Noncommutative Solitons and Quasideterminants

Abstract

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