Bicomplexes and Bäcklund transformations

A bicomplex is a simple mathematical structure, in particular associated with completely integrable models. The conditions defining a bicomplex are a special form of a parameter-dependent zero-curvature condition. We generalize the concept of a Darboux matrix to bicomplexes and use it to derive Bäcklund transformations for several models. The method also works for Moyal-deformed equations with a corresponding deformed bicomplex.