zbMATH — the first resource for mathematics

Pattern propagation in nonlinear dissipative systems. (English) Zbl 0622.76051
The paper, motivated in a large part by a similarity of pattern propagation in a wide class of continuous systems described by nonlinear parabolic equations, is devoted to a generalization of marginal-stability hypothesis that has been suggested in the theory of dendritic crystal growth. Starting with a nonlinear diffusion problem and ”amplitude” equation the authors examine different kinds of propagating solutions. They arrive at a conjecture on the validity of marginal-stability principle also away from the threshold for the pattern-forming models. Finally, three different pattern-forming models (the Hohenberg-Swift- Pomeau-Manneville equation, the reaction diffusion equations and the fourth order nonlinear parabolic equation derived in the theory of crystal growth) are solved numerically to support the hypothesis.
Reviewer: A.Tylikowski

76E15 Absolute and convective instability and stability in hydrodynamic stability
76R99 Diffusion and convection
35K55 Nonlinear parabolic equations
Full Text: DOI
[1] Busse, F.H., Hydrodynamic instabilities and the transition to turbulence, (1981), Springer Berlin · Zbl 0459.76037
[2] Koschmieder, L., Adv. chem. phys., 32, 109, (1975)
[3] Gollub, J.P., ()
[4] Swinney, H.L., Hydrodynamic instabilities and the transition to turbulence, (1981), Springer New York · Zbl 0453.00052
[5] Langer, J.S., Rev. mod. phys., 52, 1, (1980)
[6] Flicker, M.; Ross, J., J. chem. phys., 60, 3458, (1974)
[7] Feinn, D.; Ortoleva, P.; Scalf, W.; Schmidt, S.; Wolff, M., J. chem. phys., 69, 27, (1978)
[8] Fife, P.C., Mathematical aspects of reacting and diffusing systems, (1979), Springer New York · Zbl 0403.92004
[9] Pomeau, Y.; Manneville, P., J. phys. lett., 40, 609, (1979)
[10] Cross, M.C.; Daniels, P.G.; Hohenberg, P.C.; Siggia, E.D., Phys. rev. lett., 45, 898, (1980)
[11] Pomeau, Y.; Manneville, P., J. phys., 42, 1067, (1981)
[12] M.C. Cross and A.C. Newell, preprint.
[13] Newell, A.C., ()
[14] Cross, M.C.; Hohenberg, P.C.; Daniels, P.G.; Siggia, E.D., J. fluid mech., 127, 155, (1983)
[15] Kramer, L.; Ben-Jacob, E.; Brand, H.; Cross, M.C., Phys. rev. lett., 49, 1981, (1982)
[16] Dee, G.; Langer, J.S., Phys. rev. lett., 50, 383, (1983)
[17] E. Ben-Jacob, H.R. Brand and L. Kramer, unpublished.
[18] Ahlers, G.; Cannell, D., Phys. rev. lett., 50, 1583, (1983)
[19] Langer, J.S.; Müller-Krumbhaar, H., Phys. rev. A, 27, 499, (1983)
[20] Fisher, R.A.; Murray, J.D., Lectures on nonlinear-differential-equation models in biology, Ann. eugenics, 7, 355, (1977), Clarendon Oxford, For a more general discussion and other references, see
[21] Kolmagorov, A.; Petrovsky, I.; Piscounov, N., Bull. univ. Moscow, ser. internat., sec. A, 1, 1, (1937)
[22] Aronson, D.G.; Weinberger, H.F., Adv. math., 30, 33, (1978)
[23] Hagan, P.S., Studies in appl. math., 64, 57, (1981)
[24] Swift, J.; Hohenberg, P.C., Phys. rev. A, 15, 319, (1977)
[25] Stokes, A.N., Mathematical biosciences, 31, 307, (1976)
[26] Hadeler, K.P.; Rothe, F.; Magyari, E., J. mathematical biology, J. phys. A, 15, L139, (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.