On the equivalence of different approaches for generating multisoliton solutions of the KPII equation

The unexpectedly rich structure of the multisoliton solutions of the KPII equation has been explored by using different approaches, running from dressing method to twisting transformations and to the tau-function formulation. All these approaches proved to be useful in order to display different properties of these solutions and their related Jost solutions. The aim of this paper is to establish the explicit formulae relating all these approaches. In addition some hidden invariance properties of these multisoliton solutions are discussed.