An overview on the standing waves of nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity

Abstract

We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schrödinger and the NonLinear Dirac Equations (NLSE/NLDE) on metric graphs with localized nonlinearity. We first 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, proving the convergence of bound states.

arXiv comment20 pages, 10 figures; keywords: metric graphs, NLS, NLD, ground states, bound states, localized nonlinearity, nonrelativistic limit

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• Standing waves on quantum graphs
• NLS ground states on metric trees: existence results and open questions
• On the nonlinear Dirac equation on noncompact metric graphs
• Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one
• Index theorems for graph-parametrized optimal control problems