An Overview on the Standing Waves of Nonlinear Schrödinger and Dirac Equations on Metric Graphs with Localized Nonlinearity

Abstract

We present a brief overview of the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. First, we focus on the NLSE (both in the subcritical and the critical case) and, then, on the NLDE highlighting similarities and differences with the NLSE. Finally, we show how the two equations are related in the nonrelativistic limit by the convergence of the bound states.

Keywords

• metric graphs
• NLS
• NLD
• ground states
• bound states
• localized nonlinearity
• nonrelativistic limit