scholarly journals Differential Galois theory and Darboux transformations for Integrable Systems

2017 ◽  
Vol 115 ◽  
pp. 75-88
Author(s):  
Sonia Jiménez ◽  
Juan J. Morales-Ruiz ◽  
Raquel Sánchez-Cauce ◽  
María-Angeles Zurro
Keyword(s):  
Integrable Systems ◽  
Galois Theory ◽  
Differential Galois Theory ◽  
Darboux Transformations

scholarly journals DIFFERENTIAL GALOIS THEORY AND INTEGRABILITY

2009 ◽  
Vol 06 (08) ◽  
pp. 1357-1390 ◽  
Author(s):  
ANDRZEJ J. MACIEJEWSKI ◽  
MARIA PRZYBYLSKA
Keyword(s):  
Galois Group ◽  
Hamiltonian Systems ◽  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Galois Theory ◽  
Differential Galois Theory ◽  
Differential Galois Group ◽  
Variational Equations ◽  
Homogeneous Potentials ◽  
Particular Solution

This paper is an overview of our works that are related to investigations of the integrability of natural Hamiltonian systems with homogeneous potentials and Newton's equations with homogeneous velocity independent forces. The two types of integrability obstructions for these systems are presented. The first, local ones, are related to the analysis of the differential Galois group of variational equations along a non-equilibrium particular solution. The second, global ones, are obtained from the simultaneous analysis of variational equations related to all particular solutions belonging to a certain class. The marriage of these two types of the integrability obstructions enables to realize the classification programme of all integrable homogeneous systems. The main steps of the integrability analysis for systems with two and more degrees of freedom as well as new integrable systems are shown.

scholarly journals Quantum integrable systems and differential Galois theory

1997 ◽  
Vol 2 (1) ◽  
pp. 31-56 ◽  
Author(s):  
A. Braverman ◽  
P. Etingof ◽  
D. Gaitsgory
Keyword(s):  
Integrable Systems ◽  
Galois Theory ◽  
Differential Galois Theory ◽  
Quantum Integrable Systems

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).

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