|M.Sc Thesis||Department of Applied Mathematics|
|Supervisor:||Prof. Nepomniashchy Alexander|
The convective Cahn-Hilliard model has recently been proposed for the description of the evolution of crystal surfaces, which are unstable because of anisotropic surface tension.
In the simplest case of a two-dimensional crystal, the surface slope is described by the renormalized equation
where , is the growing driving force.
The purpose of our research is the investigation of the solutions of the convective Cahn-Hilliard equation for different values of the driving force , and their stability analysis.
The essential part of our investigation is devoted to the main family of odd stationary periodic solutions
that bifurcates from the trivial solution when
In the case , the main family with finite values of ?was calculated numerically.
The boundaries of the stability intervals for the main family of periodic solutions are obtained numerically for selected values of driving force ?.
The existence of the finite stability interval leads to the conclusion that for the sufficiently large values of the driving force , the evolution of the system does not lead to the minimum of the thermodynamic potential which corresponds to a sole facet, but is stopped by the formation of a periodic system of facets and corners.
During the investigation of stability we obtain the analytical prediction regarding the location of the boundaries of stability intervals in the case of longwave perturbations.
We also investigate non-stationary regimes for selected values of , such as travelling waves for ?and heteroclinic loops for ?and . In addition, the stability analysis of travelling wave solutions is done.
We investigate the dynamics of kink solutions for the values and obtain the equation of motion for the pair of kinks in the limit together with an analytical prediction for the kink velocity that depends exponentially on the distance between kinks .