Spectral anti-broadening due to four-wave mixing in optical fibers

We show that the four-wave mixing can restrict spectral broadening. This is a general physical phenomenon that occurs in one-dimensional systems of four wave packets that resonantly interact "2-to-2": $ \omega_1+\omega_2=\omega_3+\omega_4,\; k_1+k_2=k_3+k_4$, when an annihilation of one pair of waves results in the creation of another pair. In addition, for this phenomenon to occur, the group velocities $C_1,C_2,C_3,C_4$ of the packets should be in a certain order: The extreme value (max or min) of the four group velocities should be in the same pair with the middle value of the remaining three, e.g. $C_1<C_3<C_2<C_4$. This phenomenon is due to the presence of an extra invariant, in addition to the energy, momentum, and Manley-Rowe relations.