In this paper we describe the close relationship between invariant evolutions of projective curves and the Hamiltonian evolutions of Adler, Gel'fand and Dikii. We also show how KdV evolutions are related as well to invariant evolutions of projective surfaces.
@article{bwmeta1.element.bwnjournal-article-bcpv51z1p187bwm,
author = {Beffa, Gloria},
title = {The Theory of differential invariance and infinite dimensional Hamiltonian evolutions},
journal = {Banach Center Publications},
volume = {51},
year = {2000},
pages = {187-196},
zbl = {0962.37037},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv51z1p187bwm}
}
Beffa, Gloria. The Theory of differential invariance and infinite dimensional Hamiltonian evolutions. Banach Center Publications, Tome 51 (2000) pp. 187-196. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv51z1p187bwm/
[000] [1] M. Adler, On a Trace Functional for Formal Pseudo-differential Operators and the Symplectic Structure of the KdV, Inventiones Math. 50 (1979), 219-248. | Zbl 0393.35058
[001] [2] V. G. Drinfel'd and V. V. Sokolov, Lie Algebras and Equations of KdV Type, J. of Sov. Math. 30 (1985), 1975-2036. | Zbl 0578.58040
[002] [3] M. Fels and P. J. Olver Moving coframes. I. A practical algorithm, Acta Appl. Math. 51 (1998), 161-213. | Zbl 0937.53012
[003] [4] M. Fels and P. J. Olver Moving coframes. II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999), 127-208. | Zbl 0937.53013
[004] [5] I. M. Gel'fand and L. A. Dikii, A family of Hamiltonian structures connected with integrable nonlinear differential equations, in: I. M. Gelfand, Collected papers v.1, Springer-Verlag, 1987.
[005] [6] A. González-López, R. Hernandez and G. Marí Beffa, Invariant differential equations and the Adler-Gel'fand-Dikii bracket, J. Math. Phys. 38 (1997), 5720-5738. | Zbl 0892.58037
[006] [7] B. A. Kupershmidt and G. Wilson, Modifying Lax equations and the second Hamiltonian structure, Inventiones Math. 62 (1981), 403-436. | Zbl 0464.35024
[007] [8] G. Marí Beffa, Differential invariants and KdV Hamiltonian evolutions, Bull. Soc. Math. France 127 (1999) 363-391.
[008] [9] G. Marí Beffa and P. Olver, Differential Invariants for parametrized projective surfaces, Comm. Anal. Geom. 7 (1999), 807-839. | Zbl 0949.53012
[009] [10] P. Olver, Equivalence, Invariants and Symmetries, Cambridge University Press, Cambridge (1995).
[010] [11] I. McIntosh, SL(n+1)-invariant equations which reduce to equations of Korteweg-de Vries type, Proc. of the Royal Soc. of Edinburgh 115A (1990), 367-381. | Zbl 0724.35095
[011] [12] E. J. Wilczynski, Projective differential geometry of curves and ruled surfaces, B.G. Teubner, Leipzig (1906). | Zbl 37.0620.02
[012] [13] G. Wilson, On the antiplectic pair connected with the Adler-Gel'fand-Dikii bracket, Nonlinearity 5 (1992), 109-31. | Zbl 0761.58023