Spectral and linear stability of peakons in the Novikov equation

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 paper, we study the spectral and linear stability of peakon solutions of the Novikov equation. We prove spectral instability of the peakons in $L^2\left(R\right)$. To do so, we start with a linearized operator defined on $H^1\left(R\right)$ and extend it to a linearized operator defined on weaker functions in $L^2\left(R\right)$. The spectrum of the linearized operator in $L^2\left(R\right)$ is proven to cover a closed vertical strip of the complex plane. Furthermore, we prove that the peakons are spectrally unstable on $W^{1,\infty }\left(R\right)$ and linearly and spectrally stable on $H^1\left(R\right)$. The result on $W^{1,\infty }\left(R\right)$ is in agreement with previous work about linear instability and our result on $H^1\left(R\right)$ is in line with past work on orbital stability.