# Instability of $H^1$-stable peakons in the Camassa-Holm equation

## TL;DR

This paper demonstrates that despite $H^1$-orbital stability of peakons in the Camassa-Holm equation, certain perturbations grow over time, revealing an instability not captured by linear analysis.

## Contribution

It shows the instability of peakons under piecewise $C^1$ perturbations using characteristics, challenging the traditional stability understanding.

## Key findings

- Piecewise $C^1$ perturbations grow over time.
- Linearized stability analysis contradicts $H^1$-orbital stability.
- Passage from linear to nonlinear stability in $H^1$ is invalid.

## Abstract

It is well-known that peakons in the Camassa-Holm equation are $H^1$-orbitally stable thanks to the presence of conserved quantities and properties of peakons as constrained energy minimizers. By using the method of characteristics, we prove that piecewise $C^1$ perturbations to peakons grow in time in spite of their stability in the $H^1$-norm. We also show that the linearized stability analysis near peakons contradicts the $H^1$-orbital stability result, hence passage from linear to nonlinear theory is false in $H^1$.