Back
 APM  Vol.4 No.8 , August 2014
Weierstrass’ Elliptic Function Solution to the Autonomous Limit of the String Equation of Type (2,5)*
Abstract: In this article, we study the string equation of type (2,5), which is derived from 2D gravity theory or the string theory. We consider the equation as a 4th order analogue of the first Painlevé equation, take the autonomous limit, and solve it concretely by use of the Weierstrass’ elliptic function.
Cite this paper: Sasaki, Y. (2014) Weierstrass’ Elliptic Function Solution to the Autonomous Limit of the String Equation of Type (2,5)*. Advances in Pure Mathematics, 4, 494-497. doi: 10.4236/apm.2014.48055.
References

[1]   Douglas, M.R. (1990) String in Less than One-Dimensions and K-dV Hierarchies. Physics Letters B, 238, 176-180.
http://dx.doi.org/10.1016/0370-2693(90)91716-O

[2]   Moore, G. (1990) Geometry of the String Equations. Communications in Mathematical Physics, 133, 261-304.
http://dx.doi.org/10.1007/BF02097368

[3]   Moore, G. (1991) Matrix Models of 2D Gravity and Isomonodromic Deformations. Progress of Theoretical Physics Supplement, 102, 255-285. http://dx.doi.org/10.1143/PTPS.102.255

[4]   Fukuma, M., Kawai, H. and Nakayama, R. (1991) Infinite Dimensional Grassmannian Structure of Two Dimensional String Theory. Communications in Mathematical Physics, 143, 371-403.
http://dx.doi.org/10.1007/BF02099014

[5]   Kac, V. and Schwarz, A. (1991) Geometric Interpretation of Partition Functions of 2D Gravity. Physics Letters B, 257, 329-334. http://dx.doi.org/10.1016/0370-2693(91)91901-7

[6]   Schwarz, A. (1991) On Solutions to the String Equations. Modern Physics Letters A, 29, 2713-2725.
http://dx.doi.org/10.1142/S0217732391003171

[7]   Adler, M. and van Moerbeke, P. (1992) A Matrix Integral Solution to Two-Dimensional Wp-Gravity. Communications in Mathematical Physics, 147, 25-26. http://dx.doi.org/10.1007/BF02099527

[8]   van Moerbeke, P. (1994) Integrable Foudations of String Theory. In: Babelon, O., et al., Ed., Lectures on Integrable Systems, World Science Publisher, Singapore, 163-267.

[9]   Takasaki, K. (2007) Hamiltonian Structure of PI Hierarchy. SIGMA, 3, 42-116.

[10]   Ince, E.L. (1956) Ordinary Differential Equations. Dover Publications, New York.

[11]   Conte, R. and Mussette, M. (2008) The Painlevé Handbook. Springer Science + Business Media B.V., Dordrecht.

[12]   Weiss, J. (1984) On Classes of Integrable Systems and the Painlevé Property. Journal of Mathematical Physics, 25, 13-24. http://dx.doi.org/10.1063/1.526009

[13]   Kudryashov, N.A. (1997) The First and Second Painlevé Equations of Higher Order and Some Relations between Them. Physics Letters A, 224, 353-360. http://dx.doi.org/10.1016/S0375-9601(96)00795-5

[14]   Gromak, V.I., Laine, I. and Shimomura, S. (2002) Painlevé Differential Equations in the Complex Plane. Walter de Gruyter, Berlin. http://dx.doi.org/10.1515/9783110198096

[15]   Shimomura, S. (2004) Poles and α-Points of Meromorphic Solutions of the First Painlevé Hierarchy. Publications of the Research Institute for Mathematical Sciences, Kyoto University, 40, 471-485.
http://dx.doi.org/10.2977/prims/1145475811

[16]   Kimura, H. (1989) The Degeneration of the Two Dimensional Garnier System and the Polynomial Hamiltonian Structure. Annali di Matematica Pura ed Applicata, 155, 25-74.
http://dx.doi.org/10.1007/BF01765933

[17]   Suzuki, M. (2006) Spaces of Initial Conditions of Garnier System and Its Degenerate Systems in Two Variables. Journal of the Mathematical Society of Japan, 58, 1079-1117.
http://dx.doi.org/10.2969/jmsj/1179759538

[18]   Shimomura, S. (2000) Painlevé Property of a Degenerate Garnier System of (9/2)-Type and a Certain Fourth Order Non-Linear Ordinary Differential Equation. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 29, 1-17.

 
 
Top