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Metastable Patterns, Layer Collapses, and Coarsening for a One‐Dimensional Ginzburg‐Landau Equation
Author(s) -
Ward Michael J.
Publication year - 1994
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm199491151
Subject(s) - metastability , mathematics , neumann boundary condition , nonlinear system , asymptotic analysis , boundary value problem , asymptotic expansion , mathematical analysis , boundary layer , dirichlet boundary condition , boundary (topology) , motion (physics) , limit (mathematics) , equations of motion , periodic boundary conditions , physics , statistical physics , classical mechanics , mechanics , quantum mechanics
The internal layer motion associated with the Ginzburg‐Landau equationis analyzed, in the limit , for various boundary conditions on x = ±1. The nonlinearity Q ( u ) results either from a double‐well potential or a periodic potential, each having wells of equal depth. Using a systematic asymptotic method, some previous work in deriving equations of motion for the internal layers corresponding to metastable patterns is extended. The effect of the various types of boundary conditions and nonlinearities will be highlighted. A dynamical rescaling method is used to numerically integrate these equations of motion. Using formal asymptotic methods, certain canonical problems describing layer collapse events are formulated and solved numerically. A hybrid asymptotic‐numerical method, which incorporates these layer collapse events, is used to give a complete quantitative description of the coarsening process associated with the Ginzburg‐Landau equation. For the Neumann problem with a double‐well potential, the qualitative description of the coarsening process given in Carr and Pego [4] will be confirmed quantitatively. In other cases, such as for a periodic potential with Dirichlet boundary conditions, it will be shown that, through layer collapse events, a metastable pattern can tend to a stable equilibrium solution with an internal layer structure.