Abstract
By analogy with the linear vector bundle case, a non-linear partial differential equation on a manifold can be defined as a fibred submanifold Rk of a k-jet bundle. By observing that under natural conditions the first prolongation gives rise to a vector bundle over Rk, (that is, a quasilinear equation), techniques of the linear case are adapted to establish conditions for the formal integrability of the equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
GOLDSCHMIDT, H. “Existence Theorems for analytic linear partial differential equations.” Ann. of Math., II. Ser., 86, (1967), 246–270.
GOLDSCHMIDT, H. “Integrability criteria for systems of non-linear partial differential equations,” J. Diff. Geometry 1, (1967), 269–307.
KURANISHI, M. “Sheaves defined by differential equations and application to the theory of pseudo-group structures,” Amer. J. Math., 86, (1964), 379–391.
SPENCER, D. C. “Deformation of structures on manifolds defined by transitive, continuous pseudogroups: I–II,” Ann. of Math., II. Ser. 76, (1962), 306–445.
SPENCER, D. C. “Overdetermined systems of linear partial differential equations,” Bull. Amer. math. Soc., 75, (1969), 179–239.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Elliott, R.J. Quasilinear resolutions of non-linear equations. Manuscripta Math 12, 399–410 (1974). https://doi.org/10.1007/BF01171083
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01171083