Erratum to “Weierstrass’ Elliptic Function Solution to the Autonomous Limit of the String Equation of Type (2,5)” [Advances in Pure Mathematics 4 (2014), 494-497]

Yoshikatsu Sasaki^1*

¹Department of Mathematics, Hiroshima University, Higashi-Hiroshima, Japan.

DOI: 10.4236/apm.2014.412077                

Abstract

In this note, we analyze a few major claims about . As a consequence, we rewrite a major theorem, nullify its proof and one remark of importance, and offer a valid proof for it. The most important gift of this paper is probably the reasoning involved in all: We observe that a constant, namely t, has been changed into a variable, and we then tell why such a move could not have been made, we observe the discrepancy between the claimed domain and the actual domain of a supposed function that is created and we then explain why such a function could not, or should not, have been created, along with others.

Keywords

Painlevé Hierarchy, String Equation, Elliptic Function

P. 494, L. 7-: The string equation of type (q, p) should be correctly read as

P. 496, L. 13 - 14: Theorem B should be correctly read as follows:

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

where or 3.

P. 496, L. 17: In Remark, g₂ and g₃ in the elliptic function theory should be correctly read as follows:

P. 496, L. 21: In the r.h.s. of Equation (1), “” should be correctly read as “”.

P. 496, L. 3- - P. 497, L. 2: These 5 lines should be correctly read as follows:

If both of (2) and (3) are valid, then must vanish and coincides with 4 or.

Case and: In this case, we immediately obtain, ,

, where is a root of. Inversely, if these are sa-

tisfied, both of (2) and (3) are valid. can be reduced to by

. But, for brevity, now we put, and then, , i.e.

. Here and. The irrational equation satisfied by determines the

integral constant in the r.h.s. of (2) as.

Case and: In this case, we easily obtain, ,. Only is

allowed as the integral constant c in the r.h.s. of (2). Inversely, if these are satisfied, both of (2) and (3) are valid.

is reduced to by w = 3v, and then and. □

Conflicts of Interest

The authors declare no conflicts of interest.

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