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The theory of Legendrian unfoldings and first-order differential equations

Published online by Cambridge University Press:  14 November 2011

Shyuichi Izumiya
Shyuichi Izumiya
Affiliation:
Department of Mathematics, Hokkaido University, Sapporo 060, Japan
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Synopsis

We consider some properties of completely integrable first-order differential equations for real-valued functions. In order to study this subject, we introduce the theory of Legendrian unfoldings. We give a characterisation of equations with classical complete solutions in terms of Legendrian unfoldings, and also assert that the set of equations with singular solutions is an open set in the space of completely integrable equations even though such a set is thin in the space of all equations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 3 , 1993 , pp. 517 - 532
Copyright
Copyright © Royal Society of Edinburgh 1993

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References

1Arnol'd, V. I., Gusein-Zade, S. M. and Varchenko, A. N.. Singularities of Differentiable Maps, vol. I (Boston: Birkhäuser, 1986).Google Scholar
2Carathèodory, C.. Calculus of Variations and Partial Differential Equations of First Order, Part I, Partial Differential Equations of First Order (Holden-Day, 1965).Google Scholar
3Hayakawa, A., Ishikawa, G., Izumiya, S. and Yamaguchi, K.. Classification of generic integral diagram and first order ordinary differential equations (preprint).Google Scholar
4Izumiya, S.. Generic singularities for Hamilton-Jacobi equation. Adv. Stud. Pure Math. 22 (1993) (to appear).CrossRefGoogle Scholar
5Izumiya, S.. Generic properties of first order partial differential equations. Proceedings of Topology conference, Hawaii (Boston: Birkhäuser, to appear).Google Scholar
6Izumiya, S.. Completely integrable holonomic systems of first order differential equations (preprint).Google Scholar
7Kossowski, M.. First order partial differential equations with singular solution. Indiana Univ. Math. J. 35 (1986), 209–223.CrossRefGoogle Scholar
8Lychagin, V. V.. Local classification of non-linear first order partial differential equations. Russian Math. Surveys 30 (1975), 105–175.Google Scholar
9Zakalyukin, V. M.. Lagrangian and Legendrian singularities. Fund. Anal. Appl. 10 (1976), 23–31.CrossRefGoogle Scholar
10Zakalyukin, V. M.. Reconstructions of fronts and caustics depending on a parameter and versatility of mappings. J. Soviet Math. 27 (1983), 2713–2735.CrossRefGoogle Scholar