Spectral instability of periodic peaked waves in the μ-b-family of Camassa-Holm equations

Considered herein is a μ-version b-family of the Camassa-Holm equations on the circle. First, we define a linearized operator associated with these equations in $H^1\left(S\right)\cap W^{1,\infty }\left(S\right)$ and extend its domain to the larger space $L^2\left(S\right)$. Then we show that the periodic peaked waves of these equations are spectrally unstable in $L^2\left(S\right)$ for $b=2,3$. Finally, by using the method of characteristic, the time evolution of the linearized system is obtained, which is related to the spectral properties of the linearized operator in $L^2\left(S\right)$. We emphasize that this is the first result which proves the spectral instability in $L^2\left(S\right)$ of peakons for the μ-version equations.