Generalized Darboux and Darboux-Levi transformations in 2+1 dimensions

Solution generating techniques for 2+I-dimensional nonlinear integrable systems given by the integrability condition of linear problems (Lax pairs) are presented. According to certain symmetries of these linear problems a distinction between generalized Darboux and Darboux-Levi transformations is made. In the 1+1-dimensional limit the link to twisted and untwisted Kac-Moody algebras as prolongation algebras and the well-known N-soliton Ansatz is discussed. It is shown that the Moutard theorem and the dromion solutions for the Davey-Stewartson equation I are contained within this approach. Moreover, the applicability of an extended version of the generalized Darboux-Levi transformation to a Loewner-type system is demonstrated which leads to localized soli tonic solutions of a 2+1-dimensional sine-Gordon system (Konopelchenko-Rogers equations).