Non-commutative Wardʹs conjecture and integrable systems Original Research Article

Non-commutative Wardʹs conjecture is a non-commutative version of the original Wardʹs conjecture which says that almost all integrable equations can be obtained from anti-self-dual Yang–Mills equations by reduction. In this paper, we prove that wide class of non-commutative integrable equations in both image- and image-dimensions are actually reductions of non-commutative anti-self-dual Yang–Mills equations with finite gauge groups, which include non-commutative versions of Calogero–Bogoyavlenskii–Schiff equation, Zakharov system, Wardʹs chiral and topological chiral models, (modified) Korteweg–de Vries, non-linear Schrödinger, Boussinesq, N-wave, (affine) Toda, sine-Gordon, Liouville, Tzitzéica, (Wardʹs) harmonic map equations, and so on. This would guarantee existence of twistor description of them and the corresponding physical situations in image string theory, and lead to fruitful applications to non-commutative integrable systems and string theories. Some integrable aspects of them are also discussed.