On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

Acosta-Humánez, Primitivo B.; Lázaro, J. Tomás; Ruiz, Juan José Morales; Pantazi, Chara

doi:10.3934/dcds.2015.35.1767

Cited by 22 publications

(24 citation statements)

References 46 publications

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“…To accomplish our purposes, we are interested in second-order differential equations of the form see [10]. Jerald Kovacic developed in 1986 an algorithm to solve explicitly second-order differential equations with rational coefficients given in the form of equation (2.25), see [30].…”

Section: Differential Galois Theorymentioning

confidence: 99%

Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory

Acosta-Humánez¹,

Acosta-Humánez²,

Tuiran³

2018

SIGMA

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In this paper we start with proving that the Schrödinger equation (SE) with the classical 12 − 6 Lennard-Jones (L-J) potential is nonintegrable in the sense of the differential Galois theory (DGT), for any value of energy; i.e., there are no solutions in closed form for such differential equation. We study the 10−6 potential through DGT and SUSYQM; being it one of the two partner potentials built with a superpotential of the form w(r) ∝ 1/r 5 . We also find that it is integrable in the sense of DGT for zero energy. A first analysis of the applicability and physical consequences of the model is carried out in terms of the so called De Boer principle of corresponding states. A comparison of the second virial coefficient B(T ) for both potentials shows a good agreement for low temperatures. As a consequence of these results we propose the 10 − 6 potential as an integrable alternative to be applied in further studies instead of the original 12 − 6 L-J potential. Finally we study through DGT and SUSYQM the integrability of the SE with a generalized (2ν − 2) − ν L-J potential. This analysis do not include the study of square integrable wave functions, excited states and energies different than zero for the generalization of L-J potentials.

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Section: Differential Galois Theorymentioning

confidence: 99%

Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory

Acosta-Humánez¹,

Acosta-Humánez²,

Tuiran³

2018

SIGMA

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“…The problem of the integrability of planar vector fields has attracted the attention of many mathematicians during decades. Among the different approaches, the Galois theory of linear differential equations has played an important rôle in its understanding, even in the a priori (so far) simpler case of polynomial vector fields (see [4,18,1] and references therein). For instance, the application of differential Galois theory to variational equations along a given integral curve constitutes a powerful criterium of non-integrability for Hamiltonian systems (see [14]).…”

Section: Introductionmentioning

confidence: 99%

Differential Galois theory and non-integrability of planar polynomial vector fields

Acosta-Humánez

Lázaro

Ruiz

et al. 2018

Journal of Differential Equations

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We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check wether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the "Risch algorithm". In this way we point out the connection of the non integrablity with some higher transcendent functions, like the error function.

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“…A writing errata presented in one of these problems was corrected in [1]. This problem in correct form was used and studied on [8]. The differential equation system asociated of this problem called "Polyanin-Zaitsev vector field" [1], has as associated foliation a Lienard equation, that is to say, it is closely related to a problem of type Van Der Pol.…”

Section: Introductionmentioning

confidence: 99%

Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields

2018

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The analysis of dynamical systems has been a topic of great interest for researches mathematical sciences for a long times. The implementation of several devices and tools have been useful in the finding of solutions as well to describe common behaviors of parametric families of these systems. In this paper we study deeply a particular parametric family of differential equations, the so-called Linear Polyanin-Zaitsev Vector Field, which has been introduced in a general case in [1] as a correction of a family presented in [2]. Linear Polyanin-Zaitsev Vector Field is transformed into a Liénard equation and in particular we obtain the Van Der Pol equation. We present some algebraic and qualitative results to illustrate some interactions between algebra and the qualitative theory of differential equations in this parametric family.

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On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory

Cited by 22 publications

References 46 publications

Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory

Generalized Lennard-Jones Potentials, SUSYQM and Differential Galois Theory

Differential Galois theory and non-integrability of planar polynomial vector fields

Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields

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