# An overview on the standing waves of nonlinear Schr\"odinger and Dirac equations on metric graphs with localized nonlinearity

## TL;DR

This paper reviews the existence and nonexistence of standing waves in nonlinear Schrödinger and Dirac equations on metric graphs with localized nonlinearity, highlighting their similarities, differences, and the nonrelativistic limit convergence.

## Contribution

It provides a comparative overview of standing wave solutions for NLSE and NLDE on metric graphs, including the nonrelativistic limit analysis.

## Key findings

- Existence and nonexistence results for standing waves in NLSE and NLDE.
- Analysis of subcritical and critical cases for NLSE.
- Convergence of bound states from NLDE to NLSE in the nonrelativistic limit.

## Abstract

We present a brief overview on the existence/nonexistence of standing waves for the NonLinear Schr\"odinger 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.