KPII: Cauchy-Jost function, Darboux transformations and totally nonnegative matrices

Direct definition of the Cauchy-Jost (known also as Cauchy-Baker-Akhiezer) function is given in the case of a pure solitonic solution. Properties of this function are discussed in detail using the Kadomtsev-Petviashvili II equation as an example. This enables formulation of the Darboux transformations in terms of the Cauchy-Jost function and classification of these transformations. Action of Darboux transformations on Grassmanians-i. e. on the space of soliton parameters-is derived and the relation of the Darboux transformations with the property of total nonnegativity of elements of corresponding Grassmanians is discussed.