The steady states of one-dimensional Sivashinsky equations
Authors:
A. Novick-Cohen and L. A. Peletier
Journal:
Quart. Appl. Math. 50 (1992), 759-777
MSC:
Primary 35G30; Secondary 35B35, 35K35, 80A22
DOI:
https://doi.org/10.1090/qam/1193665
MathSciNet review:
MR1193665
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Abstract: Phase plane analysis is used to calculate the number of steady states for two equations which arise in the context of directional solidification: the Sivashinsky equation and the modified Sivashinsky equation.
K. Brattkus and S. H. Davis, Cellular growth near absolute stability, Phys. Rev. B38, 11452–11460 (1988)
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L. H. Ungar and R. H. Brown, Cellular interface morphologies in directional solidification. IV. The formation of deep cells, Phys. Rev. B31, 5931–5940 (1985)
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K. Brattkus and S. H. Davis, Cellular growth near absolute stability, Phys. Rev. B38, 11452–11460 (1988)
J. Carr, M. E. Gurtin, and M. Slemrod, Structural phase transitions on a finite interval, Arch. Rational Mech. Anal. 86, 317–351 (1984)
J. W. Cahn and J. H. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28, 258–267 (1958)
C. M. Elliot and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal. 96, 339–357 (1986)
J. M. Hyman, A. Novick-Cohen, and P. Rosenau, A modified asymptotic approach to modeling a dilute binary alloy, Phys. Rev. B37, 7603–7608 (1988)
T. Kawahara and S. Toh, An approximate equation for nonlinear cross-field instability, Phys. Lett. A113, 21–24 (1985)
Y. Kuramoto, Phase dynamics of weakly unstable periodic structures, Progr. Theor. Phys. 71, 1182–1196 (1984)
G. B. McFadden, R. F. Boisvert, and S. R. Coriell, Nonplanar interface morphologies during directional solidification of a binary alloy. II. Three dimensional computations, J. Crys. Growth 84, 371–388 (1987)
A. Novick-Cohen, Instability of periodic states for the Sivashinsky equation, Quart. Appl. Math. 48, 217–224 (1990)
A. Novick-Cohen, Stability properties of the modified Sivashinsky equation, private communication
A. Novick-Cohen, On Cahn-Hilliard type equations, Nonlinear Anal. TMA 15, 797–814 (1990)
A. Novick-Cohen, Instability in solidification fronts of dilute binary alloys: the Kuramoto-Sivashinsky equation, Physica 26D, 403–410 (1987)
D. S. Riley and S. H. Davis, Long-wave morphological instabilities in the directional solidification of a dilute binary alloy, SIAM J. Appl. Math. 50, 420–436 (1990)
G. I. Sivashinsky, On cellular instability in the solidification of a dilute binary alloy, Physica 8D, 243–248 (1983)
L. H. Ungar and R. H. Brown, Cellular interface morphologies in directional solidification. IV. The formation of deep cells, Phys. Rev. B31, 5931–5940 (1985)
J. D. Weeks and W. van Saarloos, Directional solidification cells with groove for small partition coefficient, preprint
S. Zheng, Asymptotic behavior of solutions to the Cahn-Hilliard equation, Appl. Anal. 23, 165–184 (1986)
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Article copyright:
© Copyright 1992
American Mathematical Society