Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2023
• Main Authors: Acosta-Humánez, Primitivo, Barkatou, Moulay, Sánchez-Cauce, Raquel, Weil, Jacques-Arthur
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2023
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211868
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory. Primitivo Acosta-Humánez, Moulay Barkatou, Raquel Sánchez-Cauce and Jacques-Arthur Weil. SIGMA 19 (2023), 016, 29 pages

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 that 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 ((3, ) 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.