  A method is shown for preventing temporal broadening of ultrafast optical
pulses in highly dispersive and fluctuating media for arbitrary signal-pulse
profiles. Pulse pairs, consisting of a strong-field control-pulse and a
weak-field signal-pulse co-propagate, whereby the specific profile of the
strong-field pulse precisely compensates for dispersive phase in the weak
pulse. A numerical example is presented in an optical system consisting of both
resonant and gain dispersive effects. Here we show signal-pulses that do not
temporally broaden across a vast propagation distance, even in the presence of
dispersion that fluctuates several orders of magnitude and in sign (for example
within a material resonance) across the pulse's bandwidth. Our approach
illustrates the potential for using cross-phase modulation to compensate for
dispersive effects on a signal-pulse and fills the gap in the current
understanding of this nonlinear phenomenon.
