# Nonlinear Instability of Half-Solitons on Star Graphs

## TL;DR

This paper proves that half-soliton states on star graphs with subcritical nonlinear Schrödinger equations are nonlinearly unstable saddle points, with small perturbations growing slowly over time, extending previous results to any number of edges and arbitrary subcritical powers.

## Contribution

It demonstrates the nonlinear instability of half-solitons on star graphs for any number of edges and subcritical powers, using normal form analysis to characterize the saddle point nature.

## Key findings

- Half-solitons are nonlinear saddle points on star graphs.
- Small perturbations grow slowly, indicating instability.
- Results extend previous work to arbitrary N and subcritical powers.

## Abstract

We consider a half-soliton stationary state of the nonlinear Schrodinger equation with the power nonlinearity on a star graph consisting of N edges and a single vertex. For the subcritical power nonlinearity, the half-soliton state is a degenerate critical point of the action functional under the mass constraint such that the second variation is nonnegative. By using normal forms, we prove that the degenerate critical point is a nonlinear saddle point, for which the small perturbations to the half-soliton state grow slowly in time resulting in the nonlinear instability of the half-soliton state. The result holds for any $N \geq 3$ and arbitrary subcritical power nonlinearity. It gives a precise dynamical characterization of the previous result of Adami {\em et al.}, where the half-soliton state was shown to be a saddle point of the action functional under the mass constraint for $N = 3$ and for cubic nonlinearity.