Manifold Ways to Darboux-Halphen System

Morales, John Alexander Cruz; Movasati, Hossein; Nikdelan, Younes; Roychowdhury, Raju; Torres, Marcus A. C.

doi:10.3842/sigma.2018.003

Cited by 4 publications

(2 citation statements)

References 26 publications

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“…See, for example, [20] for more recent discussions on this system. We introduce a new set of dependent variables…”

Section: Darboux-halphen System In Symmetric Polynomialsmentioning

confidence: 99%

$3D$-flows Generated by the Curl of a Vector Potential \& Maurer-Cartan Equations

Esen¹,

Guha²,

Gümral³

2021

Preprint

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We examine 3D flows ẋ = v(x) admitting vector identity M v = ∇×A for a multiplier M and a potential field A. It is established that, for those systems, one can complete the vector field v into a basis fitting an sl(2)-algebra. Accordingly, in terms of covariant quantities, the structure equations determine a set of equations in Maurer-Cartan form. This realization permits one to obtain the potential field as well as to investigate the (bi-)Hamiltonian character of the system. The latter occurs if the system has a time-independent first integral. In order to exhibit the theoretical results on some concrete cases, two examples are provided, namely the Gulliot system and the Darboux-Halphen system in symmetric polynomials.

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“…See, for example, [20] for more recent discussions on this system. We introduce a new set of dependent variables…”

Section: Darboux-halphen System In Symmetric Polynomialsmentioning

confidence: 99%

$3D$-flows Generated by the Curl of a Vector Potential \& Maurer-Cartan Equations

Esen¹,

Guha²,

Gümral³

2021

Preprint

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“…Because of this difference, it may be difficult to obtain various concrete expressions for these levels as in this paper. Note also that in the case of level 1, the Atkin polynomials and the normalized extremal quasimodular forms are directly connected as in (24), but this is not necessarily the case for level ≥ 2.…”

Section: Further Directionsmentioning

confidence: 99%

Extremal Quasimodular Forms of Lower Depth With Integral Fourier Coefficients

Kaminaka¹,

Kato²

2021

Kyushu J. Math.

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We show that, based on Grabner's recent results on modular differential equations satisfied by quasimodular forms, there exist only finitely many normalized extremal quasimodular forms of depth r that have all Fourier coefficients integral for each of r = 1, 2, 3 and 4, and partly classifies them, where the classification is complete for r = 2, 3 and 4. In fact, we show that there exists no normalized extremal quasimodular forms of depth four with all Fourier coefficients integral. Our result disproves a conjecture by Pellarin.

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On the orthogonality of Atkin-like polynomials and orthogonal polynomial expansion of generalized Faber polynomials

Nakaya

2024

Ramanujan J

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Manifold Ways to Darboux-Halphen System

Cited by 4 publications

References 26 publications

$3D$-flows Generated by the Curl of a Vector Potential \& Maurer-Cartan Equations

$3D$-flows Generated by the Curl of a Vector Potential \& Maurer-Cartan Equations

Extremal Quasimodular Forms of Lower Depth With Integral Fourier Coefficients

On the orthogonality of Atkin-like polynomials and orthogonal polynomial expansion of generalized Faber polynomials

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