Spectral and linear stability of peakons in the Novikov equation

TL;DR

This paper investigates the spectral and linear stability of peakon solutions in the Novikov equation, revealing spectral instability in certain function spaces while confirming stability in others, thus clarifying the stability landscape of these solutions.

Contribution

It provides a comprehensive spectral stability analysis of Novikov peakons, extending the linearized operator to weaker function spaces and establishing stability results across different norms.

Findings

Peakons are spectrally unstable in $L^2( ext{R})$.

Spectrum of the linearized operator covers a vertical strip in the complex plane.

Peakons are spectrally unstable in $W^{1, ext{infty}}( ext{R})$ and stable in $H^1( ext{R})$.

Abstract

The Novikov equation is a peakon equation with cubic nonlinearity which, like the Camassa-Holm and the Degasperis-Procesi, is completely integrable. In this article, we study the spectral and linear stability of peakon solutions of the Novikov equation. We prove spectral instability of the peakons in $L^2(\mathbb{R})$. To do so, we start with a linearized operator defined on $H^1(\mathbb{R})$ and extend it to a linearized operator defined on weaker functions in $L^2(\mathbb{R})$. The spectrum of the linearized operator in $L^2(\mathbb{R})$ is proven to cover a closed vertical strip of the complex plane. Furthermore, we prove that the peakons are spectrally unstable on $W^{1,\infty}(\mathbb{R})$ and linearly and spectrally stable on $H^1(\mathbb{R})$. The result on $W^{1,\infty}(\mathbb{R})$ are in agreement with previous work about linear stability, while our results on $H^1(\mathbb{R})$ are in agreement with the orbital stability obtained previously.