Metastable Patterns, Layer Collapses, and Coarsening for a One‐Dimensional Ginzburg‐Landau Equation

The internal layer motion associated with the Ginzburg-Landau equation is 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.

[1]  M. Ward.  Eliminating Indeterminacy in Singularly Perturbed Boundary Value Problems with Translation Invariant Potentials , 1992 .

[2]  M. Mimura,et al.  Very slow dynamics for some reaction-diffusion systems of the activator-inhibitor type , 1992 .

[3]  Peter W. Bates,et al.  Slow motion for the Cahn-Hilliard equation in one space dimension , 1991 .

[4]  Robert V. Kohn,et al.  On the slowness of phase boundary motion in one space dimension , 1990 .

[5]  Ehud Meron,et al.  Patterns of propagating pulses , 1990 .

[6]  J. Carr,et al.  Metastable patterns in solutions of ut = ϵ2uxx − f(u) , 1989 .

[7]  J. Keller,et al.  Fast reaction, slow diffusion, and curve shortening , 1989 .

[8]  Spiegel,et al.  Spatiotemporal complexity in traveling patterns. , 1988, Physical review letters.

[9]  P. Coullet,et al.  Nature of spatial chaos. , 1987, Physical review letters.

[10]  C. G. Lange.  On Spurious Solutions of Singular Perturbation Problems , 1983 .

[11]  Takao Ohta,et al.  Kink dynamics in one-dimensional nonlinear systems , 1982 .

[12]  Hiroshi Matano,et al.  Asymptotic Behavior and Stability of Solutions of Semilinear Diffusion Equations , 1979 .

[13]  J. Cahn,et al.  A microscopic theory for antiphase boundary motion and its application to antiphase domain coasening , 1979 .

[14]  R. G. Casten,et al.  Instability results for reaction diffusion equations with Neumann boundary conditions , 1978 .

[15]  R. O'Malley.  Phase-plane solutions to some singular perturbation problems , 1976 .

[16]  Nathaniel Chafee,et al.  Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions , 1975 .

[17]  M. Mimura,et al.  PATTERN DYNAMICS IN EXCITABLE REACTION-DIFFUSION MEDIA , 1990 .

[18]  Jack K. Hale,et al.  Slow-motion manifolds, dormant instability, and singular perturbations , 1989 .

[19]  C. M. Elliott,et al.  Numerical Studies of the Cahn-Hilliard Equation for Phase Separation , 1987 .

[20]  C. Knessl,et al.  A variational approach to nonlinear singularly perturbed boundary-value problems , 1987 .

[21]  E. F. Infante,et al.  A Bifurcation Problem for a Nonlinear Partial Differential Equation of Parabolic Type. , 1974 .