Spectral instability of the peaked periodic wave in the reduced Ostrovsky equation

We show that the peaked periodic traveling wave of the reduced Ostrovsky equations with quadratic and cubic nonlinearity is spectrally unstable in the space of square integrable periodic functions with zero mean and the same period. The main novelty is that we discover a new instability phenomenon: the instability of the peaked periodic waves is induced by spectrum of a linearized operator which completely covers a closed vertical strip of the complex plane.