scholarly journals Liouvillian propagators, Riccati equation and differential Galois theory

2013 ◽  
Vol 46 (45) ◽  
pp. 455203 ◽  
Author(s):  
Primitivo Acosta-Humánez ◽  
Erwin Suazo
Keyword(s):  
Riccati Equation ◽  
Galois Theory ◽  
Differential Galois Theory

scholarly journals Differential Galois Theory and Lie Symmetries

2015 ◽  
Author(s):  
David Blázquez-Sanz ◽  
◽  
Juan J. Morales-Ruiz ◽  
Jacques-Arthur Weil ◽  
◽  
...  
Keyword(s):  
Galois Theory ◽  
Lie Symmetries ◽  
Differential Galois Theory

scholarly journals Rational KdV Potentials and Differential Galois Theory

2019 ◽  
Author(s):  
Sonia Jiménez ◽  
◽  
Juan J. Morales-Ruiz ◽  
Raquel Sánchez-Cauce ◽  
María-Ángeles Zurro ◽  
...  
Keyword(s):  
Galois Theory ◽  
Differential Galois Theory

scholarly journals Second proof of the irreducibility of the first differential equation of painlevé

1990 ◽  
Vol 117 ◽  
pp. 125-171 ◽  
Author(s):  
Hiroshi Umemura
Keyword(s):  
Differential Equation ◽  
General Theory ◽  
Galois Theory ◽  
Rigorous Proof ◽  
Mathematical Foundation ◽  
Differential Galois Theory ◽  
Infinite Dimensional ◽  
Finite Dimensional

In our paper [U2], we proved the irreducibility of the first differential equation y″ = 6y2 + x of Painlevé. In that paper we explained the origin of the problem and the importance of giving a rigorous proof. We can say that our method in [U2] is algebraic and finite dimensional in contrast to a prediction of Painlevé who expected a proof depending on the infinite dimensional differential Galois theory. Even nowadays the latter remains to be established. It seems that Painlevé needed an armament with the general theory (the infinite dimensional differential Galois theory) in the controversy with R. Liouville on the mathematical foundation of the proof of the irreducibility of the first differential equation (1902-03).

Meromorphic non-integrability of a steady Stokes flow inside a sphere

2012 ◽  
Vol 34 (2) ◽  
pp. 616-627 ◽  
Author(s):  
TAKAHIRO NISHIYAMA
Keyword(s):  
Stokes Flow ◽  
Computer Algebra System ◽  
Galois Theory ◽  
Type Equation ◽  
First Integral ◽  
Special Linear Group ◽  
Differential Galois Theory ◽  
Differential Galois Group ◽  
Special Values ◽  
Type Differential Equation

AbstractThe non-existence of a real meromorphic first integral for a spherically confined steady Stokes flow of Bajer and Moffatt is proved on the basis of Ziglin’s theory and the differential Galois theory. In the proof, the differential Galois group of a second-order Fuchsian-type differential equation associated with normal variations along a particular streamline is shown to be a special linear group according to Kovacic’s algorithm. A set of special values of a parameter contained in the Fuchsian-type equation is studied by using the theory of elliptic curves. For this set, a computer algebra system is used in part of Kovacic’s algorithm.

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