Rational KdV Potentials and Differential Galois Theory

Énez, Sonia Jim; Ruiz, Juan José Morales; Sánchez-Cauce, Raquel; Zurro, María-Angeles

doi:10.3842/sigma.2019.047

Cited by 3 publications

(5 citation statements)

References 21 publications

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“…Recently, a Galoisian approach to Darboux transformations and shape invariant potentials has been proposed in [1,2,4], where it was proved that the Darboux transformation preserves the galoisian structure of the differential equation (the Darboux transformation is isogaloisian). A similar approach was presented in [25,26,27] in the context of integrable systems. There, the authors studied the behavior of the galoisian structure of some families of linear systems with respect to Darboux transformations.…”

Section: Introductionmentioning

confidence: 89%

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Darboux Transformations for orthogonal differential systems and differential Galois Theory

Acosta-Humánez¹,

Barkatou²,

Sánchez-Cauce³

et al. 2021

Preprint

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Darboux developed an algebraic mechanism to construct an infinite chain of "integrable" second order differential equations as well as their solutions. After a surprisingly long time, Darboux's results had important features in the analytic context, for instance in quantum mechanics where it provides a convenient framework for Supersymmetric Quantum Mechanics. Today, there are a lot of papers regarding the use of Darboux transformations in various contexts, not only in mathematical physics. In this paper, we develop a generalization of the Darboux transformations for tensor product constructions on linear differential equations or systems. Moreover, we provide explicit Darboux transformations for sym 2 (SL(2, C)) systems and, as a consequence, also for so(3, C K ) systems, to construct an infinite chain of integrable (in Galois sense) linear differential systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials.

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Section: Introductionmentioning

confidence: 89%

“…, where u and q have the explicit form expressed in Theorem 1. Thus, since Y = Sym 2 ( X) = Sym 2 (P m )•Sym 2 (X) = Sym 2 (P m ) • Y , the gauge transformation (20) also induces a transformation in the second symmetric power systems which sends system (27) to system (28). The following result formalizes this idea.…”

Section: Darboux Transformations Inmentioning

confidence: 99%

Darboux Transformations for orthogonal differential systems and differential Galois Theory

Acosta-Humánez¹,

Barkatou²,

Sánchez-Cauce³

et al. 2021

Preprint

Self Cite

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“…For Schödinger equations, this has been observed in [1,2,4] in a Galoisian approach. For systems, such as AKNS systems, this approach can be found in the reference book of Gu, Hu and Zhu [27, Section 1.3, p. 18] and in papers such as [29,30,31,49]. We now review this observation in a way that will allow us to construct our generalizations of Darboux transformations.…”

Section: Matrix Formalism Darboux Transformation As a Gauge Transform...mentioning

confidence: 99%

“…Morales-Ruiz, R. Sánchez-Cauce and M.-A. Zurro in [29,30,31] in the context of integrable systems and rational solitons of KdV equations. There, the authors studied the behavior of the Galoisian structure of some families of linear systems with respect to Darboux transformations.…”

Section: Introductionmentioning

confidence: 99%

“…For systems, such as AKNS systems, this approach can be found in the reference book of Gu, Hu and Zhu [27,Section 1.3,p. 18] and is used in papers such as [49] or [29,30,31]. The contribution of this paper is to show how, viewing the Darboux transformation as a gauge transformation, tools from the Galois theory of differential equations (the Tannakian approach) allow us to construct Darboux transformations for linear differential equations or systems of order higher than two when they arise as constructions on an sl(2)-system, a situation that occurs in physical systems.…”

Section: Introductionmentioning

confidence: 99%

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Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory

Acosta-Humánez¹,

Barkatou

Sánchez-Cauce

et al. 2023

SIGMA

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Darboux developed an ingenious algebraic mechanism to construct infinite chains of "integrable" second-order differential equations as well as their solutions. After a surprisingly long time, Darboux's results were rediscovered and applied in many frameworks, for instance in quantum mechanics (where they provide useful tools for supersymmetric quantum mechanics), in soliton theory, Lax pairs and many other fields involving hierarchies of equations. In this paper, we propose a method which allows us to generalize the Darboux transformations algorithmically for tensor product constructions on linear differential equations or systems. We obtain explicit Darboux transformations for third-order orthogonal systems (so(3, C K ) systems) as well as a framework to extend Darboux transformations to any symmetric power of SL(2, C)-systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials. All results in this paper have been implemented and tested in the computer algebra system Maple.

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Semiclassical quantification of some two degree of freedom potentials: A differential Galois approach

Acosta-Humánez,

Lázaro,

Morales-Ruiz

et al. 2024

Journal of Mathematical Physics

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In this work we explain the relevance of the Differential Galois Theory in the semiclassical (or WKB) quantification of some two degree of freedom potentials. The key point is that the semiclassical path integral quantification around a particular solution depends on the variational equation around that solution: a very well-known object in dynamical systems and variational calculus. Then, as the variational equation is a linear ordinary differential system, it is possible to apply the Differential Galois Theory to study its solvability in closed form. We obtain closed form solutions for the semiclassical quantum fluctuations around constant velocity solutions for some systems like the classical Hermite/Verhulst, Bessel, Legendre, and Lamé potentials. We remark that some of the systems studied are not integrable, in the Liouville–Arnold sense.

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Rational KdV Potentials and Differential Galois Theory

Cited by 3 publications

References 21 publications

Darboux Transformations for orthogonal differential systems and differential Galois Theory

Darboux Transformations for orthogonal differential systems and differential Galois Theory

Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory

Semiclassical quantification of some two degree of freedom potentials: A differential Galois approach

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