Noncommutative solitons and quasideterminants

We discuss an extension of soliton theory and integrable systems to noncommutative spaces, focusing on integrable aspects of noncommutative anti-self-dual Yang–Mills equations. We give a wide class of exact solutions by solving a Riemann–Hilbert problem for the Atiyah–Ward ansatz and present Bäcklund transformations for the G = U(2) noncommutative anti-self-dual Yang–Mills equations. We find that noncommutative determinants of one kind, quasideterminants, play crucial roles in the construction of noncommutative solutions. We also discuss the reduction of a noncommutative anti-self-dual Yang–Mills equation to noncommutative integrable equations. This work is partially based on a collaboration with C R Gilson and J J C Nimmo (of Glasgow). (comment)