View abstract

Session III.6 - Symbolic Analysis

Monday, June 19, 17:30 ~ 18:00

Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory

Jaques-Arthur Weil

Université de Limoges, France   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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 work, 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.

Joint work with Primitivo Acosta-Humanez (Universidad Autónoma de Santo Domingo, Dominican Republic), Moulay Barkatou (Université de Limoges, France), Raquel Sanchez-Cauce (Universidad Nacional de Educación a Distancia, Spain).

View abstract PDF