Geodesic
Whitham hierarchy in growth problems
Teoretičeskaâ i matematičeskaâ fizika, Tome 142 (2005) no. 2, pp. 197-217 Cet article a éte moissonné depuis la source Math-Net.Ru

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We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows distinguishing a class of exact solutions of the Laplacian growth problem in the multiply connected case. These solutions correspond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type, which are solvable by the generalized hodograph method.
Mots-clés : Saffman–Taylor problem
Keywords: Laplacian growth, Whitham equations, Schwarz function. 
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A. V. Zabrodin. Whitham hierarchy in growth problems. Teoretičeskaâ i matematičeskaâ fizika, Tome 142 (2005) no. 2, pp. 197-217. http://geodesic.mathdoc.fr/item/TMF_2005_142_2_a1/

[1] D. Bensimon, L. P. Kadanoff, S. Liang, B. I. Shraiman, C. Tang, Rev. Mod. Phys., 58 (1986), 977–999 | DOI

[2] S. Richardson, J. Fluid Mech., 56 (1972), 609–618 ; Eur. J. Appl. Math., 5 (1994), 97–122 ; Phil. Trans. R Soc. London A, 354 (1996), 2513–2553 ; Eur. J. Appl. Math., 12 (2001), 571–599 ; П. И. Этингоф, ДАН СССР, 313 (1990), 42–47 | DOI | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | MR | Zbl | MR

[3] S. Howison, Eur. J. Appl. Math., 3 (1992), 209–224 | DOI | MR | Zbl

[4] A. N. Varchenko, P. I. Etingof, Pochemu granitsa krugloi kapli prevraschaetsya v inversnyi obraz ellipsa?, Nauka, M., 1995 | MR | Zbl

[5] L. A. Galin, DAN SSSR, 47 (1945), 250–253 | MR

[6] B. Shraiman, D. Bensimon, Phys. Rev. A, 30 (1984), 2840–2842 | DOI | MR

[7] M. Mineev-Weinstein, S. P. Dawson, Phys. Rev. E, 50 (1994), R24–R27 | DOI

[8] M. Mineev-Weinstein, P. Wiegmann, A. Zabrodin, Phys. Rev. Lett., 84 (2000), 5106–5109 | DOI

[9] I. Krichever, M. Mineev-Weinstein, P. Wiegmann, A. Zabrodin, Laplacian growth and Whitham equations soliton theory, E-print nlin.SI/0311005

[10] I. M. Krichever, Funkts. analiz i ego prilozh., 22:3 (1988), 37–52 ; УМН, 44:2 (1989), 121–184 | MR | Zbl | MR | Zbl

[11] I. Krichever, Commun. Pure. Appl. Math., 47 (1992), 437–476 | DOI | MR

[12] S. P. Tsarev, DAN SSSR, 282 (1985), 534–537 | MR | Zbl

[13] M. Shiffer, D. Spenser, Funktsionaly na konechnykh rimanovykh poverkhnostyakh, IL, M., 1957

[14] J. Hadamard, Mém. présentés par divers savants à l'Acad. sci., 33 (1908) ; П. Леви, Конкретные проблемы функционального анализа, ИЛ, М., 1960 | Zbl | MR

[15] G. B. Whitham, Linear and nonlinear waves, Wiley, New York, 1974 ; H. Flashka, M. Forest, D. McLaughlin, Commun. Pure Appl. Math., 33 (1980), 739–784 | MR | Zbl | DOI | MR

[16] P. J. Davis, The Schwarz function and its applications, The Carus Math. Monographs, 17, The Math. Assotiation of America, Washington, 1974 | MR

[17] B. Gustafsson, Acta Appl. Math., 1 (1983), 209–240 | DOI | MR | Zbl

[18] R. Teodorescu, E. Bettelheim, O. Agam, A. Zabrodin, P. Wiegmann, Normal random matrix ensemble as a growth problem, E-print hep-th/0401165 | MR

[19] V. Kazakov, A. Marshakov, J. Phys. A, 36 (2003), 4629–4640 | DOI | MR