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Towards noncommutative integrable systems. (English) Zbl 1031.37046

Summary: We present a powerful method to generate various equations which possess the Lax representations on noncommutative (1+1)- and (1+2)-dimensional spaces. The generated equations contain noncommutative integrable equations obtained by using the bicomplex method and by reductions of the noncommutative (anti-)self-dual Yang-Mills equation. This suggests that the noncommutative Lax equations would be integrable and be derived from reductions of the noncommutative (anti-)self-dual Yang-Mills equation, which implies the noncommutative version of Richard Ward conjecture. The integrability and the relation to string theories are also discussed.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
81T75 Noncommutative geometry methods in quantum field theory
81T13 Yang-Mills and other gauge theories in quantum field theory
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