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 $\Lambda <k<\mathrm{}\infty$ generates the random force and "viscosity" which is positive in the $d=1\mathrm{}$ and negative in the $d\ge 2$ cases. The stable fixed point is found in the $d=1\mathrm{}$ system while the theory is asymptotically free when $d=3\mathrm{}$. No fixed point exists in the $d=2\mathrm{}$ case which explains the patterns formation obtained from computer simulations.