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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References
Brunat M. Riemann invariants. Fluid Dynam Trans, 1969, 4: 17–27
Bryant R, Chern S S, Gardner R, et al. Exterior Differential Systems. New York: Springer-Verlag, 1986
Clebsch A. Über die simultane Integration linearer partieller Differentialgleichungen. J Reine und Angew Math Crelle, 1866, 65: 257–268
Courant R, Hilbert D. Methods of Mathematical Physics, vol. II. New York: John Wiley and Sons, 1962
Deahna F. Über die Bedingungen der Integrabilitat. J Reine Angew Math, 1840, 20: 340–350
Frobenius G. Über das Pfaffsche probleme. J Reine Angew Math, 1877, 82: 230–315
Garabedian P R. Partial Differential Equations. Providence, RI: Amer Math Soc, 1998
Gardner R. Invariants of Pfaffian systems. Trans Amer Math Soc, 1967, 126: 514–533
Griffiths P A, Jensen G R. Differential Systems and Isometric Embeddings. Princeton, NJ: Princeton University Press, 1987
Grundland A M, Huard B. Riemann invariants and rank-k solutions of hyperbolic systems. J Nonlinear Math Phys, 2006, 13: 393–419
Grundland A M, Huard B. Conditional symmetries and Riemann invariants for hyperbolic systems of PDEs. J Phys A, 2007, 40: 4093–4123
Han C K. Pfaffian systems of Frobenius type and solvability of generic overdetermined PDE systems. In: Symmetries and Overdetermined Systems of Partial Differential Equations, vol. 144. New York: Springer, 2008, 421–429
Han C K. Generalization of the Frobenius theorem on involutivity. J Korean Math Soc, 2009, 46: 1087–1103
Han C K. Foliations associated with Pfaffian systems. Bull Korean Math Soc, 2009, 46: 931–940
Han C K, Lee K H. Integrable submanifolds in almost complex manifolds. J Geom Anal, 2010, 20: 177–192
Han C K, Park J D. Partial integrability of almost complex structures and the existence of solutions for quasi-linear Cauchy-Riemann equations. Pacific J Math, 2013, 265: 59–84
Jeffrey A. Quasilinear Hyperbolic Systems and Waves. London-San Francisco-Calif-Melbourne: Pitman Publishing, 1976
Monge G. Mémoire sur la théorie d’une équation aux dérivées partielles du premier ordre. In: Journal de l’École Polytechnique. Paris: 9ieme cahier, 1803, 56–99
Peradzynski Z. Riemann invariants for the nonplanar k-waves. Bull Acad Polon Sci Ser Sci Tech, 1971, 19: 717–724
Tabov J. Simple waves and simple states in R 2. J Math Anal Appl, 1997, 214: 613–632
Warner F. Foundations of Differentiable Manifolds and Lie Groups. Glenview, IL: Scott Foresman and Co, 1971
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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