Differential Galois theory and Darboux transformations for Integrable Systems

Jiménez, S.; Ruiz, Juan José Morales; Sánchez-Cauce, Raquel; Zurro, María-Angeles

doi:10.1016/j.geomphys.2016.06.016

Cited by 4 publications

(8 citation statements)

References 13 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%

“…Corollary 2 Using the fundamental matrix (25), the matrices in the differential Galois group of the so(3) system…”

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

confidence: 99%

“…A fundamental matrix for this system is given by matrix (25), where {y 1 , y 2 } is a basis of solutions of equation (7).…”

Section: Corollary 4 (Second Darboux Transformation Formentioning

confidence: 99%

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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%

“…Corollary 2 Using the fundamental matrix (25), the matrices in the differential Galois group of the so(3) system…”

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

confidence: 99%

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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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“…This paper is a contribution to the Special Issue on Algebraic Methods in Dynamical Systems. The full collection is available at https://www.emis.de/journals/SIGMA/AMDS2018.html Thus, in some sense this paper can be considered as a continuation of our previous paper [18], where we studied the invariance of the Galois group of the AKNS systems with respect to the Darboux transformations. But one of the essential differences here is that in general we can not use the Darboux invariance result in [18], because the Darboux transformation here is not a well-defined gauge transformation, i.e., it is not inversible.…”

Section: Introductionmentioning

confidence: 99%

“…The full collection is available at https://www.emis.de/journals/SIGMA/AMDS2018.html Thus, in some sense this paper can be considered as a continuation of our previous paper [18], where we studied the invariance of the Galois group of the AKNS systems with respect to the Darboux transformations. But one of the essential differences here is that in general we can not use the Darboux invariance result in [18], because the Darboux transformation here is not a well-defined gauge transformation, i.e., it is not inversible. Thus we must use the classical Darboux tranformation of the Schrödinger equation, we call it the Darboux-Crum transform; and then to verify the compatibility of this transform with the complete linear system (1.1).…”

Section: Introductionmentioning

confidence: 99%

Rational KdV Potentials and Differential Galois Theory

Énez¹,

Ruiz

Sánchez-Cauce

et al. 2019

SIGMA

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In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schrödinger equation for rational time dependent KdV potentials. In particular, we compute the fundamental matrices of the linear systems associated to the Schrödinger equation. Furthermore we prove the invariance of the Galois groups with respect to time, to generic values of the spectral parameter and to Darboux transformations.

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Integrability of the Zakharov-Shabat Systems by Quadrature

Yagasaki

2023

Commun. Math. Phys.

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Differential Galois theory and Darboux transformations for Integrable Systems

Cited by 4 publications

References 13 publications

Darboux Transformations for orthogonal differential systems and differential Galois Theory

Darboux Transformations for orthogonal differential systems and differential Galois Theory

Rational KdV Potentials and Differential Galois Theory

Integrability of the Zakharov-Shabat Systems by Quadrature

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