|M.Sc Thesis||Department of Applied Mathematics|
|Supervisor:||Prof. Nepomniashchy Alexander|
In this thesis, we consider a nonlinear two-dimensional heat conduction model for the dynamics of explosive crystallization of amorphous films in the case when some thermophysical properties of the amorphous and crystalline phases are not equal. The basic solution is found and its existence region is investigated. We provide a linear stability analysis of this solution which describes the propagation with a constant velocity of a straight front separating the crystalline and amorphous phases. It is shown that under certain conditions an oscillatory instability can occur, which is caused by disturbances corresponding to periodically varying front velocity and temperature. For the two-dimensional case and some ranges of the parameters these disturbances generate also a wavy shape of the front.
Also we provide a nonlinear analysis of the problem and derive nonlocally coupled complex Ginzburg-Landau equations for the amplitudes of the unstable modes. The equations are written in characteristic variables and involve averaged terms. The plane wave solutions of the amplitude equations are found. These solutions correspond either to traveling waves or to quasiperiodic wave, depending on parameters. The regions of stability for these solutions are determined. We find that traveling and quasiperiodic waves are never stable simultaneously.