The Atiyah--Hitchin bracket for the cubic Nonlinear Schrodinger equation. IV. the scattering potentials

TL;DR

This paper explores the Poisson structure of scattering potentials in the cubic nonlinear Schrödinger equation, completing a series that develops a formalism for understanding its integrable properties.

Contribution

It introduces a novel Poisson formalism for scattering potentials in the NLS equation, extending previous work on the equation's integrable structure.

Findings

Develops a Poisson bracket for scattering potentials

Completes the series on the formalism for the NLS equation

Provides new insights into the integrable structure of the scattering data

Abstract

This is the last in a series of four papers on Poisson formalism for the cubic nonlinear Schrodinger equation with repulsive nonlinearity. In this paper we consider scattering potentials.