×

Diffeomorphisms on \(S^1\), projective structures and integrable systems. (English) Zbl 1003.35109

Summary: We consider a projective connection as defined by the \(n\)th-order Adler-Gelfand-Dikii (AGD) operator on the circle. It is well known that the Korteweg-de Vries (KdV) equation is the archetypal example of a scalar Lax equation defined by a Lax pair of scalar \(n\)th-order differential AGD operators. In this paper we derive (formally) the KdV equation as an evolution equation of the AGD operator (at least for \(n\leq 4)\) under the action of \(\text{Vect}(S^1)\). The solutions of the AGD operator define an immersion \({\mathbf R}\to \mathbb{R} P^{n-1}\) in homogeneous coordinates. We derive the Schwarzian KdV equation as an evolution of the solution curve associated with \(\Delta^{(n)}\), for \(n\leq 4\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures

Software:

AGD
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] McIntosh, Proc. Roy. Soc. Edinburgh 115 pp 367– (1990) · Zbl 0724.35095 · doi:10.1017/S0308210500020710
[2] DOI: 10.1016/S0375-9601(00)00063-3 · Zbl 0949.37049 · doi:10.1016/S0375-9601(00)00063-3
[3] DOI: 10.1023/A:1007310811875 · Zbl 0881.17019 · doi:10.1023/A:1007310811875
[4] Gelfand, I. M. Gelfand, Collected papers 1 (1987)
[5] DOI: 10.1155/S1073792896000177 · Zbl 0851.17023 · doi:10.1155/S1073792896000177
[6] Cartan, Leçons sur la théorie des espaces à connection projective (1937)
[7] Beffa, J. Geom. Anal. 6 pp 207– (1996) · Zbl 0896.58028 · doi:10.1007/BF02921600
[8] Bakas, Phys. Lett. 213 pp 313– (1988) · doi:10.1016/0370-2693(88)91767-4
[9] Arnold, Mathematical methods of classical mechanics (1989) · doi:10.1007/978-1-4757-2063-1
[10] DOI: 10.1112/jlms/s2-36.1.176 · Zbl 0609.58011 · doi:10.1112/jlms/s2-36.1.176
[11] Alber, C. R. Acad. Sci. Paris Sér. I Math. 301 pp 777– (1985)
[12] DOI: 10.1007/BF01410079 · Zbl 0393.35058 · doi:10.1007/BF01410079
[13] DOI: 10.1007/BF01218287 · Zbl 0632.22015 · doi:10.1007/BF01218287
[14] DOI: 10.1088/0951-7715/5/1/004 · Zbl 0761.58023 · doi:10.1088/0951-7715/5/1/004
[15] DOI: 10.1007/BF01078174 · Zbl 0316.30019 · doi:10.1007/BF01078174
[16] Kuiper, Michigan Math. 2 pp 95– (1954)
[17] Kirillov, Representation theory of groups and algebras pp 33– (1993) · doi:10.1090/conm/145/1216180
[18] Kirillov, Representation theory of groups and algebras pp 1– (1993) · doi:10.1090/conm/145/1216179
[19] Kirillov, Twistor geometry and non-linear systems 970 (1980)
[20] Hitchin, Mechanics, analysis and geometry: 200 years after Lagrange (1991)
[21] DOI: 10.1016/0393-0440(94)00024-X · Zbl 0827.53024 · doi:10.1016/0393-0440(94)00024-X
[22] DOI: 10.1016/S0926-2245(99)00034-0 · Zbl 0963.37063 · doi:10.1016/S0926-2245(99)00034-0
[23] DOI: 10.1142/S0129055X00000538 · Zbl 0990.37053 · doi:10.1142/S0129055X00000538
[24] DOI: 10.1063/1.532162 · Zbl 0892.58037 · doi:10.1063/1.532162
[25] DOI: 10.1016/0375-9601(88)90510-5 · Zbl 0978.35055 · doi:10.1016/0375-9601(88)90510-5
[26] Wilczynski, Projective differential geometry of curves and ruled surfaces (1906) · JFM 37.0620.02
[27] DOI: 10.1142/S0217751X91001416 · Zbl 0741.35073 · doi:10.1142/S0217751X91001416
[28] DOI: 10.1007/BF01208274 · Zbl 0495.22017 · doi:10.1007/BF01208274
[29] DOI: 10.1007/BF01077916 · Zbl 0723.58021 · doi:10.1007/BF01077916
[30] DOI: 10.1007/BF01077813 · Zbl 0655.58018 · doi:10.1007/BF01077813
[31] Mathieu, Phys. Lett. 208 pp 101– (1988) · doi:10.1016/0370-2693(88)91211-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.