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(R) to one defined on weaker functions on L-2(R), we prove spectral and linear instability on L-2(R) of peakons in the b-Novikov equations for any b. We also consider the stability on H-1(R) and show that the peakons are spectrally or linearly stable only in the case b = 3.