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.
Highlights
Over the past 15 years, the original proposal of Bender and coworkers that systems with PT -symmetry may constitute relevant extensions of the usual Hermitian quantum mechanical models has gained considerable traction
In numerous ones among these systems, the underlying linear dynamics is of the oscillator type i.e., it involves a dimer of two oscillators, one with loss and one with gain, typically in the form of a linear dashpot
Our aim in the present work is to quantify this general theorem in the special case of the operator pencils discussed above in (7)
Summary
Over the past 15 years, the original proposal of Bender and coworkers that systems with PT -symmetry may constitute relevant extensions of the usual Hermitian quantum mechanical models has gained considerable traction. We explore the long-wavelength limit of a modified form of the oscillator problem whereby the oscillation of u involves a dashpot effect from v and that of v a gain effect from u. This type of velocity dependent coupling has been argued, for instance, to exist in the coupling of pendula in the recent experiments and associated modeling of [15]. Our aim in the present work is to quantify this general theorem in the special case of the operator pencils discussed above in (7).
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