scholarly journals Manifold Ways to Darboux-Halphen System

2018 ◽  
Author(s):  
John Alexander Cruz Morales ◽  
◽  
Hossein Movasati ◽  
Younes Nikdelan ◽  
Raju Roychowdhury ◽  
...  
Keyword(s):  
Halphen System

scholarly journals Nonintegrabilty of a Halphen system

2014 ◽  
Vol 181 (2) ◽  
pp. 1383-1395
Author(s):  
J. Llibre ◽  
C. Valls
Keyword(s):  
Halphen System

scholarly journals Bianchi-IX, Darboux–Halphen and Chazy–Ramanujan

2016 ◽  
Vol 13 (04) ◽  
pp. 1650042 ◽  
Author(s):  
Sumanto Chanda ◽  
Partha Guha ◽  
Raju Roychowdhury
Keyword(s):  
Vacuum Einstein Equation ◽  
Yang Mills ◽  
Gravitational Instanton ◽  
Self Duality ◽  
Bianchi Ix ◽  
Euler Top ◽  
Integrable Dynamical Systems ◽  
Halphen System ◽  
Special Case ◽  
Dual Case

Bianchi-IX four metrics are SU(2) invariant solutions of vacuum Einstein equation, for which the connection-wise self-dual case describes the Euler top, while the curvature-wise self-dual case yields the Ricci flat classical Darboux–Halphen system. It is possible to see such a solution exhibiting Ricci flow. The classical Darboux–Halphen system is a special case of the generalized one that arises from a reduction of the self-dual Yang–Mills equation and the solutions to the related homogeneous quadratic differential equations provide the desired metric. A few integrable and near-integrable dynamical systems related to the Darboux–Halphen system and occurring in the study of Bianchi-IX gravitational instanton have been listed as well. We explore in details whether self-duality implies integrability.

scholarly journals On the algebraic non-integrability of Halphen system

1995 ◽  
Vol 201 (2-3) ◽  
pp. 161-166 ◽  
Author(s):  
Andrzej J. Maciejewski ◽  
Jean-Marie Strelcyn
Keyword(s):  
Halphen System

Anharmonic solutions to the Riccati equation and elliptic modular functions

2018 ◽  
Vol 30 (2) ◽  
pp. 433-455
Author(s):  
Ahmed Sebbar ◽  
Oumar Wone
Keyword(s):  
Riccati Equation ◽  
Algebraic Equation ◽  
Modular Forms ◽  
Irreducible Polynomial ◽  
Polynomial Equation ◽  
Algebraic Solution ◽  
Modular Functions ◽  
Riccati Differential Equation ◽  
Differential Field ◽  
Halphen System

AbstractWe study the irreducible algebraic equationx^{n}+a_{1}x^{n-1}+\cdots+a_{n}=0,\quad\text{with ${n\geq 4}$,}on the differential field{(\mathbb{F}=\mathbb{C}(t),\delta=\frac{d}{dt})}. We assume that a root of the equation is a solution to the Riccati differential equation{u^{\prime}+B_{0}+B_{1}u+B_{2}u^{2}=0}, where{B_{0}},{B_{1}},{B_{2}}are in{\mathbb{F}}.We show how to construct a large class of polynomials as in the above algebraic equation, i.e., we prove that there exists a polynomial{F_{n}(x,y)\in\mathbb{C}(x)[y]}such that for almost{T\in\mathbb{F}\setminus\mathbb{C}}, the algebraic equation{F_{n}(x,T)=0}is of the same type as the above stated algebraic equation. In other words, all its roots are solutions to the same Riccati equation. On the other hand, we give an example of a degree 3 irreducible polynomial equation satisfied by certain weight 2 modular forms for the subgroup{\Gamma(2)}, whose roots satisfy a common Riccati equation on the differential field{(\mathbb{C}(E_{2},E_{4},E_{6}),\frac{d}{d\tau})}, with{E_{i}(\tau)}being the Eisenstein series of weighti. These solutions are related to a Darboux–Halphen system. Finally, we deal with the following problem: For which “potential”{q\in\mathbb{C}(\wp,\wp^{\prime})}does the Riccati equation{\frac{dY}{dz}+Y^{2}=q}admit algebraic solutions over the differential field{\mathbb{C}(\wp,\wp^{\prime})}, with{\wp}being the classical Weierstrass function? We study this problem via Darboux polynomials and invariant theory and show that the minimal polynomial{\Phi(x)}of an algebraic solutionumust have a vanishing fourth transvectant{\tau_{4}(\Phi)(x)}.

scholarly journals First integrals of a generalized Darboux–Halphen system

2003 ◽  
Vol 44 (4) ◽  
pp. 1751-1762 ◽  
Author(s):  
S. Chakravarty ◽  
R. Halburd
Keyword(s):  
First Integrals ◽  
Halphen System

scholarly journals On a reduction of the generalized Darboux–Halphen system

2018 ◽  
Vol 382 (7) ◽  
pp. 455-460 ◽  
Author(s):  
Sumanto Chanda ◽  
Sarbarish Chakravarty ◽  
Partha Guha
Keyword(s):  
Halphen System
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