Weierstrass’ Elliptic Function Solution to the Autonomous Limit of the String Equation of Type (2,5)*

Yoshikatsu Sasaki

Department of Mathematics, Hiroshima University, Higashi-Hiroshima, Japan.

DOI: 10.4236/apm.2014.48055                

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.

Keywords

Painlevé Hierarchy, String Equation, Elliptic Function

1. Introduction

1.1. The String Equation of Type (2,5)

Put. Consider the commutator equation of ordinary differential operators

We call it the string equation (or Douglas equation) of type, which appears in the string theory or the theory of quantum gravity in 2D [1] -[9] . In the followings, we set,.

In the case where, , the string equation is written as an ODE satisfied by the potential w of Sturm-Liouville operator, and then, by a fractional linear transformation, it is reduced to the first Painlevé equation [10] [11]

, (PI)

which is equivalent to the Hamiltonian system:

In the case where, , yields

,

where, , are integral constants. By the fractional linear transformation, ,

and putting, the string equation is reduced to

, (S)

We also call it the string equation of type (2,5). Note that (S) coincides the 4th order equation of the first Painlevé hierarchy [12] -[15]

, (2nPI)

for, where is an expression of a given meromorphic function w defined by and.

1.2. Degenerated Garnier System

Equation (S) is also obtained as follows. Consider a 2D degenerated Garnier system [16] [17] :

(dG9/2)

which is a 2D analogue of (PI) in the theory of isomonodromic deformations. If we fix one of the independent variables, we get a Hamiltonian system with only one independent variable as follows:

From the above system, eliminating, , and putting, we obtain (S). So, Equation (S) is 4th order analogue of (PI) in the double sences.

It is already known by Shimomura [18] that every solution to (S) is meromorphic on, and that every pole of every solution is double one with its residue 0.

1.3. Autonomous Limit of the First Painlevé Equation

The first Painlevé equation (PI) has the autonomous limit [11] . Replacing by with a constant, and taking limit, we obtain which is solved by the Weierstrass’ elliptic function [10] [11] . The relation between the fundamental 2-form before and after the replacement is

.

1.4. Results

It is quite natural to think that:

Conjecture. Each equation of the first Painlevé hierarchy has the autonomous limit, and which is satisfied by the Weierstrass’ elliptic function.

For, the statement is valid, i.e.

Theorem A. Replacing by, or replacing by with a constant, and taking limit, we obtain the autonomous limit of the 4th order equation of the first Painlevé hierarchy (S). Moreover, the relation between the fundamental 2-form before and after the replacement is

.

It is easy to show the above. The autonomous limit is given by

, (A)

Theorem B. The autonomous limit Equation (A) has a solution concretely described by the Weierstrass’ elliptic function as

,

where.

Remark. Modulus of the elliptic function is determined by the constants a and b. g₂ and g₃ in the elliptic function theory are as follows:

.

The next section is devoted to give the proof of Theorem B.

2. Proof of Theorem B

Put, i.e.

. (1)

Multiplying both sides of by, and integrating it, we obtain a first integral of (A)

(2)

In order to find the elliptic function solution, let w satisfy the relation:

. (3)

Substituting (3), and into (2), we have

So, if we take

then solutions of (3) satisfy (2). Now, in order to reduce to, we use the scale transformation,. Immediately we obtain, and also,.

NOTES

^*Dedicated to Professor Masafumi Yoshino on the occasion of his 60th birthday.

Conflicts of Interest

The authors declare no conflicts of interest.

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.