# Nonlinear standing waves on planar branched systems: Shrinking into metric graph

## TL;DR

This paper investigates the behavior of nonlinear standing waves on two-dimensional branched systems as they shrink into one-dimensional metric graphs, providing analytical and numerical insights into boundary condition convergence.

## Contribution

It offers a detailed analysis of the convergence of boundary conditions from fat graphs to metric graphs for the nonlinear Schrödinger equation, including numerical validation and practical applications.

## Key findings

- Convergence of boundary conditions from fat to metric graphs demonstrated.
- Numerical solutions confirm analytical estimates.
- Applicable to various physical systems like Bose-Einstein condensates and optical media.

## Abstract

We treat the stationary nonlinear Schroodinger equation on two-dimensional branched domains, so-called fat graphs. The shrinking limit when the domain becomes one-dimensional metric graph is studied by using analytical estimate of the convergence of fat graph boundary conditions into those for metric graph. Detailed analysis of such convergence on the basis of numerical solution of stationary nonlinear Schrodinger equation on a fat graph is provided. Possibility for reproducing different metric graph boundary conditions studied in earlier works is shown. Practical applications of the proposed model for such problems as Bose-Einstein condensation in networks, branched optical media, DNA, conducting polymers and wave dynamics in branched capillary networks are discussed.