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

Abstract

Our GBDT version of Bäcklund-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.

This paper essentially develops some ideas from our papers arXiv:nlin/0402055 and arXiv:1609.03451

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• Dressing for generalised linear Hamiltonian systems depending rationally on the spectral parameter and some applications
• On the solution of the inverse problem for a class of canonical systems corresponding to matrix string equations