ONSET OF OSCILLATIONS IN NONSOLITON PULSES IN NONLINEAR DISPERSIVE FIBERS

We study the modulation equations for the amplitude and phase of smoot hed rectangular pulse initial data for the defocusing nonlinear Schrod inger (NLS) equation in the semiclassical limit, and show that these e quations exhibit shock formation. In this way we identify and explain one source for the onset of pulse oscillations in nonlinear fibers who se transmission is modeled by the semiclassical NLS equation. The onse t of pulse ripples predicted here develops on the leading and trailing slopes of a smooth pulse, as a consequence of shock formation in the modulation equations. This mechanism for the onset of pulse ripples is distinct, both in the location and timescale, from the scenario pursu ed by Kodama and Wabnitz [11]: A piecewise linear pulse evolves for di stances O(1) down the fiber, beyond which oscillations develop associa ted with the vanishing of the upper step of the pulse [10]. Here we sh ow that the scenario in [11] is correct, but specific to pure rectangu lar pulses; any smoothing of this data fails to obey their scenario, b ut rather is described by the results presented here. That is, the sem iclassical limit of the NLS equation is highly unstable with respect t o smooth regularizations of rectangular data. In our analysis, the ons et of oscillations is associated with the location of the maximum grad ient of the pulse slopes, and onset occurs on the pulse slopes, at sho rt distances down the fiber proportional to the inverse of this maximu m gradient. Explicit upper and lower bounds on the initial shock locat ion are derived. We thereby deduce the onset for this source of pulse degradation scales linearly with the pulse width, and scales with the reciprocal square root of the fiber nonlinear coefficient, the pulse p ower, and the fiber dispersion coefficient.