KPII: Cauchy–Jost function, Darboux transformations and totally nonnegative matrices**To the memory of our friend and colleague Peter P Kulish

Abstract

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.