scholarly journals Lie and Lie Backlund Vector Fields and Painleve Test for a Class of Scale Invariant Partial Differential Equations of First Order

1987 ◽  
Vol 78 (2) ◽  
pp. 214-223
Author(s):  
W. - H. Steeb ◽  
N. Euler
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Fields ◽  
Painlevé Test ◽  
Scale Invariant ◽  
Partial Differential ◽  
First Order

scholarly journals SYMMETRIES AND FIRST INTEGRALS FOR NON-VARIATIONAL EQUATIONS

2007 ◽  
Vol 04 (07) ◽  
pp. 1217-1230
Author(s):  
DIEGO CATALANO FERRAIOLI ◽  
PAOLA MORANDO
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vector Fields ◽  
First Integrals ◽  
Partial Differential ◽  
Variational Equations ◽  
First Order ◽  
The Ideal

For a class of exterior ideals, we present a method associating first integrals of the characteristic distributions to symmetries of the ideal. The method is applied, under some assumptions, to the study of first integrals of ordinary differential equations and first order partial differential equations as well as to the determination of first integrals for integrable distributions of vector fields.

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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