Yeojin Chung

Ultra-short pulses in linear and nonlinear media

We consider the evolution of ultra-short optical pulses in linear and nonlinear media. For the linear case, we first show that the initial-boundary value problem for Maxwells equations in which a pulse is injected into a quiescent medium at the left endpoint can be approximated by a linear wave equation which can then be further reduced to the linear short-pulse equation (SPE). A rigorous proof is given that the solution of the SPE stays close to the solutions of the original wave equation over the time scales expected from the multiple scales derivation of the SPE. For the nonlinear case we compare the predictions of the traditional nonlinear Schrodinger equation (NLSE) approximation with those of the SPE. We show that both equations can be derived from Maxwells equations using the renormalization group method, thus bringing out the contrasting scales. The numerical comparison of both equations with Maxwells equations shows clearly that as the pulse length shortens, the NLSE approximation becomes steadily less accurate, while the SPE provides a better and better approximation.

Strongly non-Gaussian statistics of optical soliton parameters due to collisions in the presence of delayed Raman response

We study the effects of a delayed Raman response on soliton collisions in optical fibre transmission systems with multiple frequency channels. We show that the propagation of a given soliton undergoing many collisions with solitons from other frequency channels is described by a perturbed stochastic nonlinear Schrodinger equation, in which the stochastic perturbative terms are due to collision induced amplitude and frequency changes. Using the adiabatic perturbation theory we find that the distribution function of the soliton amplitude is lognormal, i.e. strongly non-Gaussian. The frequency of the soliton is also found to be a random variable that is not self-averaging. The results of our extensive numerical simulations incorporating the technique of importance sampling are in very good agreement with the theoretical predictions.

Strong collapse turbulence in a quintic nonlinear Schrödinger equation

We consider the quintic one-dimensional nonlinear Schrödinger equation with forcing and both linear and nonlinear dissipation. Quintic nonlinearity results in multiple collapse events randomly distributed in space and time, forming forced turbulence. Without dissipation each of these collapses produces finite-time singularity, but dissipative terms prevent actual formation of singularity. In statistical steady state of the developed turbulence, the spatial correlation function has a universal form with the correlation length determined by the modulational instability scale. The amplitude fluctuations at that scale are nearly Gaussian while the large-amplitude tail of the probability density function (PDF) is strongly non-Gaussian with powerlike behavior. The small-amplitude nearly Gaussian fluctuations seed formation of large collapse events. The universal spatiotemporal form of these events together with the PDFs for their maximum amplitudes define the powerlike tail of the PDF for large-amplitude fluctuations, i.e., the intermittency of strong turbulence.

Higher-order corrections to the short-pulse equation

Using renormalization group techniques, we derive an extended short-pulse equation as an approximation to a nonlinear wave equation. We investigate the new equation numerically and show that the new equation captures efficiently higher-order effects on pulse propagation in cubic nonlinear media. We illustrate our findings using one- and two-soliton solutions of the first-order short-pulse equation as initial conditions in the nonlinear wave equation.

Stationary solutions to the nonlinear Schrödinger equation in the presence of third-order dispersion

We study stationary solutions of the nonlinear Schrodinger equation in the presence of small but non-zero third-order dispersion (TOD). Using a singular perturbation theory around the ideal soliton we calculate these solutions up to the second order in the TOD coefficient. The existence and linear stability of the stationary solutions is proved for any finite order of the perturbation theory. The results obtained by our numerical simulations of the nonlinear Schrodinger equation are in very good agreement with theory. The significance of these results for fibre optic communication systems is discussed.

Diverging probability-density functions for flat-top solitary waves

We investigate the statistics of flat-top solitary wave parameters in the presence of weak multiplicative dissipative disorder. We consider first propagation of solitary waves of the cubic-quintic nonlinear Schrödinger equation (CQNLSE) in the presence of disorder in the cubic nonlinear gain. We show by a perturbative analytic calculation and by Monte Carlo simulations that the probability-density function (PDF) of the amplitude eta exhibits loglognormal divergence near the maximum possible amplitude eta(m), a behavior that is similar to the one observed earlier for disorder in the linear gain [A. Peleg, Phys. Rev. E 72, 027203 (2005)]. We relate the loglognormal divergence of the amplitude PDF to the superexponential approach of eta to eta(m) in the corresponding deterministic model with linear/nonlinear gain. Furthermore, for solitary waves of the derivative CQNLSE with weak disorder in the linear gain both the amplitude and the group velocity beta become random. We therefore study analytically and by Monte Carlo simulations the PDF of the parameter p, where p = eta/(1-epsilon(s)beta/2) and epsilon(s) is the self-steepening coefficient. Our analytic calculations and numerical simulations show that the PDF of p is loglognormally divergent near the maximum p value.

Interaction of solitons through radiation in optical fibers with randomly varying birefringence

Propagation of solitons in optical fibers is studied taking into account the polarization mode dispersion (PMD) effect. We show that the soliton interaction caused by the radiation emitted by solitons due to the PMD disorder leads to soliton jitter, and we find its statistical properties. The theoretical predictions are justified by direct numerical simulations.

Phase analysis of nonlinear femtosecond pulse propagation and self-frequency shift in optical fibers

Abstract Phase sensitive analysis of femtosecond pulse propagation in optical fibers employing frequency resolved optical gating (FROG) is presented and compared to numerical simulations employing a modified cubic nonlinear Schrodinger equation (NLSE). Phase information obtained from deconvolution of the experimental traces allows the observation and characterization of specific pulse propagation features as a function of energy and distance. The experimental observation of the phase signature of a soliton during propagation and the phase properties of the soliton self-frequency shift are described and are found to be in remarkable agreement with the simulations.

Inertial Manifolds and Gevrey Regularity for the Moore-Greitzer Model of an Axial-Flow Compressor

SummaryIn this paper, we study the regularity and long-time behavior of the solutions to the Moore-Greitzer model of an axial-flow compressor. In particular, we prove that this dissipative system of evolution equations possesses a global invariant inertial manifold, and therefore its underlying long-time dynamics reduces to that of an ordinary differential system. Furthermore, we show that the solutions of this model belong to a Gevrey class of regularity (real analytic in the spatial variables). As a result, one can show the exponentially fast convergence of the Galerkin approximation method to the exact solution, an evidence of the reliability of the Galerkin method as a computational scheme in this case. The rigorous results presented here justify the readily available low-dimensional numerical experiments and control designs for stabilizing certain states and traveling wave solutions for this model.

Collapse Turbulence in Nonlinear Schrödinger Equation

We consider a nonlinear Schrodinger equation (NLS) with dissipation and forcing in critical dimension. Without both linear and nonlinear dissipation NLS results in a finite‐time singularity (collapse) for any initial conditions. Dissipation ensures collapse regularization. If dissipation is small then multiple near‐singular collapses are randomly distributed in space and time forming collapse turbulence. Collapses are responsible for non‐Gaussian tails in the probability distribution function of amplitude fluctuations which makes turbulence strong. Power law of non‐Gaussian tails is obtained for strong NLS turbulence.