A continuous analog of the binary Darboux transformation for the Korteweg-de Vries equation

TL;DR

This paper introduces a continuous binary Darboux transformation for the KdV equation, providing explicit formulas for spectral perturbations that extend previous finite-state methods to arbitrary negative spectra.

Contribution

It develops a continuous version of the binary Darboux transformation using Riemann-Hilbert techniques, enabling spectral modifications for a broad class of potentials.

Findings

Provides explicit formulas for spectral perturbations

Extends finite-state formulas to arbitrary negative spectra

Maintains scattering data unchanged

Abstract

In the KdV context we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann-Hilbert problem and yields a new explicit formula for perturbation of the negative spectrum of a wide class of step-type potentials without changing the rest of the scattering data. This extends the previously known formulas for inserting/removing finitely many bound states to arbitrary sets of negative spectrum of arbitrary nature. In the KdV context our method offers same benefits as the classical binary Darboux transformation does.