Symmetries and analytic structure of phase separation on curved manifolds.

摘要

In this thesis, an extensive analytic study of the time-dependent Ginzburg-Landau (TDGL) or Model A equation in a variety of curved, two-dimensional surfaces is carried out, and comparisons are made to the well-known flat space case, using Lie group symmetry analysis and the calculation of the singularity structure of the generic solution to this equation. We show that while symmetry methods in general predict a non-trivial difference between flat and curved space dynamics, the singularity structure approach suggests the dynamics are largely identical, both methods however show little variation of their predictions with respect to parameter values, particularly for explicit curvature coupling. Specifically, the singularity structure is shown to be that of a movable logarithmic branch point for all of the curved manifolds examined, as well as flat space. Numerical simulations using a measure that focuses on phase separation were performed on the equation on flat space, a torus and an elliptic cylinder and they demonstrate that while the symmetry methods do capture significant local differences in the dynamics of phase separation, the singularity structure methodology better reflects the underlying similarity of both flat and curved space phase separation flowing from the deeper unity of all such variations on the TDGL equation, namely, the cubic nonlinearity and second-order parabolic structure. Numerical experiments are also performed on Model A in the case of a fully variable geometry whose dynamics are coupled to those of the order parameter, where the great complexity of the equations renders them largely intractable to analytic techniques. Lastly, the techniques of Painlevé analysis and Lie group theory are also applied to a family of financial models for options based around the Black-Scholes equation.