Large-scale properties of unstable systems governed by the Kuramoto-Sivashinksi equation

The dynamic renormalization-group method is developed for investigation of correlations generated by the Kuramoto equation with random initial conditions. It is shown that elimination of modes from domain Λ<k<∞ generates the random force and "viscosity" which is positive in the $d=1$ and negative in the d≥2 cases. The stable fixed point is found in the $d=1$ system while the theory is asymptotically free when $d=3$. No fixed point exists in the $d=2$ case which explains the patterns formation obtained from computer simulations.