Noncommutative Integrable Field

We study the noncommutative generalization of (euclidean) integrable models in two-dimensions, specifically the sine-and sinh-Gordon and the U (N) principal chiral models. By looking at tree-level amplitudes for the sinh-Gordon model we show that its na¨ıve noncommutative generalization is not integrable. On the other hand, the addition of extra constraints, obtained through the generalization of the zero-curvature method, renders the model integrable. We construct explicit non-local non-trivial conserved charges for the U (N) principal chiral model using the Brezin-Itzykson-Zinn-Justin-Zuber method.