Onset of Oscillations in Nonsoliton Pulses in Nonlinear Dispersive Fibers

We study the modulation equations for the amplitude and phase of smoothed rectangular pulse initial data for the defocusing nonlinear Schrodinger (NLS) equation in the semiclassical limit, and show that these equations exhibit shock formation. In this way we identify and explain one source for the onset of pulse oscillations in nonlinear fibers whose transmission is modeled by the semiclassical NLS equation. The onset 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 pursued by Kodama and Wabnitz [11]: A piecewise linear pulse evolves for distances O(1) down the fiber, beyond which oscillations develop associated with the vanishing of the upper step of the pulse [10]. Here we show that the scenario in [11] is correct, but specific to pure rectangular pulses: any smoothing of this data fails to obey their scenario, but rather is described by the results presented here. That is, the semiclassical limit of the NLS equation is highly unstable with respect to smooth regularizations of rectangular date. In our analysis, the onset of oscillations os associated with the location of the maximum gradient of the pulse slopes, and onser occurs on the pulse slopes, at short distances down the fiber proportional to the inverse of this maximum gradient. Explicit upper and lower bounds on the initials shock location 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 power, and the fiber dispersion coefficient.