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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Abstract

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

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

Dates
First available in Project Euclid: 30 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.die/1427744096

Mathematical Reviews number (MathSciNet)
MR3328129

Zentralblatt MATH identifier
1340.35209

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

Citation

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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