Random input problem for the nonlinear Schrödinger equation

We consider the random input problem for a nonlinear system modeled by the integrable one-dimensional self-focusing nonlinear Schrödinger equation (NLSE). We concentrate on the properties obtained from the direct scattering problem associated with the NLSE. We discuss some general issues regarding soliton creation from random input. We also study the averaged spectral density of random quasilinear waves generated in the NLSE channel for two models of the disordered input field profile. The first model is symmetric complex Gaussian white noise and the second one is a real dichotomous (telegraph) process. For the former model, the closed-form expression for the averaged spectral density is obtained, while for the dichotomous real input we present the small noise perturbative expansion for the same quantity. In the case of the dichotomous input, we also obtain the distribution of minimal pulse width required for a soliton generation. The obtained results can be applied to a multitude of problems including random nonlinear Fraunhoffer diffraction, transmission properties of randomly apodized long period Fiber Bragg gratings, and the propagation of incoherent pulses in optical fibers.

Figure 1

The plot of the average spectral density $\overline{n}\left(T\right)$ vs $DT$ for a complex symmetric WGN input as defined by Eq. (25).

Figure 2

(Color online) The plot of $\overline{n}\left(\xi \right)$ vs $\xi$ for the dichotomous input with a fixed support of length $T=1$. Solid line is the case $c_0⪡c_1$, where we set $c_1=1$, $c_0=0.1$; dashed line is the same for the case $c_0⪢c_1$, where we set $c_0=1$, $c_1=0.1$. The initial probabilities $P_0\left(0\right)=P_1\left(0\right)=1∕2$.