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

In the Korteweg-de Vries equation (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.

Automated summary

In the context of the Korteweg-de Vries equation, we propose a continuous version of the binary Darboux transformation based on the Riemann-Hilbert problem. Our method provides a new explicit formula for perturbing the negative spectrum of step-type potentials while preserving the rest of the scattering data. This extends the applicability of previous formulas for inserting/removing bound states to arbitrary sets of negative spectrum.