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Discrete Painlevé equations and their appearance in quantum gravity. (English) Zbl 0742.35047
Summary: We discuss an algorithmic approach for both deriving discrete analogues of Painlevé equations as well as using such equations to characterize “similarity” reductions of spatially discrete integrable evolution equations. As a concrete example we show that a discrete analogue of Painlevé I can be used to characterize “similarity” solutions of the Kac-Moerbeke equation. It turns out that these similarity solutions also satisfy a special case of Painlevé IV equation. In addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions. As an example we use the WKB method in the presence of two turning points of the third order to parametrize (at the continuous limit) the solution of Painlevé I in terms of the solution of discrete Painlevé I. Finally we show that these results are useful for investigating the partition function of the matrix model in 2D quantum gravity associated with the measure exp[-t 1 z 2 -t 2 z 4 -t 3 z 6 ].

MSC:
35Q40PDEs in connection with quantum mechanics
35Q53KdV-like (Korteweg-de Vries) equations
References:
[1]Painlevé, P.: Bull. Soc. Math. Fr.28, 214 (1900); Acta Math.25, 1 (1902); Gambier, B.: Acta Math.33, 1 (1909)
[2]Ince, E. L.: Ordinary Differential Equations, (1927). New York: Dover 1956
[3]Barouch, E., McCoy, B. M., Wu, T. T.: Phys. Rev. Lett.31, 1409 (1973); Wu, T. T., McCoy, B. M., Tracy, C. A., Barouch, E.: Phys. Rev.B13, 316 (1976) · doi:10.1103/PhysRevLett.31.1409
[4]Jimbo, M., Miwa, T., Mori, Y., Sato, M.: PhysicaD1, 80 (1980); Jimbo, M., Miwa, T.: Proc. Jpn. Acad.A56, 405 (1980) · Zbl 1194.82007 · doi:10.1016/0167-2789(80)90006-8
[5]Ablowitz, M. J., Segur, H.: Phys. Rev. Lett.38, 1103 (1977) · doi:10.1103/PhysRevLett.38.1103
[6]Flaschka, H., Newell, A. C.: Commun. Math. Phys.76, 67 (1980) · Zbl 0439.34005 · doi:10.1007/BF01197110
[7]Ueno, K.: Proc. Jpn. Acad.A56, 97 (1980); Jimbo, M., Miwa, T., Ueno, K.: PhysicaD2, 306 (1981); Jimbo, M., Miwa, T.: PhysicaD2, 407 (1981);4D, 47 (1981); Jimbo, M.: Prog. Theor. Phys.61, 359 (1979) · Zbl 0487.34004 · doi:10.3792/pjaa.56.97
[8]Fokas, A. S., Xin Zhou,: On the Solvability of Painlevé II and IV, Clarkson University, preprint INS #148, January 1990
[9]Brézin, E., Kazakov, V.: Phys. Lett.B236, 144 (1990) · doi:10.1016/0370-2693(90)90818-Q
[10]Douglas, M., Shenker, S.: Strings in Less than One Dimension, Rutgers preprint RU-89-34
[11]Gross, D., Migdal, A.: Phys. Rev. Lett.64, 127 (1990) · Zbl 1050.81610 · doi:10.1103/PhysRevLett.64.127
[12]Gross, D., Migdal, A.: A Nonperturbative Treatment of Two-Dimensional Quantum Gravity, Princeton preprint PUPT-1159 (1989)
[13]Douglas, M.: Strings in Less than One Dimension and the Generalized KdV Hierachies, Rutgers preprint RU-89-51
[14]Witten, E.: Two Dimensional Gravity and Intersection Theory on Moduli Space, IAS preprint, IASSNS-HEP-90/45
[15]Moore, G.: Geometry of the String Equations, Yale preprint YCTP-P4-90
[16]Moore, G.: Matrix Models of 2D Gravity and Isomonodromic Deformation, YCTP-P17-90, RU-90-53
[17]Kac, M., von Moerbeke, P.: Adv. Math.16, 160–164 (1975) · Zbl 0306.34001 · doi:10.1016/0001-8708(75)90148-6
[18]Fokas, A. S., Manakov, S. V.: Phys. Lett.A150, 369 (1990) · doi:10.1016/0375-9601(90)90233-E
[19]Nijhoff, F. W., Papageorgiou, V. G.: Similarity Reductions of Integrable Lattices and Discrete Analogues of Painlevé II Equation, INS #155, October 1990
[20]McCoy, M., Perk, H. H., Shrock, R. E.: Correlation Functions of the Transverse Ising Chain at the Critical Field for Large Temporal and Spatial Variations, ITP S13-83-5 (1983)
[21]Fokas, A. S., Mugan, U., Ablowitz, M. J.: PhysicaD30, 247 (1988) · Zbl 0649.34006 · doi:10.1016/0167-2789(88)90021-8
[22]Kapaev, A. A.: Asymptotics of Solutions of the Painlevé Equation of the First Kind, Differential Equations,24 (1988) (Russian)
[23]Its, A. R., Kitaev, A. V.: Mathematical Aspects of 2D Quantum Gravity, MPLA,5 (25), 2079 (1990) · Zbl 1020.81848 · doi:10.1142/S0217732390002377
[24]David, F.: Loop Equations and Non-Perturbative Effects in Two-Dimensional Quantum Gravity. MPLA,5 (13), 1019–1029 (1990) · Zbl 1020.81825 · doi:10.1142/S0217732390001141
[25]Brézin, E., Marinari, E., Parisi, G.: A Non-Perturbative Ambiguity Free Solution of a String Model, Roma preprint ROM2F-90-09
[26]Bessis, D., Itzykson, C., Zuber, I. B.: Quantum Field Theory Techniques in Graphical Enumeration. Adv. Appl. Math.1, 109–157 (1980) · Zbl 0453.05035 · doi:10.1016/0196-8858(80)90008-1
[27]Its, A., Novokshenov, V. Yu.: The Isomonodromic Deformation Method in the Theory of Painlevé Equations. Lecture Notes Mathematics, vol.1191. Berlin, Heidelberg, New York: Springer 1986
[28]Kapaev, A. A.: Quasinonlinear Solution of the EquationsP 1 2 , Zap. Sem. LOMI187, 88–110 (1990)
[29]Fokas, A. S., Its, A. R., Kitaev, A. V.: The Isomonodromic Approach in the Theory of Two Dimensional Quantum Gravity. Uspehi Matem. Nauk.45 (6), 135–136 (1990) (Russian)