Weierstrass ’ Elliptic Function Solutions to the Autonomous Limit of the String Equation

In this article, we study the string equation of type (2, 2n + 1), which is derived from 2D gravity theory or the string theory. We consider the equation as a 2n-th order analogue of the first Painlev éequation, take the autonomous limit, and solve it concretely by use of the Weierstrass’ elliptic function.

[ , ] 1, : , : for a couple of positive integers ( , ) q p .The above equation is called the string Equation (or Douglas equation) of type ( , ) q p , and appears in the string theory or the theory of 2D quantum gravity [1]- [8].In the followings, we set ( , ) (2, 2 1) q p n = + for a positive integer n .In the case where ( , ) (2,3) q p = , the string equation is written as an ODE satisfied by the potential w of Sturm-Liouville operator 2 Q D w = + , and then, by a fractional linear transformation, it is reduced to the first Painlevé equation [9]- [11].In fact, the string equation of type (2, 3) is written as an equation . In the case where ( , ) (2,5) q p = , the string equation [ , ] 1, : , : is similarly reduced to

The First Painlevé Hierarchy
Now we recall the definition of the first Painlevé hierarchy.Consider the serial equations , where 10 C is an integral constant.In the followings, each ij C is also an integral con- stant.
is a differential polynomial of 2n-th order, i.e. each ( 6) is an ordinary differential equation of 2n-th order.
Theorem B [14].At each pole 0 z z = , the meromorphic solution to (6) has the form The author proved a theorem similar to Theorem A for the second Painlevé hierarchy [15], and, in its proof, auxiliary differential polynomials play important roles.So, for the first Painlevé hierarchy as well, the auxiliary differential polynomials should exist.Recall them.

Autonomous Limits
The first Painlevé equation has the autonomous limit [9].Replacing ( , ) w z by is a doubly periodic meromorphic function with two fundamental periods by replacing ( , , ) w z a by , and taking the limit 0 ε → .Note that the Equation ( 8) is obtained as a section of the most degenerated 2D Garnier system [18] (see also [19] [20]).The following theorem is not trivial but natural if we consider Theorem D together with Theorem B.

Results
A result similar to Theorem D is valid for n ∈  .Theorem 1.1.The autonomous limit of the string equation of type (2, 2 1) n + is given by where b ∈  is a complex parameter.
with the weight [14] wt defined by wt( ) 1 z = and wt( ) taking the limit 0 ε → , we obtain the conclusion. For the autonomous limit Equation (10), each auxiliary differential polynomial obtained in Theorem C has clear meaning.
Theorem 1.2.The differential polynomial is the first integral of (10).
Proof.By definition, is a solution to (10) with suitable parameters.Moreover, we can prove the theorem as follows: Theorem 1.4.For each integer k satisfying is a solution to (10) with suitable parameters.The proofs of these two theorems are given in the next section.,we obtain the conclusion.

Proof of Theorem 1.4
Theorem 1.4 immediately follows from the following lemma.
Lemma.For every positive integer k , ( 1,..., ) is described by some polynomial of ℘, and its degree is as follows: ( + ℘ is a polynomial in ℘ of degree n , and all terms but one of top degree have Y.Sasaki integral constants.Therefore, if the term of top degree vanishes, we can make all terms vanish with suitable selection of integral constants.Thus, Theorem 1.4 is established.

Discussion
The results of this article are summarized as follows: we obtained the autonomous limit of the string equation of type ( Another remark should be given.T. Oshima and H. Sekiguchi [21] studied the commutator equation [ , ] 0 Q P = of partial differential operators , Q P invariant under the action of a Weyl group, and obtained many of elliptic function solutions.Note that the autonomous limit of [ , ] 1 Q P = means [ , ] 0 Q P = .The fact implies that, in view of the string theory, the first Painlevé equation is not only a nonautonomization but also a quantization of the Weierstrass' elliptic function.Relation between their solutions and our special solutions should be studied in the future.It may yield a new kind of quantization of KdV equation or hierarchy.Autonomous limit is a kind of approximation of the differential equation.Therefore, the solutions of the autonomous limit equation gives us information on the asymptotics of the nonautonomous equation, as well as does on the first Painlevé equation.Moreover, if all of the solutions to the autonomous limit equation are determined, it contributes the argument on the irreducibility of the string equation in the sence of the differential Galois theory, as well as on the irreduciblity of the first Painlevé equation.

1 . 1 D
The String Equation of Type (2, 2n + 1) Let or D ′ stand for the differentiation w.r.t.z , and − stand for the inverse operator of D .Consider the commutator equation of ordinary differential operators 2 2 order equation of the first Painlevé hierarchy.
an expression of a given meromorphic function w defined by 0 are derived from the singular manifold equation for the KdV hierarchy, and we call them the first Painlevé hierarchy[10] [12][13].For example, satisfied by the Weierstrass' elliptic function, i.e.

 1 . 3 .
Now we extend Theorem E to the case where Theorem Weierstrass' elliptic function ( ) z ℘ [17]similar result is valid, i.e.Theorem D[17].The 4th order equation of the first Painlevé hierarchy with suitable parameters