Effects of randomness, dissipation and interaction on solitons of the cubic nonlinear Schrödinger equation and related nonlinear wave models
Nguyen, Quan Minh
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We investigate the propagation of solitons of the perturbed nonlinear Schrödinger equation (NLSE) via asymptotic perturbation techniques and numerical simulations. The dissertation consists of several inter-related projects [22, 98, 103, 108, 109] that are focused on the effects of nonlinear processes and randomness on dynamics of pulses of light in optical waveguides. We particularly consider two of the most important nonlinear processes affecting pulse dynamics in multichannel optical waveguides: weak cubic loss and delayed Raman response. In the presence of weak cubic loss , we obtain the analytic expressions for the amplitude and frequency shifts in a single two-soliton collision and show that the impact of a fast three-soliton collision is given by the sum of the two-soliton interactions. Furthermore, we show that amplitude dynamics in an N-channel waveguide system is described by a Lotka-Volterra model for N competing species. We find the conditions on the time slot width and the soliton's equilibrium amplitude value under which the transmission is stable. The predictions of the reduced Lotka-Volterra model are confirmed by numerical solution of a coupled-NLSE model, which takes into account intra-pulse and inter-pulse effects due to cubic nonlinearity and cubic loss. These results uncover an interesting analogy between the dynamics of energy exchange in pulse collisions and population dynamics in Lotka-Volterra models. In the presence of delayed Raman response [103,108,109], we show that the dynamics of pulse amplitudes in an N-channel transmission system in differential phase shift keying (DPSK) scheme is described by an N-dimensional predator-prey model. We find the equilibrium states with non-zero amplitudes and prove their stability by obtaining the Lyapunov function. We then show that stable transmission can be achieved by a proper choice of the frequency profile of linear amplifier gain. We also investigate the impact of Raman self- and collsion-induced frequency shifts on the dynamics and establish the stability of the equilibrium state with respect to these perturbations. In multichannel optical fiber transmission, the interplay between Raman-induced energy exchange in pulse collisions and bit pattern randomness in the on-off-keying (OOK) transmission scheme leads to the disorder in the linear gain coefficient . In this case, the disorder can lead to relatively high bit-error-rate (BER, i.e., error probability per bit) values and intermittent dynamics of pulse parameters [96, 104]. We thus study the combined effects of randomness and high-order nonlinearity on pulse dynamics in optical waveguides. We focus attention on the dynamics of solitary waves of the cubic-quintic nonlinear Schrödinger equation (CQNLSE) in the presence of disorder in the cubic gain coefficient . We show by analytic calculation and by Monte Carlo simulations that the probability density function (PDF) of the solitary waves amplitude exhibits loglognormal divergence near the maximum possible amplitude. We relate this exotic divergence of the PDF to the super-exponential approach of the amplitude to its maximum possible value, and investigate the extent to which this statistical behavior is general in at-top solitary wave models.