Nonlinear pulse propagation is investigated in the neighborhood of the zero-dispersion wavelength in monomode fibers. When the amplitude is sufficiently large to generate breathers (N > 1 solitons), it is found that the pulses break apart if lambda - lambda(0) is sufficiently small, owing to the third-order dispersion. Here lambda(0) denotes the zero-dispersion wavelength. By contrast, the solitary-wave (N = 1) solution appears well behaved for arbitrary lambda - lambda(0). Implications for communication systems and pulse compression are discussed.