It is known that a near-linear solution (a train of densely packed kinks) of the underdamped sine-Gordon equation with a dc driving term has a region of modulational instability in the tachyonic regime, in which the trains velocity exceeds the limit velocity of the sine-Gordon model. In the present work a nonlinear analysis of this instability is developed for the slightly overcritical case. A system of coupled evolution equations for two complex amplitudes of the growing disturbances is derived, and it is demonstrated that the instability gives rise to a pair of coupled small-amplitude waves of modulation traveling along the underlying wave train. A physical implementation is proposed in terms of an I-V characteristic of a long Josephson junction described by this model. It is shown that, at the point where the instability sets in, the I-V characteristic suffers a break. A corresponding jump of the differential resistance is found.