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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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References

Alikakos, N., Bates, P. W. & Fusco, G. 1991 Slow motion for the Cahn–Hilliard equation in one space dimension. J. Differ. Equations 90, 81–135.CrossRefGoogle Scholar
Alikakos, N. & McKinney, W. 1990 Remarks on the equilibrium theory for the Cahn—Hilliard equation in one space dimension. In Reaction–Diffusion Equations (eds. Brown, K. J. & Lacey, A. A.), Oxford Science Publications, pp. 75–93.CrossRefGoogle Scholar
Allen, S. & Cahn, J. 1979 Microscopic theory of anti-phase boundary motion and its application to anti-phase domain coarsening. Acta. Metall. 27, 1084–1095.Google Scholar
Barles, G., Bronsard, L. & Souganidis, P. E. 1992 Front propagation for reaction-diffusion equations of bistable type. Ann. Henri. Poinc. Ann. Non-Lineaire 9 (5), 479–496.Google Scholar
Brenan, K., Campbell, S. & Petzold, L. 1989 Numerical Solution of Initial-Value Problems in Differential Algebraic Equations. North-Holland.Google Scholar
Brown, D. 1985 A numerical method for singular perturbation problems with turning points. In Numerical Boundary Value Problems for ODE'S (eds. Ascher, U. & Russel, B.), Birkhaeuser, pp. 193–206.CrossRefGoogle Scholar
Carr, J., Gurtin, M. & Slemrod, M. 1984 Structural phase transitions on a finite interval. Arch. Rat. Mechanics and Analysis 86, 317–351.CrossRefGoogle Scholar
Carr, J. & Pego, R. 1989a Slow Phase Separation in One Dimension; Lecture Notes in Physics vol. 344 (eds. Rasele, M. et al. ), Springer-Verlag, pp. 216–226.Google Scholar
Carr, J. & Pego, R. 1989b Metastable patterns in solutions of Comm. Pure Appl. Math. 42, 523–576.Google Scholar
Doedel, E. 1986 Auto User Manual: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. Applied Mathematics Report, California Institute of Technology.Google Scholar
Elliot, C. M. & French, D. A. 1987 Numerical studies of the Cahn–Hilliard equation for phase separation. IMA J. Appl. Math. 38, 97–128.CrossRefGoogle Scholar
Elphick, C., Meron, E. & Spiegel, E. 1990 Patterns of propagating pulses. SIAM J. Appl. Math. 50 (2), 490–503.CrossRefGoogle Scholar
Fusco, G. & Hale, J. 1989 Slow motion manifold, dormant instability and singular perturbation. J. Dynamics and Diff. Equations 1, 75–94.CrossRefGoogle Scholar
Kath, W., Knessl, C. & Matkowsky, B. 1987 A variational approach to nonlinear singularly perturbed boundary value problems. Stud. Appl. Math. 77, 61–88.Google Scholar
Kreiss, G. & Kreiss, H. 1986 Convergence to steady state of solutions of Burgers equation. Appl. Num. Math. 2, 161–179.CrossRefGoogle Scholar
Lange, C. 1983 On spurious solutions of singular perturbation problems. Stud. Appl. Math. 68, 227–257.CrossRefGoogle Scholar
Neu, J. Unpublished Notes.Google Scholar
Ohta, T. & Mimura, M. 1990 Pattern dynamics in excitable reaction-diffusion media. In Formation, Dynamics and Statistics of Patterns, Vol. 1 (eds. Kawasaki, K. et al. ), World Scientific, 55–112.CrossRefGoogle Scholar
O'malley, R. Jr 1976 Phase-plane solutions to some singular perturbation problems. J. Math. Anal. and Appl. 54, 449–466.CrossRefGoogle Scholar
Reyna, L. & Ward, M. J. 1993 On the exponentially slow motion of a viscous shock. Comm. Pure Appl. Math. (Accepted.)Google Scholar
Reyna, L. & Ward, M. J. 1994 Metastable internal layer dynamics for the viscous Cahn–Hilliard equation. Method & Appl. of Anal. (Submitted.)Google Scholar
Rosenblat, S. & Szeto, R. 1980 Multiple solutions of nonlinear boundary value problems. Stud. Appl. Math. 63, 99–117.Google Scholar
Rubinstein, J. & Sternberg, P. 1992 Nonlocal reaction-diffusion equations and nucleation. IMA J. Appl. Math. 48, 249–264.CrossRefGoogle Scholar
Ward, M. J. 1992 Eliminating indeterminacy in singularly perturbed boundary value problems with translation invariant potentials. Stud. Appl. Math. 87 (2), 95–135.Google Scholar
Ward, J. M. 1994 Metastable patterns, layer collapses, and coarsening for a one-dimensional Ginzburg-Landau equation. Stud. Appl. Math. 91 (1), 51–93.CrossRefGoogle Scholar