Closed loop solitons and sigma functions: Classical and quantized elasticas with genera one and two

Journal of Geometry and Physics

Volumn 39, Issue 1, 2001, Pages 51-62

  Matsutani, Shigeki ^a  

^a 8 21 1 Higashi Linkan ^*  (Japan)

Author keywords

02.30. f; 05.20. y; 05.45.Yv; 11J89; 14H45; 35Q53; 46.70.Hg; 53A04; Differential geometry; Elastica; Hyperelliptic function; Sigma function; Space curve

Indexed keywords

EID: 0009888442     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/s0393-0440(00)00074-7     Document Type: Article

References (21)

1. H.F. Baker, Abelian Functions - Abel's Theorem and the Allied Theory including the Theory of the Theta Functions, Cambridge University Press, Cambridge, 1897 (republished, 1995).
2. Baker H.F. On the hyperelliptic sigma functions. Am. J. Math. XX:1898;301-384.
3. Baker H.F. On a system of differential equations leading to periodic functions. Acta Math. 27:1903;135-156.
4. Buchstaber V.H., Enolskii V.Z., Leykin D.V. Klein function, hyperelliptic Jacobians and applications. Rev. Math. Math. Phys. 10:1997;3-120.
5. P.G. Drazin, R.S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, 1989.
6. Grant D. Formal groups in genus two. J. Reine Angew. Math. 411:1990;96-121.
7. Ishimori Y. On the modified Korteweg-de Vries soliton and the loop soliton. J. Phys. Soc. Jpn. 50:1981;2741-2742.
8. Konno K., Ichikawa Y., Wadati M. A loop soliton propagating along a stretched rope. J. Phys. Soc. Jpn. 50:1981;1025-1026.
9. A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, Cambridge, 1927.
10. D. Mumford, Tata Lectures on Theta II, Progress in Mathematics, Vol. 43, Birkhäuser, Cambridge, 1984.
11. Matsutani S. Geometrical construction of the Hirota bilinear form of the modified Korteweg-de Vries equation on a thin elastic rod: Bosonic classical theory. Int. J. Mod. Phys. A. 22:1995;3109-3123.
12. Matsutani S. Statistical mechanics of elastica on plane: origin of MKdV hierarchy. J. Phys. A. 31:1998;2705-2725.
13. 3 J. Geom. Phys. 29:1999;243-259.
14. S. Matsutani, On the moduli of a quantized elastica in P and KdV flows: study of hyperelliptic curves as an extension of Euler's perspective of elastica I, mathDG/9808099.
15. S. Matsutani, Y. Ônishi, Hyperelliptic solutions of KdV and KP equations: reevaluation of Baker's study on hyperelliptic sigma functions, nlin.SI/0007001.
16. Ônishi Y. Complex multiplication formulae for curves of genus three. Tokyo J. Math. 21:1998;381-431.
17. Y. Ônishi, Chodaenkansuu-ron (Introduction to Hyperelliptic Function), 1998, unpublished (in Japanese).
18. Y. Ônishi, Determinatal expressions for some Abelian functions in genus two, Preprint, 2000.
19. Truesdell C. The influence of elasticity on analysis: the classic heritage. Bull. Am. Math. Soc. 9:1983;293-310.
20. A. Weil, Number Theory: An Approach through History; From Haammurapi to Legendre, Birkhäuser, Cambridge, 1983.
21. E.T. Wittaker, G.N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1927.