  We theoretically investigate the solitary waves and their switching dynamics
in a $\mathcal{PT}$-symmetric directional fiber coupler, exhibiting Kerr
nonlinearity, by developing a variational analysis. We analyze the fundamental
switching characteristics of the $\mathcal{PT}$-symmetric Kerr coupler in the
picosecond timescale by considering the coupled-mode equation for the
unperturbed nonlinear Schr\"{o}dinger equation, which we compare to its
conventional counterpart. The impacts of higher-order perturbations (intrapulse
Raman scattering, self-steepening, and third-order dispersion) are investigated
in detail in the femtosecond timescale. In all cases, the variational method
successfully predicts each of the numerically observed switching
characteristics. Our semianalytical treatment has the potential to provide
physical insights into complex switching dynamics in various nonlinear coupler
configurations from different areas of physics.
