|Ph.D Thesis||Department of Applied Mathematics|
|Supervisor:||Prof. Nepomniashchy Alexander|
The pattern-forming systems have been a subject of extensive research during several decades. A significant progress has been achieved in studying universal amplitude equations (specifically, the complex Ginzburg-Landau equation) governing the pattern formation. However, the control of patterns excited in a subcritical way is still hardly investigated.
We consider the subcritical complex Ginzburg-Landau equation under the action of global nonlinear feedback control.
In the limit of large dispersion and nonlinear frequency shift, the exact periodic solutions are investigated both in the focusing and defocusing cases. A new class of oscillatory periodic solutions, revealed by a direct numerical simulation of complex Ginzburg-Landau equation, is studied in the focusing case. It is shown that a global feedback control can extend existence and stability regions of the stationary solutions. The analytical results are justified by direct simulations of original equation.
Solitary-wave solutions of the complex Ginzburg-Landau equation with the global feedback control are studied in a wide region of parameters using an approximate approach. The simple shape of solitary-wave solutions allows us to construct a simplified low-dimensional model of ordinary differential equations based on a modified variational principle. The derived finite-dimensional model is used for studying the action of instantaneous feedback control and feedback control with a delay.
Next, we apply a modified variational principle to a subcritical Ginzburg-Landau equation coupled to a diffusion equation for a Goldstone mode. We study the stability of a traveling-wave solution in the framework of the obtained low-dimensional model.
The validity of low-dimensional models is verified by direct numerical simulations.