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The dynamic properties of a nonlinear directional coupler made of Kerr materials are investigated using a normal-mode theory based on even and odd supermodes. This analysis provides a good platform to study the pulse propagation in nonlinear multimode waveguides, since the device is simple and can be regarded as a bimodal waveguide. Although the nonlinear directional coupler has already been well investigated using the standard coupled-mode theory, this paper presents new knowledge concerning the device operation from the standpoint of supermodes. Under the normal-mode model, power exchange between the two supermodes is performed through non-phase-matched four-wave mixing (FWM) and new frequency components are created on propagation, leading to spectral broadening of optical pulses. It is found that the group-delay mismatch between the two modes induces instability in collaboration to the nonlinearity when the duration of input pulses is of the order of sub-picoseconds. The dynamic properties of the nonlinear directional coupler are quite similar to those of the nonlinear MMI coupler that were more recently investigated by the present authors.