# Nonlinear instability of inhomogeneous steady states solutions to the HMF Model

## TL;DR

This paper proves the nonlinear instability of inhomogeneous steady states in the Hamiltonian Mean Field (HMF) model, extending linear instability analysis to nonlinear regimes using adapted mathematical techniques.

## Contribution

It introduces a novel proof of nonlinear instability for inhomogeneous steady states in the HMF model, building on and extending existing linear and nonlinear analysis methods.

## Key findings

- Proves nonlinear instability of inhomogeneous steady states in the HMF model.
- Extends linear instability criteria to nonlinear regimes.
- Provides mathematical techniques applicable to similar Hamiltonian systems.

## Abstract

In this work we prove the nonlinear instability of inhomogeneous steady states solutions to the Hamiltonian Mean Field (HMF) model. We first study the linear instability of this model under a simple criterion by adapting the techniques developed in [19]. In a second part, we extend to the inhomogeneous case some techniques developed in [14, 17, 18] and prove a nonlinear instability result under the same criterion.