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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 Painlevéequation, take the autonomous limit, and solve it concretely by use of the Weierstrass’ elliptic function.

1.1. The String Equation of Type (2, 2n + 1)

Let

for a couple of positive integers

In the case where

i.e.

is written as an equation

by replacement

In the case where

is similarly reduced to

by replacement

Now we recall the definition of the first Painlevé hierarchy. Consider the serial equations

for

If

If

Again, it essentially coinsides with (5), i.e. the 4th order equation of the first Painlevé hierarchy.

As proved by K. Takasaki [

Note that S. Shimomura [

Theorem A [

Theorem B [

for some positive integer

The author proved a theorem similar to Theorem A for the second Painlevé hierarchy [

Theorem C [

The first Painlevé equation has the autonomous limit [

where

where

For

Theorem D [

is reduced to the autonomous equation

by replacing

Note that the Equation (8) is obtained as a section of the most degenerated 2D Garnier system [

Theorem E [

A result similar to Theorem D is valid for

Theorem 1.1. The autonomous limit of the string equation of type

where

Proof. Note that

with the weight [

taking the limit

For the autonomous limit Equation (10), each auxiliary differential polynomial obtained in Theorem C has clear meaning.

Theorem 1.2. The differential polynomial

Proof. By definition,

Now we extend Theorem E to the case where

Theorem 1.3. Weierstrass’ elliptic function

Moreover, we can prove the theorem as follows:

Theorem 1.4. For each integer

is a solution to (10) with suitable parameters.

The proofs of these two theorems are given in the next section.

Let all of

with suitable

with suitable

Theorem 1.4 immediately follows from the following lemma.

Lemma. For every positive integer

Proof. Using

So, if

Thus, we have

Note that

The results of this article are summarized as follows: we obtained the autonomous limit of the string equation of type

Of course, poles of these solutions are uniform, i.e. every pole

Another remark should be given. T. Oshima and H. Sekiguchi [

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.

The author wishes to acknowledge Prof. T. Oshima for his helpful comment.

Yoshikatsu Sasaki, (2016) Weierstrass’ Elliptic Function Solutions to the Autonomous Limit of the String Equation. Journal of Applied Mathematics and Physics,04,857-862. doi: 10.4236/jamp.2016.44093