On the spectral stability of kinks in some PT-symmetric variants of the   classical Klein-Gordon Field Theories
  In the present work we consider the introduction of PT-symmetric terms in the context of classical Klein-Gordon field theories. We explore the implication of such terms on the spectral stability of coherent structures, namely kinks. We find that the conclusion critically depends on the location of the kink center relative to the center of the PT-symmetric term. The main result is that if these two points coincide, the kink's spectrum remains on the imaginary axis and the wave is spectrally stable. If the kink is centered on the "lossy side" of the medium, then it becomes stabilized. On the other hand, if it becomes centered on the "gain side" of the medium, then it is destabilized. The consequences of these two possibilities on the linearization (point and essential) spectrum are discussed in some detail. 
A. Demirkaya, M. Stanislavova, A. Stefanov, T. Kapitula and P.G.   Kevrekidis
nlin.PS math-ph math.MP