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 previously been explored using different approaches ranging from the dressing method to twisting transformations and the $\tau$-function formulation. All these approaches proved useful for displaying different properties of these solutions and the corresponding Jost solutions. The aim of our investigation is to establish explicit formulas relating all these approaches. We discuss some hidden invariance properties of these multisoliton solutions.