Inverse scattering for reflectionless Schr\"odinger operators with integrable potentials and generalized soliton solutions for the KdV equation

TL;DR

This paper characterizes reflectionless Schrödinger operators with integrable potentials, solves the inverse scattering problem for reconstructing these potentials, and derives generalized soliton solutions for the KdV equation.

Contribution

It provides a complete characterization and reconstruction method for reflectionless Schrödinger operators with integrable potentials, extending soliton solutions of the KdV equation.

Findings

Complete characterization of reflectionless Schrödinger operators

Solution to inverse scattering problem for these operators

Derivation of generalized soliton solutions for KdV

Abstract

We give a complete characterisation of the reflectionless Schr\"odinger operators on the line with integrable potentials, solve the inverse scattering problem of reconstructing such potentials from the eigenvalues and norming constants, and derive the corresponding generalized soliton solutions of the Korteweg--de Vries equation