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

TL;DR

This paper investigates the nonlinear Dirac equation on graphs with localized nonlinearities, establishing existence and multiplicity of bound states and their convergence to nonlinear Schrödinger bound states in the nonrelativistic limit.

Contribution

It introduces the analysis of the nonlinear Dirac equation on metric graphs with localized nonlinearities, including existence, multiplicity, and nonrelativistic limit results.

Findings

Existence and multiplicity of bound states for the NLD on graphs.

Bound states converge to NLS bound states in the nonrelativistic limit.

Results are valid in the $L^2$-subcritical case.

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.