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

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. We discover that the spectrum of a linearized operator at the peaked periodic wave completely covers a closed vertical strip of the complex plane. In order to obtain this instability, we prove an abstract result on spectra of operators under compact perturbations. This justifies the truncation of the linearized operator at the peaked periodic wave to its differential part for which the spectrum is then computed explicitly.