The long-wavelength properties of the Kuramoto-Sivashinsky equation are studied in 2+1 dimensions using numerical and analytic techniques. It is shown that this equation is not in the universality class of the Kardar-Parisi-Zhang model. Its roughening exponents are (up to logarithmic corrections) like those of the free-field theory, with dimension 2 being the marginal dimension for roughening. Assuming that the solution has logarithmic corrections, we derive a scaling relation for the exponents of the logarithmic terms. This solution is consistent order by order with the Dyson-Wyld diagrams. We explain why previous renormalization-group treatments failed.

• Received 3 December 1991

DOI:https://doi.org/10.1103/PhysRevA.46.3220