Spectral and linear stability of peakons in the Novikov equation

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.