ON THE SPECTRAL STABILITY OF PERIODIC WAVES OF THE KLEIN-GORDON EQUATION

Abstract. The object of study is the Klein-Gordon equation in 1 + 1 dimensions, with integer power non-linearities. In particular, of inter-est is the spectral stability/instability (with respect to perturbations of the same period) of traveling-standing periodic solitons, which are of cnoidal (p = 2), dnoidal (p = 3) or more general type (p = 5). The cor-responding linearized problem for this two-parameter family of solutions fits the general abstract framework of spectral stability for second order Hamiltonian systems, recently developed by the last two authors and Bronski-Johnson-Kapitula. It is worth noting that the spatial periodic-ity however, forces a relation between the speed and the phase, which results in some unique challenges in the computations of the quantities involved in the stability index. Our results generalize recent work on the simpler case of standing waves of Natali-Pastor, [9] and Natali-Cardoso, [10]. 1