# Normalized solutions for nonlinear Schr\"odinger equations on graphs

## TL;DR

This paper investigates normalized solutions for nonlinear Schrödinger equations on graphs, analyzing their behavior under varying mass constraints and supporting findings with numerical experiments.

## Contribution

It establishes the existence of normalized solutions on graphs using variational methods and describes their asymptotic behaviors as mass constraints vary.

## Key findings

- Existence of normalized solutions on finite and locally finite graphs.
- Behavior of solutions as mass tends to zero or infinity.
- Numerical illustrations confirming theoretical results.

## Abstract

We are concerned with the nonlinear Schr\"odinger equation with an $L^2$ mass constraint on both finite and locally finite graphs and prove that the equation has a normalized solution by employing variational methods. We also pay attention to the behaviours of the normalized solution as the mass constraint tends to $0^+$ or $+\infty$ and give clear descriptions of the limit equations. Finally, we provide some numerical experiments on a finite graph to illustrate our theoretical results.