Differential and Integral Equations

On the spectral stability of periodic waves of the Klein-Gordon equation

Aslihan Demirkaya, Sevdzhan Hakkaev, Milena Stanislavova, and Atanas Stefanov

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The object of study is the Klein-Gordon equation in $1+1$ dimensions, with integer power non-linearities. In particular, of interest 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 corresponding 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 periodicity 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].

Article information

Differential Integral Equations, Volume 28, Number 5/6 (2015), 431-454.

First available in Project Euclid: 30 March 2015

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B35: Stability 35L71: Semilinear second-order hyperbolic equations 35C07: Traveling wave solutions


Demirkaya, Aslihan; Hakkaev, Sevdzhan; Stanislavova, Milena; Stefanov, Atanas. On the spectral stability of periodic waves of the Klein-Gordon equation. Differential Integral Equations 28 (2015), no. 5/6, 431--454. https://projecteuclid.org/euclid.die/1427744096

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