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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

Citations:

Zbl 1309.35117
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