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 Schrödinger, 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 Dbrane configurations of D0-D4 brane systems in open N=2 string theories. The BPS conditions of the reduced D-brane configurations would lead to integrabilities of the corresponding noncommutative equations. Possible applications to the D-brane dynamics are also discussed. E-mail: [email protected]