# The one-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: limit distributions

## TL;DR

This paper demonstrates that the one-dimensional Kuramoto-Sivashinsky (KS) PDE exhibits the same universal limit distributions as the KPZ equation in a nonequilibrium steady state, confirmed through extensive numerical simulations.

## Contribution

It establishes that the 1D KS PDE belongs to the KPZ universality class by numerically showing identical limit distributions in a steady state.

## Key findings

- KS height fluctuations follow Tracy-Widom and Baik-Rains distributions
- Numerical simulations confirm universality class membership
- Statistical properties are consistent across different initial conditions

## Abstract

Tracy-Widom and Baik-Rains distributions appear as universal limit distributions for height fluctuations in the one-dimensional Kardar-Parisi-Zhang (KPZ) \textit{stochastic} partial differential equation (PDE). We obtain the same universal distributions in the spatiotemporally chaotic, nonequilibrium, but statistically steady state (NESS) of the one-dimensional Kuramoto-Sivashinsky (KS) \textit{deterministic} PDE, by carrying out extensive pseudospectral direct numerical simulations to obtain the spatiotemporal evolution of the KS height profile $h(x,t)$ for different initial conditions. We establish, therefore, that the statistical properties of the 1D KS PDE in this state are in the 1D KPZ universality class.