Spatial periodic solutions of the extended Cahn-Hilliard equation

The present thesis studies the problem of existence and stability of spatial periodic solutions of the extended Cahn–Hilliard equation. The extended Cahn–Hilliard equation is a well-known model that describes the process of phase transition in diblock copolymer melts and can be derived using the general Landau theory of phase separations together with some approximations for a short range and long range interactions in copolymer subchains [7], [31], [35]. In this thesis we will present studies of the existence of periodic steady states of the extended Cahn–Hilliard equation in a full parameter space. We will analytically describe steady states in the case of weak nonlinearity (solutions are close to trivial) using perturbation theory. Besides single-wave solutions, described by Liu and Goldenfeld in [32], we found regions where two-wave solutions coexist along with the general type solutions. Numerical studies were done to find periodic steady states in general situation without any assumptions regarding parameters. We will also present linear stability analysis of described above steady states for bounded disturbances using Floquet boundary conditions. Stability diagram will be shown and comparison with the results of [32] will be presented.