On ∂-problem and integrable equations

Using the ∂-problem and dual ∂-problem , we derive bilinear relations which allows us to construct integrable hierarchies in different parametrizations, their Darboux-Bäcklund transformations and to analyze constraints for them in a very simple way. Scalar KP, BKP and CKP hierarchies are considered as examples. There are different methods to construct integrable equations and to analyze their properties (see e.g. [1-4]). The ∂-dressing method proposed in [5] is, perhaps, one of the most effective of them. Recently, it has been applied successfully to several important problems in soliton theory (see e.g. [6-11]). In this letter we would like to attract an attention to one more profitable aspect of the ∂-dressing method. Namely, starting with ∂-problem and dual ∂-problem, we derive two important bilinear relations for the so-called Cauchy-Baker-Akhiezer (CBA) functions associated with different kernels R of the ∂-problem. These relations provide us simple variational relations for CBA functions and ∂-kernel R. In a simple unified manner, they generate integrable hierarchies in different parametrizations and corresponding bilinear Hirota identities. These bilinear relations are convenient also for analysis of different constraints. It is shown how scalar BKP and CKP hierarchies arise ∗E-mail: [email protected]