  We introduce a model of a dual-core optical waveguide with opposite signs of
the group-velocity-dispersion (GVD) in the two cores, and a phase-velocity
mismatch between them. The coupler is embedded into an active host medium,
which provides for the linear coupling of a gain-loss type between the two
cores. The same system can be derived, without phenomenological assumptions, by
considering the three-wave propagation in a medium with the quadratic
nonlinearity, provided that the depletion of the second-harmonic pump is
negligible. This linear system offers an optical realization of the
charge-parity ($\mathcal{CP}$) symmetry, while the addition of the intra-core
cubic nonlinearity breaks the symmetry. By means of direct simulations and
analytical approximations, it is demonstrated that the linear system generates
expanding Gaussian states, while the nonlinear one gives rise to broad
oscillating solitons, as well as a general family of stable stationary gap
solitons.
