Approximate solutions of the Kuramoto–Sivashinsky equation for periodic boundary value problems and chaos

An approach to find approximate solutions of the Kuramoto–Sivashinsky (KS) equation for boundary value problems (BVP) is developed. Attention is focused to periodic boundary value problems. This approach is used to find the approximate solutions for stress-free and rigid boundary conditions. In the first case, it is shown that the spatial pattern of the solutions fluctuates chaotically for small times. But it becomes asymptotically regular. The time-averaged solutions are also regular. In contrast, the solution for rigid boundary conditions exhibit robust chaos.