Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 016, 29 pages      arXiv:2101.07470      https://doi.org/10.3842/SIGMA.2023.016
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver

Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory

Primitivo Acosta-Humánez a, Moulay Barkatou b, Raquel Sánchez-Cauce c and Jacques-Arthur Weil b
a) Instituto de Matemática & Escuela de Matemática, Universidad Autónoma de Santo Domingo, Dominican Republic
b) XLim - Université de Limoges & CNRS, Limoges, France
c) Department of Artificial Intelligence, Universidad Nacional de Educación a Distancia (UNED), Madrid, Spain

Received July 21, 2022, in final form March 20, 2023; Published online March 31, 2023

Abstract
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 ($\mathfrak{so}(3, C_K)$ systems) as well as a framework to extend Darboux transformations to any symmetric power of $\mathrm{SL}(2,\mathbb{C})$-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.

Key words: Darboux transformations; differential Galois group; differential Galois theory; Frenet-Serret formulas; orthogonal differential systems; rigid solid problem; Schrödinger equation; shape invariant potentials; supersymmetric quantum mechanics; symmetric power; tensor product.

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