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

TL;DR

This paper defines the Cauchy-Jost function for pure solitonic solutions of KP-II, explores its properties, and links Darboux transformations to total nonnegativity in Grassmannians, advancing soliton theory and matrix analysis.

Contribution

It introduces a direct definition of the Cauchy-Jost function for solitons and connects Darboux transformations with total nonnegativity in Grassmannians.

Findings

Darboux transformations can be expressed via the Cauchy-Jost function.

The action of Darboux transformations on Grassmannians is characterized.

Total nonnegativity of matrices relates to soliton parameter spaces.

Abstract

Direct definition of the Cauchy-Jost (known also as Cauchy-Baker-Akhiezer) function in the case of pure solitonic solution is given and properties of this function are discussed in detail using the Kadomtsev-Petviashvili II equation as 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 relation of the Darboux transformations with property of total nonnegativity of elements of corresponding Grassmanians is discussed.