"Sato theory"의 두 판 사이의 차이
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| 1번째 줄: | 1번째 줄: | ||
<h5>introduction</h5> | <h5>introduction</h5> | ||
| − | * Sato’s Grassmannian and its determinant bundle became a “universal” setting where moduli spaces of curves (or maps or bundles) of arbitrary genus could | + | * Sato’s Grassmannian and its determinant bundle became a “universal” setting where moduli spaces of curves (or maps or bundles) of arbitrary genus could be mapped and made to interact |
* tau function = the section of a determinant line bundle over an infinite-dimensional Grassmannian | * tau function = the section of a determinant line bundle over an infinite-dimensional Grassmannian | ||
| − | * | + | * Sato found that character polynomials (Schur functions) solve the KP hierarchy and, based on this observation, he created the theory of the infinite-dimensional (universal) Grassmann manifold<br> and showed that the Hirota bilinear equations are nothing but the Plucker relations for this Grassmann manifold. |
| 18번째 줄: | 18번째 줄: | ||
<math>2(\frac{\partial^2}{\partial x^2})\log K(x,t)</math> | <math>2(\frac{\partial^2}{\partial x^2})\log K(x,t)</math> | ||
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| 45번째 줄: | 39번째 줄: | ||
<h5>relation to Kac-Moody algebras</h5> | <h5>relation to Kac-Moody algebras</h5> | ||
| − | the totality of tau-functions of the KdV hierarchy is the group orbit of the highest weight vector (=1) of the basic representation of A_1^1 | + | * the totality of tau-functions of the KdV hierarchy is the group orbit of the highest weight vector (=1) of the basic representation of A_1^1 |
| − | + | * applications of vertex operators are precisely Ba¨cklund transformations | |
| − | applications of vertex operators are precisely Ba¨cklund transformations | + | * This implies that the affine Lie algebra A(1) 1 is the infinitesimal transformation group for solutions of the KdV hierarchy. |
| − | + | * Frenkel–Kac had already used free fermions to construct basic representations. In this approach, the tau-functions are defined as vacuum expectation values. | |
| − | Frenkel–Kac had already used free fermions to construct basic representations. In this approach, the -functions are defined as vacuum expectation values. | ||
| 98번째 줄: | 91번째 줄: | ||
* [http://yokoemon.web.fc2.com/KANT2010/Notes/Yamazaki.pdf Sato theory, p-adic tau function and arithmetic geometry] | * [http://yokoemon.web.fc2.com/KANT2010/Notes/Yamazaki.pdf Sato theory, p-adic tau function and arithmetic geometry] | ||
| + | * Algebraic Geometrical Methods in Hamiltonian Mechanics http://www.jstor.org/stable/37539 | ||
* | * | ||
* Segal, Graeme, and George Wilson. 1985. Loop groups and equations of KdV type. Publications Mathématiques de L’Institut des Hautes Scientifiques 61, no. 1 (12): 5-65. doi:[http://dx.doi.org/10.1007/BF02698802 10.1007/BF02698802]. | * Segal, Graeme, and George Wilson. 1985. Loop groups and equations of KdV type. Publications Mathématiques de L’Institut des Hautes Scientifiques 61, no. 1 (12): 5-65. doi:[http://dx.doi.org/10.1007/BF02698802 10.1007/BF02698802]. | ||
| 113번째 줄: | 107번째 줄: | ||
* Nakamura, Yoshimasa. 1994. “A tau-function of the finite nonperiodic Toda lattice”. <em>Physics Letters A</em> 195 (5-6) (12월 12): 346-350. doi:[http://dx.doi.org/10.1016/0375-9601%2894%2990040-X 10.1016/0375-9601(94)90040-X]. | * Nakamura, Yoshimasa. 1994. “A tau-function of the finite nonperiodic Toda lattice”. <em>Physics Letters A</em> 195 (5-6) (12월 12): 346-350. doi:[http://dx.doi.org/10.1016/0375-9601%2894%2990040-X 10.1016/0375-9601(94)90040-X]. | ||
* Poppe, C. 1989. “General determinants and the tau function for the Kadomtsev-Petviashvili hierarchy”. <em>Inverse Problems</em> 5 (4): 613-630. doi:[http://dx.doi.org/10.1088/0266-5611/5/4/012 10.1088/0266-5611/5/4/012]. | * Poppe, C. 1989. “General determinants and the tau function for the Kadomtsev-Petviashvili hierarchy”. <em>Inverse Problems</em> 5 (4): 613-630. doi:[http://dx.doi.org/10.1088/0266-5611/5/4/012 10.1088/0266-5611/5/4/012]. | ||
| − | * | + | * Freeman, N. C., 와/과J. J. C. Nimmo. 1983. “Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: The wronskian technique”. <em>Physics Letters A</em> 95 (1) (4월 11): 1-3. doi:[http://dx.doi.org/10.1016/0375-9601%2883%2990764-8 10.1016/0375-9601(83)90764-8] |
* http://dx.doi.org/10.1016/0375-9601(94)90040-X | * http://dx.doi.org/10.1016/0375-9601(94)90040-X | ||
2011년 4월 20일 (수) 05:30 판
introduction
- Sato’s Grassmannian and its determinant bundle became a “universal” setting where moduli spaces of curves (or maps or bundles) of arbitrary genus could be mapped and made to interact
- tau function = the section of a determinant line bundle over an infinite-dimensional Grassmannian
- Sato found that character polynomials (Schur functions) solve the KP hierarchy and, based on this observation, he created the theory of the infinite-dimensional (universal) Grassmann manifold
and showed that the Hirota bilinear equations are nothing but the Plucker relations for this Grassmann manifold.
\(K(x,t)=1+e^{2a(x-4a^2t+\delta)}\)
\(2(\frac{\partial^2}{\partial x^2})\log K(x,t)\)
\(K(x,t)=1+A_1e^{2a_1(x-4a_1^2t+\delta_1)}+A_2e^{2a_2(x-4a_2^2t+\delta_2)}+A_3e^{2a_1(x-4a_1^2t+\delta_1)+{2a_2(x-4a_2^2t+\delta_2)}\)
\(2(\frac{\partial^2}{\partial x^2})\log K(x,t)\)
KdV hierarchy
The totality of soliton equations
organized in this way is called a hierarchy of soliton
equations; in the KdV case, it is called the KdV
hierarchy. This notion of hierarchy was introduced by
M Sato. He tried to understand the nature of the
bilinear method of Hirota. First, he counted the
number of Hirota bilinear operators of given degree
for hierarchies of soliton equations. For the number of
bilinear equations,M Sato and Y Sato made extensive
computations and made many conjectures that involve
eumeration of partitions.
Wronskian determinant
relation to Kac-Moody algebras
- the totality of tau-functions of the KdV hierarchy is the group orbit of the highest weight vector (=1) of the basic representation of A_1^1
- applications of vertex operators are precisely Ba¨cklund transformations
- This implies that the affine Lie algebra A(1) 1 is the infinitesimal transformation group for solutions of the KdV hierarchy.
- Frenkel–Kac had already used free fermions to construct basic representations. In this approach, the tau-functions are defined as vacuum expectation values.
history
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://eom.springer.de
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- Discrete Integrable Systems http://dx.doi.org/10.1007/b94662
- Book review on Soliton equations and their algebro-geometric solutions. Vol. I. (1+1)-dimensional continuous models
- 2011년 books and articles
- http://library.nu/search?q=
- http://library.nu/search?q=
expositions
- Sato theory, p-adic tau function and arithmetic geometry
- Algebraic Geometrical Methods in Hamiltonian Mechanics http://www.jstor.org/stable/37539
- Segal, Graeme, and George Wilson. 1985. Loop groups and equations of KdV type. Publications Mathématiques de L’Institut des Hautes Scientifiques 61, no. 1 (12): 5-65. doi:10.1007/BF02698802.
articles
- Eilbeck, J C, V Z Enolski, and J Gibbons. 2010. Sigma, tau and Abelian functions of algebraic curves. Journal of Physics A: Mathematical and Theoretical 43, no. 45 (11): 455216. doi:10.1088/1751-8113/43/45/455216.
- Eilbeck, J. C., V. Z. Enolski, S. Matsutani, Y. Onishi, and E. Previato. 2010. Abelian Functions for Trigonal Curves of Genus Three. International Mathematics Research Notices (7). doi:10.1093/imrn/rnm140. http://imrn.oxfordjournals.org/content/2007/rnm140.short.
- Kajiwara, Kenji, Marta Mazzocco, 와/과Yasuhiro Ohta. 2007. “A remark on the Hankel determinant formula for solutions of the Toda equation”. Journal of Physics A: Mathematical and Theoretical 40 (42): 12661-12675. doi:10.1088/1751-8113/40/42/S11.
- Matsutani, Shigeki. 2000. Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on Hyperelliptic Sigma Functions. nlin/0007001 (July 1). doi:doi:10.1088/0305-4470/34/22/312. http://arxiv.org/abs/nlin/0007001.
- Nakamura, Yoshimasa. 1994. “A tau-function of the finite nonperiodic Toda lattice”. Physics Letters A 195 (5-6) (12월 12): 346-350. doi:10.1016/0375-9601(94)90040-X.
- Poppe, C. 1989. “General determinants and the tau function for the Kadomtsev-Petviashvili hierarchy”. Inverse Problems 5 (4): 613-630. doi:10.1088/0266-5611/5/4/012.
- Freeman, N. C., 와/과J. J. C. Nimmo. 1983. “Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: The wronskian technique”. Physics Letters A 95 (1) (4월 11): 1-3. doi:10.1016/0375-9601(83)90764-8
- http://dx.doi.org/10.1016/0375-9601(94)90040-X
question and answers(Math Overflow)
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