|M.Sc Student||Glasman Evgeniya|
|Subject||Instabilities and Defects in Wavy Patterns|
|Department||Department of Applied Mathematics||Supervisors||Professor Emeritus Alexander Nepomniashchy|
|Mr. A. Golovin|
In this thesis, we consider a system of complex Ginzburg-Landau equation and Burgers equation. This model, which governs instabilities of the combustion front with a sequential reaction, is a typical example of a pattern-forming system with a Goldstone mode, caused by the translational symmetry. In the longwave limit we derive a system of two coupled Burgers equations, which describes the nonlinear evolution of longwave disturbances of the combustion front.
We show that the coupled system reveals several new types of instabilities, which are studied both analytically and numerically. In some limiting cases, we derive secondary amplitude equations governing these types of instabilities.
We find a class of exact solutions, which correspond to domain walls between two traveling waves with opposite wave-numbers.
Furthermore, a more general family of domain walls has been simulated numerically, by means of a finite-difference numerical code.
We discuss the dynamics of both line defects (domain walls) and point defects (spiral waves). We study also the motion and interaction of spiral waves. We have found numerically for the system of complex Ginzburg - Landau equation and Burgers equation the region of existence of bound states of spiral waves, by means of a pseudospectral code.