Yang--Baxter maps of KdV, NLS and DNLS type on division rings

Abstract

We construct nocommutative set-theoretical solutions to the Yang--Baxter equation related to the KdV, the NLS and the derivative NLS equations. In particular, we construct several Yang--Baxter maps of KdV type and we show that one of them is completely integrable in the Liouville sense. Then, we construct a noncommutative KdV type Yang--Baxter map which can be squeezed down to the noncommutative discrete potential KdV equation. Moreover, we construct Darboux transformations for the noncommutative derivative NLS equation. Finally, we consider matrix refactorisation problems for noncommutative Darboux matrices associated with the NLS and the derivative NLS equation and we construct noncommutative maps. We prove that the latter are solutions to the Yang--Baxter equation.