Picard–Vessiot theory and integrability

Ruiz, Juan José Morales

doi:10.1016/j.geomphys.2014.07.006

Cited by 22 publications

(10 citation statements)

References 54 publications

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“…In addition to the fact that they did not give details of their computation in [14,Prop. 13], see also [34], it would be difficult for us to elaborate on the dependence of the different set of coefficients, namely K 3 , · · · , K 0 from the (1.2).…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Galoisian approach to complex oscillation theory of some Hill equations

Chiang¹,

Yu²

2019

Math. Scand.

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We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponential of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. That are solutions obtainable by suitable differential field extensions construction. In particular, we have established a full correspondence between solutions of non-oscillatory type and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation whose potential is a combination of four exponential functions in the Bank-Laine theory. The differential equation is a periodic form of biconfluent Heun equation. We further show that these Liouvillian solutions exhibit novel single and double orthogonality and a Fredholm integral equation over suitable integration regions in C that mimic single/double orthogonality for the corresponding Liouvillian solutions of the Lamé and Whittaker-Hill equations, discovered by Whittaker and Ince almost a century ago.

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Section: Proof Of Theorem 12mentioning

confidence: 99%

Galoisian approach to complex oscillation theory of some Hill equations

Chiang¹,

Yu²

2019

Math. Scand.

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“…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]). Ayoul and Zung [2] extended this method to the study of some kind of non Hamiltonian fields.…”

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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“…Many scholars have devoted themselves to this topic and developed a lot of ways to study the existence of first integrals for given systems, such as the Lax pairs [6], the Painlevé analysis [7], the Lie symmetries [8] and the Darboux integrability theory [9]. Inspired by Ziglin's works [10], Morales-Ruiz et al [11][12][13][14][15][16] applied the differential Galois theory to the non-integrability of Hamiltonian systems with great success. Roughly speaking, the Morales-Ramis theory shows that if the Hamiltonian system with n degrees of freedom admits n meromorphic first integrals which are in involution and independent, then the identity component of the differential Galois group of the normal variational equation should be commutative.…”

Section: Introductionmentioning

confidence: 99%

Meromorphic Non-Integrability of Several 3D Dynamical Systems

Huang

Shi

2017

Entropy

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Abstract:In this paper, we apply the differential Galoisian approach to investigate the meromorphic non-integrability of a class of 3D equations in mathematical physics, including Nosé-Hoover equations, the Lü system, the Rikitake-like system and Rucklidge equations, which are well known in the fields of molecular dynamics, chaotic theory and fluid mechanics, respectively. Our main results show that all these considered systems are, in fact, non-integrable in nearly all parameters.

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Picard–Vessiot theory and integrability

Cited by 22 publications

References 54 publications

Galoisian approach to complex oscillation theory of some Hill equations

Galoisian approach to complex oscillation theory of some Hill equations

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

Meromorphic Non-Integrability of Several 3D Dynamical Systems

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