Discrete canonical system and non‐Abelian Toda lattice: Bäcklund–Darboux transformation, Weyl functions, and explicit solutions

Abstract

A version of the iterated Bäcklund-Darboux transformation, where Darboux matrix takes a form of the transfer matrix function from the system theory, is constructed for the discrete canonical system and Non-Abelian Toda lattice. Results on the transformations of the Weyl functions, insertion of the eigenvalues, and construction of the bound states are obtained. A wide class of the explicit solutions is given. An application to the semi-infinite block Jacobi matrices is treated.

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

“…Thus, GBDT is a convenient tool to construct wave functions and explicit solutions of the nonlinear wave equations as well as to solve various direct and inverse problems. GBDT and its applications were treated or included as important examples in the papers [22,23,48,55,56,57,58,59,61,63,64,66,67,68,70,71,72,73,75] (see also [28,29,30,31,32,33,37]). Here we consider self-adjoint and skew-self-adjoint Dirac-type systems including the singular case corresponding to soliton-positon interaction and solve direct and inverse problems.…”

Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators

“…also [29]). Finally, connections with nonabelian completely integrable systems are discussed in [5,6,44,48], [49,Chapters 9,10].…”

Continuous and discrete dynamical Schrödinger systems: explicit solutions

“…The original equation, which was studied by Darboux, is the Schrödinger equation Later, and especially in the last 40 years, this transformation was greatly modified, generalized and applied to a variety of linear and nonlinear equations (see, e.g., [5,14,19,20,30]). It was shown recently in [28,29] that the GBDT version of Bäcklund-Darboux transformation (for GBDT see [22][23][24][25][26]30] and references therein) may be successfully applied to the construction of explicit solutions of dynamical systems as well.…”

“…Jacobi matrices corresponding to explicit solutions of matrix Toda lattices were considered in [26,Appendix]. Using some modification of the results from [26,Appendix], we construct here explicit solutions of discrete dynamical Schrödinger systems. We present also direct proofs of the corresponding modified results from [26,Appendix], whereas in [26,Appendix] several essential facts are proved indirectly (via the theory of discrete canonical systems developed in the previous sections of [26]) and some details of the proofs are omitted.…”

“…We note that formulas (4.12) and (4.13) coincide (after removal of tildes) with the formulas (4.2) and (4.3) which define J . According to [26,Appendix], we have b k = b * k . Slightly modifying the proof of [26, Theorem A.1], one may derive that under condition…”