On the spectral stability of kinks in 2D Klein-Gordon model with parity-time-symmetric perturbation

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In a series of recent works by Demirkaya et al. stability analysis for the static kink solutions to the 1D continuous and discrete Klein-Gordon equations with a PT-symmetric perturbation has been analysed. We consider the linear stability problem for the static kink in 2D Klein-Gordon field taking into account spatially localized PT-symmetric perturbation. The perturbation is in the form of viscous friction, which does not affect the static solutions to the unperturbed Klein-Gordon equation. Small dynamic perturbation around the static kink solution is considered to formulate the linear stability problem. The effect of the small perturbation on the solutions to the corresponding eigenvalue problem is analysed. The main result is presented in the form of a theorem describing the behavior of the eigenvalues corresponding to the extended and localised eigenmodes as the functions of the perturbation parameter.