# On the nodal structure of nonlinear stationary waves on star graphs

## TL;DR

This paper studies stationary solutions of the nonlinear Schrödinger equation on star-shaped quantum graphs, classifying solutions based on their nodal structure and exploring their relevance for spectral analysis.

## Contribution

It proves the existence of solutions vanishing at the star's center and classifies them by nodal domains, extending nodal counting to nonlinear quantum graphs.

## Key findings

- Existence of solutions vanishing at the star center.
- Classification of solutions by nodal domains on each edge.
- Discussion of solutions' relevance for spectral curve calculations.

## Abstract

We consider stationary waves on nonlinear quantum star graphs, i.e. solutions to the stationary (cubic) nonlinear Schr\"odinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e. the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting such as the classic Sturm oscillation theorem.