Darboux transformations for orthogonal differential systems and differential Galois theory
Darboux transformations for orthogonal differential systems and differential Galois theory
Date
2023-03-31
Authors
Primitivo Acosta
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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 (so(3,CK) 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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Citation
Acosta, P. (2023, marzo 31). Darboux transformations for orthogonal differential systems and differential Galois theory. Universidad Autónoma de Santo Domingo, UASD