Abstract
Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of overdetermined partial differential equations: Given a system of quasi-linear PDEs of first order for one unknown function we find a necessary and sufficient condition for the existence of solutions in terms of the second jet of the coefficients. This generalizes to certain quasi-linear systems of first order for several unknown functions.
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Han, CK., Park, JD. Method of characteristics and first integrals for systems of quasi-linear partial differential equations of first order. Sci. China Math. 58, 1665–1676 (2015). https://doi.org/10.1007/s11425-014-4942-8
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DOI: https://doi.org/10.1007/s11425-014-4942-8
Keywords
- overdetermined PDE system
- quasi-linear first order PDEs
- first integrals
- Pfaffian systems
- Frobenius theorem