Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-14T23:13:09.527Z Has data issue: false hasContentIssue false
  • English
  • Français

On the geometry of the Painlevé V equation and a Bäcklund transformation

Published online by Cambridge University Press:  17 February 2009

W. K. Schief
W. K. Schief
Affiliation:
School of Mathematics, The University of New South Wales, Sydney, NSW 2052, Australia.
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.

It is shown that an integrable class of helicoidal surfaces in Euclidean space E3 is governed by the Painlevé V equation with four arbitrary parameters. A connection with sphere congruences is exploited to construct in a purely geometric manner an associated Bäcklund transformation.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 141 - 148
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Alekseev, G. A., “Soliton configurations of interacting massless fields”, Sov. Phys. Dokl. 28 (1983) 133–135.Google Scholar
[2]Bobenko, A. I., “Surfaces in terms of 2 by 2 matrices. Old and new integrable cases”, in Harmonic Maps and Integrable Systems (eds. Fordy, A. and Wood, J.), (Vieweg, 1994) 83–128.CrossRefGoogle Scholar
[3]Eisenhart, L. P., A Treatise on the Differential Geometry of Curves and Surfaces (Dover, New York, 1960).Google Scholar
[4]Gromak, V. I., “Bäcklund transformations of Painlevés equations and their applications”, in The Painlevé Property. One Century Later (ed. Conte, R.), (Springer, New York, 1999) 687–734.CrossRefGoogle Scholar
[5]Konopelchenko, B. G. and Rogers, C., “On 2 + 1-dimensional nonlinear systems of Loewner-type”, Phys. Lett. A 158 (1991) 391–397.CrossRefGoogle Scholar
[6]Schief, W. K., “On Laplace-Darboux-type sequences of generalized Weingarten surfaces”, J. Math. Phys., 41 (2000) 855–870.CrossRefGoogle Scholar
[7]Schief, W. K., “On a 2 + 1-dimensional integrable Ernst-type equation”, Proc. R. Soc. London A 446 (1994) 381–398.Google Scholar
[8]Schief, W. K., “The Painlevé III, V and VI transcendents as solutions of the Einstein-Weyl equations”, Phys. Lett. A 267 (2000) 265–275.CrossRefGoogle Scholar
[9]Schief, W. K. and Rogers, C., “On a Laplace sequence of nonlinear integrable Ernst-type equations”, in Algebraic Aspects of Integrable Systems (eds. Fokas, A. S. and Gelfand, I. M.), (Birkhäuser, Boston, 1997) 315–321.CrossRefGoogle Scholar
[10]Sibgatullin, N. R., “A proof of Geroch's hypothesis for electromagnetic and neutrino fields in the general theory of relativity”, Sov. Phys. Dokl. 28 (1983) 552–554.Google Scholar