A. Geyer and D.E. Pelinovsky
Spectral instability of the peaked periodic wave
in the reduced Ostrovsky equation
Proceedings of the American Mathematical Society 148 (2020), 5109-5125
Abstract:
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.
Keywords:
Reduced Ostrovsky equation; peaked periodic waves; spectral instability;