13th International Conference on the Exact Renormalization Group 2026 (ERG2026)

Scaling regimes of the 1D Kuramoto-Sivashinsky equation

4 Sept 2026, 14:00

20m

Large Lecture Theatre (Jubilee Building)

Talk Parallel Session

Speaker

Liubov Gosteva (UGA)

Description

The Kuramoto-Sivashinsky (KS) equation describes various phenomena that exhibit instabilities, such as flames and reaction-diffusion systems. It resembles another famous non-equilibrium stochastic model - the Kardar-Parisi-Zhang (KPZ) equation, the main difference being the negative "viscosity" in the KS equation.
It is known that the KS equation falls into the KPZ universality class, that is, at large length and time scales its statistical properties are those of the KPZ fixed point, with dynamical exponent $z=3/2$. At small scales and short times, another regime with $z=2$ was observed, which corresponds to the Edwards-Wilkinson UV fixed point of the KPZ equation.
In this work, we show, using FRG, that a third regime emerges at small scales and large times, which corresponds to the Inviscid Burgers fixed point of the KPZ equation with $z=1$ [1]. In the KPZ equation, this regime is observed if one tunes the bare viscosity to a very small value. In the KS equation, this regime is intrinsic and induced by the dynamics: the effective viscosity must cross zero when flowing from a negative bare value to a positive KPZ value at large scales.
Our analysis is based on the 2-point correlation function of the KS equation, that we calculate in a wide range of momenta and frequencies, which allows us to identify the regions where these three scaling regimes occur. This is done with the help of the "two grids" method, which was developed in the context of the KPZ equation [2].

[1] L. Gosteva, D. Roy, N. Wschebor, L. Canet, "Inviscid scaling in the Kuramoto-Sivashinsky equation from functional renormalization group and direct numerical simulations", to be submitted (2026).
[2] L. Gosteva, N. Wschebor, L. Canet, "Unveiling the different scaling regimes of the one-dimensional Kardar–Parisi–Zhang–Burgers equation using the functional renormalisation group", Journal of Statistical Mechanics: Theory and Experiment 2025, 114002 (2025).