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

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.

, We call it the string equation (or Douglas equation) of type ( ) , q p , which appears in the string theory or the theory of quantum gravity in 2D [1]- [9].In the followings, we set 2 q = , 2 1 p g = + .In the case where 2 q = , 3 p = , the string equation is written as an ODE satisfied by the potential w of Sturm-Liouville operator which is equivalent to the Hamiltonian system: In the case where 2 q = , 5 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] [ ] 4 0

Degenerated Garnier System
Equation (S) is also obtained as follows.Consider a 2D degenerated Garnier system [16] [17]: H q q t p q p p p q t q q q t t q H q p p p q q q q t q t q 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 2 t z ≡ as follows: From the above system, eliminating 1 q , 1 p , 2 p and putting 2 w q = , 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.

Autonomous Limit of the First Painlevé Equation
The first Painlevé equation (PI) has the autonomous limit [11].Replacing ( ) , , , w v z H by ( ) with a constant b ∈  , and taking limit 0 ε → , 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 ( )

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 2 n = , the statement is valid, i.e.

Theorem A. Replacing ( )
, , w z a by ( ) , or replacing ( ) , , , , , q q p p z H by ( ) with a constant b ∈  , and taking limit 0 ε → , we obtain the au- tonomous 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 The next section is devoted to give the proof of Theorem B.

Proof of Theorem B
Put ( ) ( ) (1) Multiplying both sides of 0 ϕ = by w′ , and integrating it, we obtain a first integral of (A) In order to find the elliptic function solution, let w satisfy the relation: then, by a fractional linear transformation, it is reduced to the first Painlevé equation[10] [11] an expression of a given meromorphic function w defined by The autonomous limit Equation (A) has a solution concretely described by the Weierstrass' elliptic function Modulus of the elliptic function is determined by the constants a and b. g 2 and g 3 in the elliptic function theory are as follows: then solutions of (3) satisfy(2).Now, in order to reduce