Spectral instability of peakons for the b-family of Novikov equations

In this paper, we are concerned with a one-parameter family of peakon equations with cubic nonlinearity parametrized by a parameter usually denoted by the letter b. This family is called the “b-Novikov” since it reduces to the integrable Novikov equation in the case $b=3$. By extending the corresponding linearized operator defined on functions in $H^1\left(R\right)$ to one defined on weaker functions on $L^2\left(R\right)$, we prove spectral and linear instability on $L^2\left(R\right)$ of peakons in the b-Novikov equations for any b. We also consider the stability on $H^1\left(R\right)$ and show that the peakons are spectrally or linearly stable only in the case $b=3$.