these divergences cancel in each order. The proof uses a fluctuation-dissipation theorem (FDT), which is an exact result for KPZ in 1+1 dimensions.