Resolving weak internal layer interactions for the Ginzburg–Landau equation

Abstract

The internal layer behaviour, in one spatial dimension, associated with two classes of Ginzbug–Landau equation with double-well nonlinearities and small diffusivities is investigated. The problems that are examined are the Ginzburg–Landau equation with and without a constant mass constraint. For the constrained problem, steady-state internal layer solutions are constructed using a formal projection method. This method is also used to derive a differential-algebraic system describing the slow dynamics of the constrained internal layer motion. The dynamics of a two-layer evolution is studied in detail. For the unconstrained problem, a nonlinear WKB-type transformation is introduced that magnifies exponentially weak layer interactions and leads to well-conditioned steady problems. A conventional singular perturbation method, without the need for exponential asymptotics, is used on the resulting transformed problem as an alternative method to construct equilibrium solutions and metastable patterns. Exponentially sensitive steady-state internal layer solutions as well as a one-layer evolution are computed accurately using the transformed problem. © 1994, Cambridge University Press. All rights reserved.