On the GBDT Version of the Bäcklund-Darboux Transformation and its Applications to Linear and Nonlinear Equations and Weyl Theory

Abstract

Abstract. A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systems depending rationally 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.

Dynamical canonical systems and their explicit solutions

“…We could also study (in the spirit of [28]) blow up solutions with singularities, which appear, if we omit the requirement S(0) > 0. We mention that GBDT was successfully applied for the construction of explicit solutions of nonlinear dynamical systems as well (see, e.g., various references in [10,26,29,32]).…”

Dynamical and spectral Dirac systems: response function and inverse problems

“…explicit recovery of the potential Explicit construction of the Weyl function (direct problem) and explicit recovery of the potential V of the spectral Dirac system (1.1) from the Weyl function (inverse problem) were dealt with in [23,31,32,51]. We shall use the formulation of these results (for the case m 1 = m 2 = k) from [51, Subsection 5.1.1].…”

“…In that case φ is the Weyl function ϕ H of some spectral Dirac system with a pseudo-exponential potential, and so matrices α and ϑ i are recovered from ϕ H following the procedure from [51,Theorem 5.4] (see also references therein).…”

Generalized Bäcklund-Darboux transformation (GBDT): conservation laws, rational extensions and bispectrality

“…(see the paper [13], the reviews [14,15] and Chapter 7 in [16] as well as numerous references therein). The coefficients q k and q sk of the transformed (rational) matrix function G(x, z) are expressed via the coefficients q k and q sk of G(x, z) and matrix functions S(x), Π 1 (x) and Π 2 (x).…”