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 paper, we study the spectral and linear stability of peakon solutions of the Novikov equation. We prove spectral instability of the peakons in L2(R)$L<^>2(\mathbb {R})$. To do so, we start with a linearized operator defined on H1(R)$H<^>1(\mathbb {R})$ and extend it to a linearized operator defined on weaker functions in L2(R)$L<^>2(\mathbb {R})$. The spectrum of the linearized operator in L2(R)$L<^>2(\mathbb {R})$ is proven to cover a closed vertical strip of the complex plane. Furthermore, we prove that the peakons are spectrally unstable on W1,infinity(R)$W<^>{1,\infty }({\mathbb {R}})$ and linearly and spectrally stable on H1(R)$H<^>1(\mathbb {R})$. The result on W1,infinity(R)$W<^>{1,\infty }({\mathbb {R}})$ is in agreement with previous work about linear instability and our result on H1(R)$H<^>1(\mathbb {R})$ is in line with past work on orbital stability.