On new classes of explicit solutions of Dirac, dynamical Dirac and Dirac--Weyl systems with non-vanishing at infinity potentials, their properties and applications (2002.04975v1)

Abstract: Our GBDT version of B\"acklund-Darboux transformation is applied to the construction of wide classes of new explicit solutions of self-adjoint and skew-self-adjoint Dirac systems, dynamical Dirac and Dirac--Weyl systems. That is, we construct explicit solutions of systems with non-vanishing at infinity potentials. In particular, the cases of steplike potentials and power growth of potentials are treated. It is essential (especially, for dynamical case) that the generalised matrix eigenvalues are used in GBDT instead of the usual eigenvalues (and those matrix eigenvalues are not necessarily diagonal). The connection of Dirac--Weyl system with graphene theory is discussed. Explicit expressions for Weyl--Titchmarsh functions are derived.