Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-12-02T21:30:13.105Z Has data issue: false hasContentIssue false
  • English
  • Français

An integrable system of partial differential equations on the special linear group

Published online by Cambridge University Press:  17 February 2009

Peter J. Vassiliou
Peter J. Vassiliou
Affiliation:
Centre for Mathematics and its Applications, Australian National University, ACT 0200, Australia; e-mail: [email protected]. On leave from the School of Mathematics and Statistics, University of Canberra, ACT, 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.

We give an intrinsic construction of a coupled nonlinear system consisting of two first-order partial differential equations in two dependent and two independent variables which is determined by a hyperbolic structure on the complex special linear group regarded as a real Lie group G. Despite the fact that the system is not Darboux semi-integrable at first order, the construction of a family of solutions depending.upon two arbitrary functions, each of one variable, is reduced to a system of ordinary differential equations on the 1-jets. The ordinary differential equations in question are of Lie type and associated with G.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 1 , July 2002 , pp. 83 - 93
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Anderson, I. M., Fels, M. E. and Torre, C. G.. “Group invariant solutions without transversality”, Commun. Math. Phys. 212 (2000) 653–686.CrossRefGoogle Scholar
[2]Arnold, V. I. and Khesin, B. A., Topological Methods in Hydrodynamics (Springer, 1998).Google Scholar
[3]Bryant, R. L., “An introduction to Lie groups and symplectic geometry”, in Geometry and quantum field theory (Park City, UT 1991), IAS/Park City Math. Ser. 1, (Amer. Math. Soc., Providence, RI, 1995) 5–181.CrossRefGoogle Scholar
[4]Bryant, R. L., Chern, S. S., Gardner, R. B., Goldschmidt, H. L. and Griffiths, P. A., Exterior Differential Systems, Mathematical Sciences Research Institute Publications 18 (Springer, New York, 1991).CrossRefGoogle Scholar
[5]Bryant, R. L., Griffiths, P. A. and Hsu, L., “Hyperbolic exterior differential systems and their conservation laws I, II”, Selecta Math. (N.S.) 1 (1995) 21–112, 265–323.CrossRefGoogle Scholar
[6]Doyle, P. W., “Characteristic solutions of scalar Newton equations in one space dimension”, Internat. J. Non-Linear Mech. 33 (1998) 1013–1026.CrossRefGoogle Scholar
[7]Doyle, P. W., “Symmetry and ordinary differential constraints”, Internat. J. Non-Linear Mech. 34 (1999) 1089–1102.Google Scholar
[8]Nucci, M. C., “The role of symmetries in solving differential equations”, Mathl. Comput. Modelling 25 (1997) 181–193.CrossRefGoogle Scholar
[9]Vassiliou, P. J., “Vessiot structure for manifolds of (p, q)-hyperbolic type: Darboux integrability and symmetry”, Trans. Amer. Math. Soc. 353 (2001) 1705–1739.Google Scholar
[10]Vassiliou, P. J., “Tangential characteristic symmetries and first order hyperbolic systems”, Special issue “Computational geometry for differential equations”, Appl. Algebra Engrg. Comm. Comput 11 (2001) 377–395.CrossRefGoogle Scholar
[11]Vassiliou, P. J., “Intrinsic geometry of first order partial differential equations in the plane”, Research report #2000/4, School of Mathematics and Statistics, University of Canberra,http://beth.canberra.edu.au/peterv.Google Scholar
[12]Vessiot, E., “Sur une théorie nouvelles des problèmes généraux d'intégration”, Bull. Soc. Math. France 52 (1924) 336–395.CrossRefGoogle Scholar
[13]Winternitz, P., “Lie groups and solutions of differential equations”, in Nonlinear phenomena (Oaxtepec, 1982), Lecture Notes in Phys. 189, (Springer, Berlin, 1983) 263–331.CrossRefGoogle Scholar
[14]Yang, K., Elementary Exterior Differential Systems and its Applications to Equivlence Problems (Kluwer, 1992).CrossRefGoogle Scholar