Pseudo-Exponential-Type Solutions of Wave Equations Depending on Several Variables

Abstract

Abstract. Using matrix identities, we construct explicit pseudo-exponential-type solutions of linear Dirac, Loewner and Schrödinger equations depending on two variables and of nonlinear wave equations depending on three variables.

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]).…”

“…were constructed in [10]. However, the dependence of L on both variables is essential in that construction and the procedure itself is more complicated.…”

Dynamics of electrons and explicit solutions of Dirac–Weyl systems

“…However, a more complicated than ψ(x, y) = e iky ψ(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].…”

“…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]. 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].…”