Invariant differential equations and the Adler-Gel'fand-Dikii bracket
| dc.contributor.author | González López, Artemio | |
| dc.contributor.author | Hernández Heredero, Rafael | |
| dc.contributor.author | Beffa, Gloria Marí | |
| dc.date.accessioned | 2023-06-20T20:09:58Z | |
| dc.date.available | 2023-06-20T20:09:58Z | |
| dc.date.issued | 1997-11 | |
| dc.description | © 1997 American Institute of Physics. A.G-L. and R.H.H. would like to acknowledge the partial financial support of the DGES under Grant No. PB95-0401. | |
| dc.description.abstract | In this paper we find an explicit formula for the most general vector evolution of curves on RPn−1 invariant under the projective action of SL(n, R). When this formula is applied to the projectivization of solution curves of scalar Lax operators with periodic coefficients, one obtains a corresponding evolution in the space of such operators. We conjecture that this evolution is identical to the second KdV Hamiltonian evolution under appropriate conditions. These conditions give a Hamiltonian interpretation of general vector differential invariants for the projective action of SL(n, R), namely, the SL(n, R) invariant evolution can be written so that a general vector differential invariant corresponds to the Hamiltonian pseudo-differential operator. We find common coordinates and simplify both evolutions so that one can attempt to prove the equivalence for arbitrary n . | |
| dc.description.department | Depto. de Física Teórica | |
| dc.description.faculty | Fac. de Ciencias Físicas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | DGES | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/32862 | |
| dc.identifier.doi | 10.1063/1.532162 | |
| dc.identifier.issn | 0022-2488 | |
| dc.identifier.officialurl | http://dx.doi.org/10.1063/1.532162 | |
| dc.identifier.relatedurl | http://scitation.aip.org | |
| dc.identifier.relatedurl | http://arxiv.org/abs/hep-th/9603199 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/59725 | |
| dc.issue.number | 11 | |
| dc.journal.title | Journal of mathematical physics | |
| dc.language.iso | eng | |
| dc.page.final | 5738 | |
| dc.page.initial | 5720 | |
| dc.publisher | American Institute of Physics | |
| dc.relation.projectID | PB95-0401 | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 51-73 | |
| dc.subject.keyword | Korteweg-devries type | |
| dc.subject.ucm | Física-Modelos matemáticos | |
| dc.subject.ucm | Física matemática | |
| dc.title | Invariant differential equations and the Adler-Gel'fand-Dikii bracket | |
| dc.type | journal article | |
| dc.volume.number | 38 | |
| dcterms.references | [AH] Ackerman, M., and Hermann, R., Sophus Lie’s 1884 Differential Invariant Paper, Math Sci Press, Brookline, Mass. (1975). [A] Adler, M., On a Trace Functional for Formal Pseudo-differential Operators and the Symplectic Structure of the KdV, Inventiones Math. 50, 219–48 (1979). [DS] Drinfel’d, V.G. and Sokolov, V.V., Lie Algebras and Equations of KdV Type, J. of Sov. Math. 30, 1975–2036 (1985). [GD] Gel’fand, I.M. and Dikii, L.A. A family of Hamiltonian structures connected with integrable nonlinear differential equations, in I.M. Gel’fand, Collected papers, Vol. 1, Springer–Verlag (1987). [H] Halphen, G.-H., Sur les invariants diff´erentiels,, in Oeuvres, Vol. 2, Gauthier– Villars, Paris (1913). [KW] Kupershmidt, B.A. and Wilson, G., Modifying Lax Equations and the Second Hamiltonian Structure, Inventiones Math. 62, 403–36 (1981). [L] Lie, S., Über Differentialinvarianten, in Gesammelte Abhandlungen, Vol. 6, B.G. Teubner, Leipzig (1927); see [AH] for an English translation. [OST] Olver, P.J., Sapiro, G., and Tannenbaum, A., Differential invariant signatures and flows in computer vision: a symmetry group approach, in Geometry-Driven Diffusion in Computer Vision, B.M. ter Haar Romeny, ed., Kluwer Acad. Publ., Dordrecht, The Netherlands (1994). [O1] Olver, P.J., Applications of Lie Groups to Differential Equations, Second Edition, Springer–Verlag, New York (1993). [O2] Olver, P.J., Equivalence, Invariance and Symmetry, Cambridge University Press, Cambridge, UK (1995). [OK] Ovsienko, V.Yu. and Khesin, B.A., Symplectic leaves of the Gel’fand–Dikii brackets and homotopy classes of non- degenerate curves, Funct. Anal. and Appl. 24 n. 1, 33– 40 (1990). [T] Tresse, A., Sur les invariants diff´erentiels des groupes continus de transformations, Acta Math. 18, 1–88 (1894). [W] Wilczynski, E.J., Projective differential geometry of curves and ruled surfaces, B.G. Teubner, Leipzig (1906). [Wi] Wilson, G., On the antiplectic pair connected with the Adler–Gel’fand–Dikii bracket, Nonlinearity 5, 109–31 (1992). | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 7f260dbe-eebb-4d43-8ba9-d8fbbd5b32fc | |
| relation.isAuthorOfPublication.latestForDiscovery | 7f260dbe-eebb-4d43-8ba9-d8fbbd5b32fc |
Download
Original bundle
1 - 1 of 1
