Dynamical canonical systems and their explicit solutions

Abstract

Dynamical canonical systems and their connections with the classical (spectral) canonical systems are considered. We construct Bäcklund-Darboux transformation and explicit solutions of the dynamical canonical systems. We study also those properties of the solutions, which are of interest in evolution and control theories. MSC(2010): 35B06, 37C80, 37D99.

Generalised canonical systems related to matrix string equations: corresponding structured operators and high-energy asymptotics of the Weyl functions

“…Canonical systems are important objects of analysis, being perhaps the most important class of the one-dimensional Hamiltonian systems and including (as subclasses) several classical equations. They have been actively studied in many already classical as well as in various recent works (see, e.g., [1,5,7,12,13,18,20,[22][23][24][25][30][31][32][36][37][38]41,42] and numerous references therein).…”

On a class of canonical systems corresponding to matrix string equations: general-type and explicit fundamental solutions and Weyl--Titchmarsh theory

“…Various versions of Bäcklund-Darboux transformations and related dressing and commutation methods [6,7,16,18,22,27,31,32,54] are fruitful tools in the construction of explicit solutions of linear and integrable nonlinear equations. Bäcklund-Darboux transformations for canonical and dynamical canonical systems, respectively, were constructed in [39] and [42]. More precisely, GBDT (generalized Bäcklund-Darboux transformation) was constructed for these systems.…”

Dynamics of electrons and explicit solutions of Dirac–Weyl systems

“…However, a more complicated than ψ(x, y) = e ikyψ (x) dependence of the solutions ψ on y is of interest, and so we will apply some generalizations of our GBDT version (see [8,15,16,22] and references therein) of Bäcklund-Darboux transformation. Such generalizations (for the cases of linear systems of several variables) are given, for instance, in the papers [4,19,20]. In particular, explicit solutions of nonstationary Dirac systems are constructed in [4].…”

“…In particular, explicit solutions of nonstationary Dirac systems are constructed in [4]. Yet, taking into account that u in (1.3) does not depend on y, it seems more useful to modify here our approach to dynamical systems formulated in [20,21]. In Section 2 we present GBDT for system (2.2) which is somewhat more general than the Dirac-Weyl system (1.3).…”