Evaluation of the time correlation function for the Kuramoto-Sivashinsky equation with a projection operator method

We evaluate the time correlation function for the Kuramoto-Sivashinsky (KS) equation with a projection operator method. For this purpose, the memory term is estimated separately using three methods of approximation: the Markov approximation, and the first- and infinite-order approximations of the memory term. Comparing the results obtained with these approximations and those obtained from numerical simulations, we have found that among the three approximations the first-order approximation yields the best results in the evaluation of the time correlation function.