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We discuss the response of both moving and trapped solitary wave solutions ofa nonlinear two-component nonlinear Schr\"odinger system in 1+1 dimensions toan anti-$\mathcal{PT}$ external periodic complex potential. The dynamicalbehavior of perturbed solitary waves is explored by conducting numericalsimulations of the nonlinear system and using a collective coordinatevariational approximation. We present case examples corresponding to choices ofthe parameters and initial conditions involved therein. The results of thecollective coordinate approximation are compared against numerical simulationswhere we observe qualitatively good agreement between the two. Unlike the casefor a single-component solitary wave in a complex periodic$\mathcal{PT}$-symmetric potential, the collective coordinate equations do nothave a small oscillation regime, and initially the height of the two componentschanges in opposite directions often causing instability. We find that thedynamic stability criteria we have used in the one-component case is proven tobe a good indicator for the onset of dynamic instabilities in the presentsetup.

journal of physics. a, mathematical and theoretical

Behavior of solitary waves of coupled nonlinear Schr\"odinger equations  subjected to complex external periodic potentials with anti-$\mathcal{PT}$  symmetry