The inverse scattering transform for weak Wigner-von Neumann type potentials

TL;DR

This paper extends the inverse scattering transform for the Korteweg-de Vries equation to include initial data with Wigner-von Neumann type potentials, using Hankel operator theory, broadening the class of solvable problems.

Contribution

It introduces a novel extension of the inverse scattering transform to weak Wigner-von Neumann type potentials, leveraging Hankel operators.

Findings

Successfully extended inverse scattering to new potential class

Demonstrated applicability to initial data with specific asymptotics

Provided analytical framework for future studies

Abstract

In the context of the Cauchy problem for the Korteweg-de Vries equation we extend the inverse scattering transform to initial data that behave at plus infinity like a sum of Wigner-von Neumann type potentials with small coupling constants. Our arguments are based on the theory of Hankel operators.