Inverse scattering problem for third order differential operators on the whole axis

TL;DR

This paper solves the inverse scattering problem for third order differential operators on the entire real axis, deriving a closed system of equations and identifying reflectionless potentials similar to solitons.

Contribution

It introduces a novel approach to the inverse scattering problem for third order operators, including a closed system of equations and explicit reflectionless solutions.

Findings

Derived a closed system of equations for the inverse problem

Identified the form of reflectionless potentials analogous to solitons

Provided a method to reconstruct potentials from scattering data

Abstract

Inverse scattering problem for an operator, which is a sum of the operator of the third derivative and of an operator of multiplication by a real function, is solved. The main closed system of equations of inverse problem is obtained. This system contains scattering coefficients and bound state elements as independent parameters. Form of the simplest reflectionless potential (analogous to soliton) is found.