Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limit

Abstract

In this paper we study the nonlinear Dirac (NLD) equation on noncompact metric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-type conditions at the vertices. Precisely, we discuss existence and multiplicity of the bound states (arising as critical points of the NLD action functional) and we prove that, in the $L^2$-subcritical case, they converge to the bound states of the NLS equation in the nonrelativistic limit.

34 pages, 4 figures. Keywords: nonlinear Dirac equations, metric graphs, nonrelativistic limit, variational methods, bound states, linking. Some minor revisions have been made with respect the previous version