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

TL;DR

This paper investigates how small parity-time-symmetric perturbations, modeled as viscous friction, affect the spectral stability of static kink solutions in a 2D Klein-Gordon model, extending previous 1D analyses.

Contribution

It provides a theoretical analysis of the eigenvalue behavior under PT-symmetric perturbations in 2D Klein-Gordon models, which was not previously studied.

Findings

Eigenvalues' behavior is characterized as functions of the perturbation parameter.

The stability of static kinks is influenced by the localized PT-symmetric perturbation.

A theorem describing the eigenvalue dynamics under perturbation is established.

Abstract

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 $\mathcal{PT}$-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 $\mathcal{PT}$-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.