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

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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.

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A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I

A. B. Shabat, V. E. Zakharov (1975)

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KP solitons, total positivity, and cluster algebras

Lauren N. Williams, Yuji Kodama (2011)

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to the KP equation provides a method to construct soliton solutions. The regular soliton solutions that one obtains in this way come from points of the totally non-negative part of the Grassmannian. In this paper we explain how the theory of total positivity and cluster algebras provides a framework for understanding these soliton solutions to the KP equation. We then use this framework to give an explicit construction of certain soliton contour graphs, and solve the inverse problem for soliton solutions coming from the totally positive part of the Grassmannian.