scholarly journals Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields

2005 ◽  
Vol 57 (2) ◽  
pp. 415-460 ◽  
Author(s):  
Sunao ŌUCHI
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Fields ◽  
Normal Forms ◽  
Borel Summability ◽  
Partial Differential ◽  
First Order ◽  
Formal Solutions

A remark on Liouville vector fields and a theorem of Manouchehri

1999 ◽  
Vol 19 (4) ◽  
pp. 895-899
Author(s):  
MARC CHAPERON
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Recent Article ◽  
Vector Fields ◽  
Normal Forms ◽  
Local Version ◽  
Volume Form ◽  
Partial Differential ◽  
First Order ◽  
Symplectic Forms

In a recent article, Manouchehri proved a ‘Sternberg theorem’ for Liouville vector fields and noticed that it provided normal forms for implicit differential equations and first-order partial differential equations. We establish local and global versions of Moser's celebrated result on volume and symplectic forms when they admit a non-trivial one parameter (pseudo-) group of homotheties—by definition, a Liouville field is the generator of such a flow. The local version implies that two germs of Liouville fields of a symplectic or volume form are conjugate by a diffeomorphism germ which preserves the form if and only if they are conjugate. This contains Manouchehri's theorem.

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