Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation

Aslihan  Demirkaya; Panayotis G.  Kevrekidis; Milena Stanislavova; Atanas Stefanov

doi:10.3934/proc.2015.0359

Article Contents

2015, Volume 2015: 359-368. Doi: 10.3934/proc.2015.0359 open access

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Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation

1.
Department of Mathematics, University of Hartford, 200 Bloomfield Avenue, West Hartford, CT 06117
2.
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515
3.
Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045–7523
4.
Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd, Lawrence, KS 66045-7523

Received: August 31, 2014

Revised: March 31, 2015

Published: October 2015

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Abstract

In the present work, we introduce a new $\mathcal{PT}$-symmetric variant of the Klein-Gordon field theoretic problem. We identify the standing wave solutions of the proposed class of equations and analyze their stability. In particular, we obtain an explicit frequency condition, somewhat reminiscent of the classical Vakhitov-Kolokolov criterion, which sharply separates the regimes of spectral stability and instability. Our numerical computations corroborate the relevant theoretical result.

Keywords:

linear stability,
standing waves.

Mathematics Subject Classification: Primary: 35B35, 35B40, 35L70; Secondary: 37L10, 37L15, 37D10.

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