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Normal forms of implicit differential equations and of Liouville fields. (Formes normales d’équations differéntielles implicites et de champs de Liouville.) (French) Zbl 0854.35021

Summary: Consider the partial differential equation \(f(x, y(x), dy(x))= 0\), where \(f\) is a smooth real function on \(\mathbb{R}^n\times \mathbb{R}\times (\mathbb{R}^n)^*\). Near each singularity of the characteristic foliation, a Liouville field is associated to the equation; we classify hyperbolic germs of Liouville fields up to symplectic transformations, hence we deduce normal forms for partial differential equations up to transformations which preserve the standard contact form of \(\mathbb{R}^{2n+ 1}\). For \(n= 1\), a theorem of Davydov enables us to deduce normal forms for such equations up to transformations of the \(x\), \(y\) plane.

MSC:

35F20 Nonlinear first-order PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
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[1] Takens, Publications mathématiques I.H.E.S 43 pp 47– (1974)
[2] DOI: 10.2307/2372774 · Zbl 0083.31406 · doi:10.2307/2372774
[3] Chaperon, Astérisque 107?108 pp 259– (1983)
[4] Chaperon, Astérisque 138?139 pp none– (1986)
[5] DOI: 10.1007/BF01078387 · Zbl 0582.34056 · doi:10.1007/BF01078387
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