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On local equivalence for vector field systems

Published online by Cambridge University Press:  17 April 2009

P.J. Vassiliou
P.J. Vassiliou
Affiliation:
Institute for Mathematics and its Applications University of Minnesota Minneapolis, MN 55455, United States of America
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Abstract

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We give sufficient conditions for C∞ vector field systems on Rn with genus g = 1 to be diffeomorphic to a contact structure. The diffeomorphism is explicitly constructed and used to give the most general integral submanifolds for the systems. Finally the implications of these results for integrable hyperbolic partial differential equations in the plane is discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 42 , Issue 2 , October 1990 , pp. 215 - 229
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Bryant, R.L., Chern, S.S., Griffiths, P.A., ‘Exterior Differential Systems’, in Proceedings of the 1980 Beijing Symposium (Science Press, 1982).Google Scholar
[2]Bryant, R.L., Ph.D. Thesis, University of North Carolina, Chapel Hill (1979).Google Scholar
[3]Cartan, E., Les Systemes Differentiels Exterieurs et leur Applications Geometrique (Hermann, Paris, 1945).Google Scholar
[4]Gardner, R.B., ‘Invariants of Pfaffian systems’, Trans. Amer. Math. Soc. 126 (1965), 514–533.CrossRefGoogle Scholar
[5]Goursat, E., Lecons sur le probleme de Pfaff (Hermann, Paris, 1922).Google Scholar
[6]Goursat, E., Lecons sur l'intégration des équations aux dérivés partielles du second ordre á deux variable independantes, Vol II (Hermann, Paris, 1898).Google Scholar
[7]Hermann, R., ‘The theory of equivalence of Pfaffian systems and input systems under feedback’, Math. Systems Theory 15 (1982), 343–356.CrossRefGoogle Scholar
[8]Hermann, R., ‘Invariants for feedback equivalence and Cauchy characteristic multifoliations of nonlinear control systems’, Acta. Appl. Math. 11 (1988), 125–153.CrossRefGoogle Scholar
[9]Kumpera, A.K., ‘Invariants differentiels d’un pseudogroupe de Lie I & II’, J. Differential Geom. 10 (1975), 347–416.Google Scholar
[10]Pfaff, G.F., ‘Abh. der K.P.’, Akademie der Wissenschaften zu Berlin, (1814–1815), pp. 76–136.Google Scholar
[11]Pommaret, J.F., Systems of Partial Differential Equations and Lie Pseudogroups (Gordon and Breach, 1978).Google Scholar
[12]Vassiliou, P., ‘An exactly integrable hyperbolic partial differential equation and the nilpotent Lie group G6,15’ (to appear).Google Scholar
[13]Vassiliou, P., ‘On the solution of partial differential equations by the method of Darboux’ (to appear).Google Scholar
[14]Vassiliou, P., ‘Coupled systems of nonlinear wave equations and finite dimensional Lie algebras I & II’, Acta Appl. Math. 8 (1987), 107–147, 149–163.CrossRefGoogle Scholar
[15]Vessiot, E., ‘Sur les équations aux dérivés partielles du second ordre, F(x, y, z, p, q, r, s, t) = 0 integrable par la méthode de Darboux’, J. Math. Pure et appl. 18 (1939), 1–61.Google Scholar
[16]Vessiot, E., ‘Sur les équations aux dérivés partielles du second ordre, F(x, y, z, p, q, r, s, t) = 0 integrable par la méthode de Darboux’, J. Math. Pure et appl. 21 (1942), 1–65.Google Scholar
[17]Vessiot, E., ‘Sur l’integration des faisceaux de transformations infinitésimales dans le cas ou, le degree du faisceau étant n, celui du faisceau dérivé est n + 1’, Annales de l’Ecole Normale Superieure (3) 45 (1928), 189–253.CrossRefGoogle Scholar