A PT-symmetric dual-core system with the sine-Gordon nonlinearity and derivative coupling

We propose a system of sine-Gordon equations, with the $\mathcal{PT}$ symmetry represented by balanced gain and loss in them. The equations are coupled by sine-field terms and first-order derivatives. The sinusoidal coupling stems from local interaction between adjacent particles in the coupled Frenkel-Kontorova (FK) chains and related sine-lattices, while the cross-derivative coupling, which was not considered before, is induced by \emph{three-particle} interactions, provided that the particles in the parallel FK\ chains move in different directions. Nonlinear wave structures are then studied in this model. In particular, kink-kink (KK) and kink-antikink (KA) complexes are explored by means of analytical and numerical methods. It is predicted analytically and confirmed numerically that the complexes are unstable for one sign of the sinusoidal coupling, and stable for another. Stability regions are delineated in the underlying parameter space. Unstable complexes split into free kinks/antikinks that may propagate or become stationary, depending on whether they are subject to gain or loss, respectively.