Hamiltonian Systems and Sturm–Liouville Equations: Darboux Transformation and Applications

Abstract

Abstract. We introduce GBDT version of Darboux transformation forHamiltonian and Shin-Zettl systems as well as for Sturm-Liouville equations (including indefinite Sturm-Liouville equations). These are the first results on Darboux transformation for general-type Hamiltonian and for Shin-Zettl systems. The obtained results are applied to the corresponding transformations of the Weyl-Titchmarsh functions and to the construction of explicit solutions of dynamical systems, of two-way diffusion equations and of indefinite Sturm-Liouville equations. The energy of the explicit solutions of dynamical systems is expressed (in a quite simple form) in terms of the parameter matrices of GBDT. The insertion of non-real eigenvalues into the spectrum of indefinite SturmLiouville operators is studied.Mathematics Subject Classification. Primary 34B24, 34L15, 47E05; Secondary 34B20, 74H05.

Dynamics of electrons and explicit solutions of Dirac–Weyl systems

“….) The following auxiliary result used also in [21] (see [21, f-la (2.10)]) is a particular case of much more general GBDTformulas [16,22]. We note that GBDT is based on the method of operator identities (see relations (2.3) and (2.23), and various references in [22,23]).…”

“…In [21,Section 7] we constructed GBDT (generalized Bäcklund-Darboux transformation) for the dynamical system…”

“…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).…”

The binary Darboux transformation revisited and KdV solitons on arbitrary short‐range backgrounds

“…For example, the book 9 pays much of attention to binary Darboux transformations but double commutation is not mentioned. The recent work19 briefly mentions Refs. 9 and 18 but without discussing connections.…”

Norming constants of embedded bound states and bounded positon solutions of the Korteweg-de Vries equation

“…the book[19] pays much of attention to binary Darboux transformations but double commutation is not mentioned. The recent[28] briefly mentiones[19] and[15] but without discussing connections.…”