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Fourier-Bessel analysis of patterns in a circular domain. (English) Zbl 1076.35535
Summary: This paper explores the use of the Fourier-Bessel analysis for characterizing patterns in a circular domain. A set of stable patterns is found to be well-characterized by the Fourier-Bessel functions. Most patterns are dominated by the principal Fourier-Bessel mode \([n,m]\) which has the largest Fourier-Bessel decomposition amplitude when the control parameter \(R\) is close to the corresponding nontrivial root \((\rho_{n,m})\) of the Bessel function. Moreover, when the control parameter is chosen to be close to two or more roots of the Bessel function, the corresponding principal Fourier-Bessel modes compete to dominate the morphology of the patterns.

35Q53 KdV equations (Korteweg-de Vries equations)
33C90 Applications of hypergeometric functions
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
Full Text: DOI
[1] Percec, V.; Ahn, C.-H.; Ungar, G.; Yeardley, D.J.P.; Möller, M.; Sheiko, S.S., Controlling polymer shape through the self-assembly of dendritic side-groups, Nature, 391, 161-164, (1998)
[2] Koneripalli, N.; Singh, N.; Levicky, R.; Bates, F.S.; Gallagher, P.D.; Satija, S.K., Confined block copolymer thin films, Macromoleculars, 28, 2897-2904, (1995)
[3] Bowman, C.; Newell, A.C., Natural patterns and wavelets, Rev. mod. phys., 70, 289-301, (1998)
[4] Cross, M.C.; Hohenberg, P.C., Pattern formation outside of equilibrium, Rev. mod. phys., 65, 851-1112, (1993) · Zbl 1371.37001
[5] H. Furukawa, in: K. Kawasaki, M. Suzuki (Eds.), Formation, Dynamics, and Statistics of Patterns, Vol. 2, World Scientific, Singapore, 1993, pp. 266-308.
[6] Cahn, J.W.; Hilliard, J.E., Free energy of a non-uniform system. I. interfacial free energy, J. chem. phys., 28, 258-267, (1958)
[7] Kuramoto, Y.; Tsuzuki, T., Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. theoret. phys., 55, 356-369, (1976)
[8] J.D. Gunton, M.S. Miguel, P.S. Sahni, The dynamics of first-order phase transitions, in: C. Domb, J.L. Lebowitz (Eds.), Phase Transitions and Critical Phenomena, Vol. 8, Academic Press, London, 1983, pp. 267-482.
[9] S.G. Guan, C.-H. Lai, G.W. Wei, unpublished.
[10] Wei, G.W., Discrete singular convolution for the solution of the fokker – planck equations, J. chem. phys., 110, 8930-8942, (1999)
[11] Y. Meyer, Wavelets and Operators, Cambridge Studies in Advanced Mathematics, Vol. 37, Cambridge University Press, Cambridge, 1992. · Zbl 0776.42019
[12] Daubechies, I., Orthonormal bases of compactly supported wavelets, Commun. pure appl. math., 41, 909-996, (1988) · Zbl 0644.42026
[13] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia, PA, 1992. · Zbl 0776.42018
[14] Mallat, S., Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. am. math. soc., 315, 68-87, (1989)
[15] C.K. Chui, An Introduction to Wavelets, Academic Press, San Diego, CA, 1992. · Zbl 0925.42016
[16] Wei, G.W., Quasi-wavelets and quasi-interpolating wavelets, Chem. phys. lett., 296, 215-222, (1998)
[17] Wei, G.W., Discrete singular convolution for the sine-Gordon equation, Physica D, 137, 247-259, (2000) · Zbl 0944.35087
[18] Rosenblat, S.; Davis, S.H.; Homsy, G.M., Nonlinear Marangoni convection in bounded layers. part 1. circular cylindrical containers, J. fluid mech., 120, 91-122, (1982) · Zbl 0498.76038
[19] G.N. Watson, Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1966. · Zbl 0174.36202
[20] Cahn, J.W.; Hilliard, J.E., Spinodal decomposition—a reprise, Acta metall., 19, 151-161, (1971)
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