UC Berkeley

This dissertation focuses on the study of spatially modulated structures in pattern forming systems. The work is motivated by recent interest in spatially localized states observed in convective systems. Weakly nonlinear analysis is applied to derive the modulation equations and systematic studies, both analytical and numerical, are then performed on the simplified equations. The following is a summary of this work:

Weakly Subcritical Patterns

The transition from subcritical to supercritical periodic patterns is described by the one-dimensional cubic-quintic Ginzburg-Landau equation with cubic nonlinear gradient terms. The coefficients are real indicating that the system is spatially reversible. Properties of the equation such as well-posedness, gradient structure, and bifurcation behavior depend significantly on the coefficients of the cubic nonlinear gradient terms. In this system, periodic patterns may in turn become unstable through one of two different mechanisms, an Eckhaus instability or an oscillatory instability. Dynamics and bifurcations near the instability thresholds are analyzed. Among the stationary solutions, the front solution which connects the zero state to a spatially periodic state plays the most important role. The location of the front in the parameter $\mu$ is treated as a Maxwell point. The spatially modulated solutions which bifurcate from the periodic solutions demonstrate protosnaking behavior near this point. These results are used to shed light on both variational and non-variational systems exhibiting homoclinic snaking.

Localized Patterns in Rotating Convection and Magnetoconvection

In two-dimensional rotating convection and magnetoconvection, the formation of spatially localized patterns is strongly affected by the interaction between convection and a large scale mode: zonal velocity in rotating convection and magnetic flux in magnetoconvection. A nonlocal fifth order Ginzburg-Landau theory is developed to describe the localization near a codimension-two point. The study of the fifth-order Ginzburg-Landau theory gives us better understanding of the appearance of spatially modulated solutions and their subsequent bifurcation behavior. These results are used to explain the properties of spatially localized convectons in the full convection problem. The effect of boundary conditions is also analyzed which shows how the bifurcation picture is modified in the presence of mixed boundary conditions.

Exact Solutions of the Cubic-Quintic Swift-Hohenberg Equation

Meromorphic exact solutions of the cubic-quintic Swift-Hohenberg equation are studied and a one-parameter family of real exact solutions is derived. The solutions are of two types, differing in their symmetry properties, and are connected via an exact heteroclinic solution. These exact solutions are used as initial conditions for numerical continuation which shows that some of these lie on secondary branches while others fall on isolas. The approach substantially enhances our understanding of the solution space of this equation.