## On the spectral stability of periodic waves of the Klein-Gordon equation.(English)Zbl 1340.35209

The paper is devoted to the stability of the traveling standing waves of the form $$e^{i\omega t}e^{iq(x-ct)}\phi_{\omega ,c}(x-ct)$$ to the Klein-Gordon equation $$u_{tt}-u_{xx}+u-| u| ^{p-1}u=0$$ for $$p=2,3,5$$. The authors extend the results of M. Stanislavova [Stud. Appl. Math. 134, No. 1, 1–23 (2015; Zbl 1309.35117)]. They construct traveling-standing periodic solitons to the above equation. Then, they set up the linearized problem and give the general abstract result for second-order in time Hamiltonian systems. A specific computation of the index gives the condition for stability (or the instability) of the corresponding waves.

### MSC:

 35L70 Second-order nonlinear hyperbolic equations 35B35 Stability in context of PDEs 35Q53 KdV equations (Korteweg-de Vries equations) 35C15 Integral representations of solutions to PDEs 35L71 Second-order semilinear hyperbolic equations 35C07 Traveling wave solutions

Zbl 1309.35117