Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Special points on first order partial differential equations and the deformations of solutions

Author: Marek Kossowski
Journal: Trans. Amer. Math. Soc. 302 (1987), 171-184
MSC: Primary 35A30; Secondary 35B32, 35F20, 58A20, 58C27, 58F14
MathSciNet review: 887504
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The object of this paper is to identify four cases of special behavior in a class of first order PDE for a real valued function. (The class of PDE may be thought of as perturbation of PDE with singular solutions.) In each case we show how invariants of the PDE determine properties of solutions. The properties of solutions examined here are the structure of critical points and singularities induced by cotangent projection. These properties are described in the sense of constructing local models for solutions and characterizing their behavior under small deformations. We will find two cases where deformations exhibit bifurcation phenomena, and describe generic deformations.

References [Enhancements On Off] (What's this?)

  • [A] V. I. Arnol′d, Geometrical methods in the theory of ordinary differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 250, Springer-Verlag, New York-Berlin, 1983. Translated from the Russian by Joseph Szücs; Translation edited by Mark Levi. MR 695786
  • [B] Robert L. Bryant, Shiing Shen Chern, and Phillip A. Griffiths, Exterior differential systems, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) Science Press, Beijing, 1982, pp. 219–338. MR 714337
  • [G] R. B. Gardner, Lectures on exterior differential systems, UNC, 1980.
  • [GG] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, Vol. 14. MR 0341518
  • [GJ] J. Guckenhiemer, Catastrophes and $ PDE$, Ann. Inst Fourier (Grenoble) 23 (1973).
  • [GS] Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, American Mathematical Society, Providence, R.I., 1977. Mathematical Surveys, No. 14. MR 0516965
  • [K] Marek Kossowski, First order partial differential equations with singular solution, Indiana Univ. Math. J. 35 (1986), no. 1, 209–223. MR 825637, 10.1512/iumj.1986.35.35012
  • [L] V. V. Lychagin, Local classification of non-linear first order P.D.E., Russian Math. Surveys 30 (1975), 105-175.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35A30, 35B32, 35F20, 58A20, 58C27, 58F14

Retrieve articles in all journals with MSC: 35A30, 35B32, 35F20, 58A20, 58C27, 58F14

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society