Local attractors of one of the original versions of the Kuramoto–Sivashinsky equation

We study two rather similar evolutionary partial differential equations. One of them was obtained by Sivashinsky and the other by Kuramoto. The Kuramoto version was taken as the basic version of the equation that became known as the Kuramoto–Sivashinsky equation. We supplement each version of the Kuramoto–Sivashinsky equation with natural boundary conditions and, for the proposed boundary-value problems, study local bifurcations arising near a homogeneous equilibrium when they change stability. The analysis is based on the methods of the theory of dynamical systems with an infinite-dimensional phase space, namely, the methods of integral manifolds and normal forms. For all boundary-value problems, asymptotic formulas are obtained for solutions that form integral manifolds. We also point out boundary conditions under which the dynamics of solutions of the corresponding boundary-value problems of the two versions of the Kuramoto–Sivashinsky equation are significantly different.