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

TL;DR

This paper introduces a new PT-symmetric Klein-Gordon model, identifies its standing wave solutions, and derives an explicit stability criterion, supported by numerical validation.

Contribution

It presents a novel PT-symmetric Klein-Gordon equation, analyzes its standing wave solutions, and establishes a clear spectral stability criterion.

Findings

Derived an explicit frequency condition for stability

Identified regimes of spectral stability and instability

Numerical results confirm theoretical predictions

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.