Abstract:
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.
