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Resolving weak internal layer interactions for the Ginzburg–Landau equation

Published online by Cambridge University Press:  26 September 2008

Luis G. Reyna  and
Michael J. Ward
Luis G. Reyna
Affiliation:
Mathematical Sciences, IBM Thomas Watson Research Center, Yorktown Heights, New York 10598, USA
Michael J. Ward
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
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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.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 5 , Issue 4 , December 1994 , pp. 495 - 523
Copyright
Copyright © Cambridge University Press 1994

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