Hostname: page-component-76fb5796d-9pm4c Total loading time: 0 Render date: 2024-04-29T01:47:58.025Z Has data issue: false hasContentIssue false
  • English
  • Français

Diffeomorphisms on S1, projective structures and integrable systems

Published online by Cambridge University Press:  17 February 2009

Partha Guha
Partha Guha
Affiliation:
S.N. Bose National Centre for Basic Sciences, JD Block, Sector-3, Salt Lake, Calcutta-700091, India; e-mail: guha@BOSON.bose.res.in. Institut des Hautes Etudes Scientifiques, 35, Route de Chartres, 91440-Bures-sur-Yvette, France.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we consider a projective connection as defined by the nth-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 nth-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 ≤ 4) under the action of Vect(S1). The solutions of the AGD operator define an immersion R → RPn−1 in homogeneous coordinates. In this paper we derive the Schwarzian KdV equation as an evolution of the solution curve associated with Δ(n), for n ≤ 4.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 169 - 180
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Adler, M., “On a trace functional for formal pseudo-differential operators and the symplectic structure of the KdV”, Invent. Math. 50 (1979) 219–248.CrossRefGoogle Scholar
[2]Alber, M. S. and Alber, S. J., “Hamiltonian formalism for finite-zone solutions of integrable equations”, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985) 777–781.Google Scholar
[3]Alber, M. S. and Alber, S. J., “Hamiltonian formalism for nonlinear Schrödinger equations and sine-Gordon equations”, J. London Math. Soc. (2) 36 (1987) 176–192.CrossRefGoogle Scholar
[4]Arnold, V. I., Mathematical methods of classical mechanics, Graduate Texts in Math. 60, 2nd ed. (Springer, 1989).CrossRefGoogle Scholar
[5]Bakas, I., “The Hamiltonian structure of the spin-4 operator algebra”, Phys. Lett.B 213 (1988) 313–318.CrossRefGoogle Scholar
[6]Beffa, G. M., “On the Poisson geometry of the Adler-Gelfand-Dikii brackets”, J. Geom. Anal. 6 (1996) 207–232.CrossRefGoogle Scholar
[7]Cartan, E., Leçons sur la théorie des espaces à connection projective (Gauthier-Villars, Paris, 1937).Google Scholar
[8]Gargoubi, H. and Ovsienko, V. Yu., “Space of linear differential operators on the real line as a module over the Lie algebra of vector fields”, Internat. Math. Res. Notices 5 (1996) 235–251.CrossRefGoogle Scholar
[9]Gelfand, I. M. and Dikii, L. A., “A family of Hamiltonian structures connected with integrable nonlinear differential equations”, in I. M. Gelfand, Collected papers, vol. 1, (Springer, 1987).Google Scholar
[10]Gonzalez-Lopez, A., Heredero, R. H. and Beffa, G. M., “Invariant differential equations and the Adler-Gelfand-Dikii bracket”, J. Math. Phys. 38 (1997) 5720–5738.CrossRefGoogle Scholar
[11]Guha, P., “Projective connections, AGD manifold and integrable systems”, Rev. Math. Phys. 12 (2000) 1391–1406.CrossRefGoogle Scholar
[12]Guha, P., “Diffeomorphism, periodic KdV and C. Neumann system”, Differential Geom. Appl. 12 (2000) 1–8.CrossRefGoogle Scholar
[13]Guieu, L. and Ovsienko, V. Yu., “Structures symplectiques sur les espaces de courbes projectives et affines”, J. Geom. Phys. 16 (1995) 120–148, (French) [Symplectic structures on spaces of projective and affine curves].CrossRefGoogle Scholar
[14]Hitchin, N., “Vector fields on the circle”, in Mechanics, analysis and geometry: 200 years after Lagrange (ed. Francaviglia, M.), (Elsevier Science Publishers, 1991).Google Scholar
[15]Kirillov, A. A., “Infinite dimensional Lie groups; their orbits, invariants and representations. The geometry of moments”, in Twistor geometry and non-linear systems (eds. Dold, A. and Eckmann, B.), Lect. Notes in Math. 970, (Springer, 1980).Google Scholar
[16]Kirillov, A. A., “The orbit method. I. Geometric quantization”, in Representation theory of groups and algebras, (Amer. Math. Soc., Providence, RI, 1993) 1–32.Google Scholar
[17]Kirillov, A. A., “The orbit method. II. Infinite-dimensional Lie groups and Lie algebras”, in Representation theory of groups and algebras, (Amer. Math. Soc., Providence, RI, 1993) 33–63.CrossRefGoogle Scholar
[18]Kuiper, N. H., “Locally projective spaces of dimension one”, Michigan Math. 2 (1954) 95–97.Google Scholar
[19]Lazutkin, V. F. and Pankratova, T. F., “Normal forms and the versal deformations for Hill's equation”, Funct. Anal. Appl. 9 (1975) 41–48.Google Scholar
[20]Marcel, P., Ovsienko, V. and Roger, C., “Extension of the Virasoro algebras and generalized Sturm-Liouville operators”, Lett. Math. Phys. 40 (1997) 31–39.CrossRefGoogle Scholar
[21]Mathieu, P., “Extended classical conformal algebras and the second Hamiltonian structure of Lax equations”, Phys. Lett.B 208 (1988) 101–106.CrossRefGoogle Scholar
[22]McIntosh, I., “SL(n + 1) invariant equations which reduce to equations of Korteweg-de Vries type”, Proc. Roy. Soc. Edinburgh 115 (1990) 367–381.CrossRefGoogle Scholar
[23]Nijhoff, F., Hone, A. and Joshi, N., “On a Schwarzian PDE associated with the KdV hierarchy”, Phys. Lett. A 267 (2000) 147–156.CrossRefGoogle Scholar
[24]Ovsienko, V. Yu. and Khesin, B. A., “KdV superequation as an Euler equation”, Funct. Anal. Appl. 21 (1987) 329–331.CrossRefGoogle Scholar
[25]Ovsienko, V. Yu. and Khesin, B. A., “Symplectic leaves of the Gelfand-Dikii brackets and homotopy classes of nonflattening curves”, Funktsional. Anal. i Prilozhen. 24 (1990) 38–47; translation in Funct. Anal. Appl. 24 (1990), 33–40, (Russian).CrossRefGoogle Scholar
[26]Segal, G., “Unitary representations of some infinite dimensional groups”, Comm. Math. Phys. 80 (1981) 301–342.CrossRefGoogle Scholar
[27]Segal, G., “The geometry of the KdV equation”, Internat. J. Modern Phys.A 16 (1991) 2859–2869.CrossRefGoogle Scholar
[28]Wilczynski, E. J., Projective differential geometry of curves and ruled surfaces (Teubner, Leipzig, 1906).Google Scholar
[29]Wilson, G., “On the quasi-Hamiltonian formalism of the KdV equation”, Phys. Lett.A 132 (1988) 445–450.CrossRefGoogle Scholar
[30]Wilson, G., “On the antiplectic pair connected with the Adler-Gelfand-Dikii bracket”, Nonlinearilty 5(1992) 109–131.CrossRefGoogle Scholar
[31]Witten, E., “Coadjoint orbits of the Virasoro group”, Comm. Math. Phys. 114 (1988) 1–53.CrossRefGoogle Scholar