On the GBDT version of the B\"acklund-Darboux transformation and its applications to the linear and nonlinear equations and Weyl theory

TL;DR

This paper develops a general theorem for the GBDT version of the Bäcklund-Darboux transformation, applying it to solve linear and nonlinear equations, including Dirac systems and the N-wave equation, with new explicit solutions.

Contribution

It introduces a comprehensive theorem for GBDT Bäcklund-Darboux transformations and applies it to derive explicit solutions for various Dirac and wave systems.

Findings

Explicit solutions for Dirac-type systems with singularities

Construction of wave functions for radial Dirac equations

Applications to N-wave and nonlinear equations

Abstract

A general theorem on the GBDT version of the B\"acklund-Darboux transformation for systems rationally depending on the spectral parameter is treated and its applications to nonlinear equations are given. Explicit solutions of direct and inverse problems for Dirac-type systems, including systems with singularities, and for the system auxiliary to the $N$-wave equation are reviewed. New results on explicit construction of the wave functions for radial Dirac equation are obtained.