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]

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.

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.

Cite this paper

Sasaki, Y. (2014) 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].*Advances in Pure Mathematics*, **4**, 680-681. doi: 10.4236/apm.2014.412077.

Sasaki, Y. (2014) 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].

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.

[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.