Noncommutative Solitons and Quasideterminants

Details Noncommutative Solitons and Quasideterminants Noncommutative Solitons and Quasideterminants

2010-12-29

arxiv.org/abs/1101.0005v2

Physics

High Energy Physics

High Energy Physics - Theory

22 pages. new year version based on invited talks at the RIMS International Conference on Geometry related to Integrable Syste

Scientific paper

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).

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