UC Berkeley

This dissertation studies nonlinear partial differential equations (PDEs) describing pattern formation, using a combination of analytical and numerical techniques. A major part focuses on spatially localized states in

the 1:1 forced complex Ginzburg-Landau equation

(FCGLE)

\begin{equation*}

A_t=(\mu +i\nu)A-(1+i\beta )|A|^{2}A +(1+i\alpha )\nabla^2A+\gamma, \quad A=U+iV\in\mathbb{C}, \quad U,V\in\mathbb{R},

\end{equation*}

which is the normal form for a 1:1 resonantly forced Hopf bifurcation in spatially extended systems.

One-dimensional (1D) steady localized states can be fruitfully studied using tools from dynamical systems theory.

%The steady-state ordinary differential equation (ODE) is treated as a dynamical system in the space variable using numerical continuation techniques, and time evolutions of these steady solutions are performed using direct numerical simulation for PDEs.

In particular, the localized states consisting of a Turing pattern embedded in a background equilibrium are shown to grow along its bifurcation curve via a new mechanism called defect-mediated snaking (DMS). In this growth mechanism new rolls are nucleated from the center of the wavetrain, in contrast to standard homoclinic snaking observed in the generalized Swift-Hohenberg equation. The temporal dynamics of localized states outside the snaking region are mediated by successive phase slips resulting from the Eckhaus instability. The spatial dynamics of DMS are explained by an asymptotic theory near a saddle-center bifurcation in a planar reversible map.

Aside from DMS, a new class of steady localized states consisting of phase-winding states, namely spatially periodic states with $U$ and $V$ out of phase, are shown to grow via collapsed snaking mediated by a central pacemaker defect rather than standard homoclinic snaking. Parameter regimes exhibiting localized spatiotemporal chaos (STC) are also identified, and weak STC is interpreted in terms of its underlying coherent structures.

Two-dimensional (2D) spatially localized states that exist in the same parameter regimes as their 1D counterparts include localized ring patterns, planar and circular localized hexagons, as well as 2D localized STC. These solutions are studied in detail using numerical continuation and direct numerical simulations.

The final chapter derives a nonlocal pattern equation for weakly nonlinear Rayleigh-B enard convection with large aspect ratio using the Bogoliubov method coupled with a diagrammatic technique. A Lyapunov functional for the nonlocal pattern equation is found and a necessary condition for finding stable localized states in variational PDEs is formulated.