On Reductions of Noncommutative Anti-Self-Dual Yang-Mills Equations

In this paper, we show that various noncommutative integrable equations can be derived from noncommutative anti-self-dual Yang-Mills equations in the split signature, which include noncommutative versions of Korteweg-de Vries, Non-Linear Schroedinger, N-wave, Davey-Stewartson and Kadomtsev-Petviashvili equations. U(1) part of gauge groups for the original Yang-Mills equations play crucial roles in noncommutative extension of Mason-Sparling’s celebrated discussion. The present results would be strong evidences for noncommutative Ward’s conjecture and imply that these noncommutative integrable equations could have the corresponding physical pictures such as reduced configurations of D0-D4 brane systems in open N=2 string theories. Possible applications to the D-brane dynamics are also discussed.