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.

arXiv comment34 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

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• $L^2$-critical NLS on noncompact metric graphs with localized nonlinearity: topological and metric features
• Standing waves on quantum graphs
• NLS ground states on metric trees: existence results and open questions
• Competing nonlinearities in NLS equations as source of threshold phenomena on star graphs
• On the nonlinear Dirac equation on noncompact metric graphs
• Ground states of the $L^2$-critical NLS equation with localized nonlinearity on a tadpole graph
• Spectral properties of relativistic quantum waveguides
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