Abstract
This is a report on recent work that examines the behaviour of a class of nonlinear partial differential equations which axe considered to provide a good qualitative model of significant aspects of pattern formation and defects in a diverse range of physical systems. This work was done in collaboration with C. Bowman, R. Indik, A. C. Newell at the University of Arizona and with T. Passot at the Observatoire de Nice. The details of the formal and numerical results mentioned in this introduction will appear in [15] and details of the analytical results mentioned in the last section will appear in [9].
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References
V.I. Arnold, S.M. Gusein-Zade, and A.N. Varchenko. Singularities of Differentiable Mappings Volume 1. Birkhäuser, 1985.
F.T. Arrechi, S. Boccaletti, G. Giacomelli, P.L. Ramazza, and S. Residori. Patterns, space-time chaos, and topological defects in nonlinear optics. Physica D, 61: 25–39, 1992.
F. Bethuel, H. Brezis, and F. Hélein. Ginzburg-Landau Vortices. Birkhäuser, 675 Massachusetts Avenue, Cambridge, MA 02139, 1994.
F.H. Busse. Rep. Prog. Phys., 41: 1929–1967, 1978.
R.E. Caflisch, N.M. Ercolani, T.Y. Hou, and Y. Landis. Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems. Commun. Pure Appld. Math., 46: 453–499, 1993.
P. Collet and J.P. Eckmann. Instabilities and Fronts in Extended Systems. Princeton University Press, Princeton, NJ, 1990.
M. C. Cross and P. C. Hohenberg. Pattern formation outside of equilibrium. Reviews of Modern Physics, 65 (3): 851–1112, 1993.
M. C. Cross and A. C. Newell. Convection patterns in large aspect ratio systems. Physica D, 10: 299, 1984.
N.M. Ercolani, R. Indik, A.C. Newell, and T. Passot. The geometry of the phase diffusion equation, in preparation.
L.C. Evans. Partial Differential Equations. Berkeley Mathematics Lecture Notes, Dept. of Mathematics, Univ. of California, Berkeley, CA 94720, 1994.
J.B. Geddes. Patterns and nonlinear optics. Thesis; University of Arizona Program in Applied Mathematics, 1994.
S. Kondo and R. Asai. A reaction-diffusion wave on the skin of the marine angelfish pomacanthus. Nature, 376: 765–768, 1995.
E.H. Lieb and M. Loss. Symmetry of the ginzburg-landau minimizer in a disc. Mathematical Research Letters, 1:701 - 715, 1994.
R. Neubecker, G.-L. Oppo, B. Thuering, and T. Tschudi. Pattern formation in a liquid-crystal light valve with feedback, including polarization, saturation, and internal threshold effects. Physical Revieu A, 52, 1995.
A.C. Newell, T. Passot, C. Bowman, N.M. Ercolani, and R. Indik. Defects are weak and self-dual solutions of the cross-newell phase diffusion equation for natural patterns. Physica D, 97, 1996.
Q. Ouyang and H.L. Swinney. Transition to chemical turbulence. Chaos, 1: 413, 1991.
T. Passot and A.C. Newell. Physica D, 74: 301–352, 1994.
M. Seul, L.R. Monar, L. O’Gorman, and R. Wolf. Morphology and local structure in labyrinthine strip domain phase. Science, 254: 1557–1696, 1991.
G. B. Whitham. Linear and Nonlinear Waves. Wiley Interscience. 1974.
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Ercolani, N.M. (1997). Singularities and Defects in Patterns Far from Threshold. In: Alber, M., Hu, B., Rosenthal, J. (eds) Current and Future Directions in Applied Mathematics. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2012-1_15
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DOI: https://doi.org/10.1007/978-1-4612-2012-1_15
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