Benjamin P. Luce

Global Bifurcation of Shilnikov Type in a Double-Gyre Ocean Model

Abstract The dynamics of an idealized wind-driven double-gyre circulation in an ocean basin are studied from a dynamical systems point of view in an effort to better understand its variability. While previous analyses of this circulation have mostly dealt with local bifurcations of steady states and limit cycles, this study demonstrates the importance of considering global bifurcations as well. In one case, a coherent picture of the global dynamics spanning a range of parameters from where there are only stable steady-state solutions to where there is chaotic eddy shedding is presented. A simple but novel use of power spectra along with dynamical projections of the dynamics suggests that just beyond the regime in which there are only stable steady states, the system exhibits a complicated global bifurcation known as the “Shilnikov phenomenon.”

Homoclinic explosions in the complex Ginzburg-Landau equation

A bifurcation sequence of the complex Ginzburg-Landau (CGL) equation in an even subspace of spatially periodic solutions is analyzed with local three-dimensional bifurcation theory near homoclinic orbits. Bifurcation to chaos occurs via “homoclinic explosions”, similar to bifurcation to chaos in the Lorenz model, and a similar global bifurcation picture is developed. An analysis via one-dimensional maps is developed to predict attractors. It is shown that the two mode Galerkin truncation can mimic some bifurcations of the full CGL equationdeceptively, and also that the truncation fails to mimic many features of full PDE precisely because the truncation lacks many of the same homoclinic orbits.

OBSERVATION OF CHIRPED SOLITON DYNAMICS AT LAMBDA = 1.55 MU M IN A SINGLE-MODE OPTICAL FIBER WITH FREQUENCY-RESOLVED OPTICAL GATING

We present an experimental observation of the dynamics of an initially chirped optical soliton at 1.55microm that is propagating through a single-mode optical fiber, using frequency-resolved optical gating (FROG). FROG permits observation of both the amplitude and the phase profiles of ultrashort pulses, providing complete information on the pulse evolution. The features that are detected, which include what is believed to be the first experimental observation of phase slips, are in quantitative agreement with numerical simulations that employ the nonlinear Schrödinger equation.

Nonlinear waves and solitons in physical systems

Abstract Advances in nonlinear science have been plentiful in recent years. In particular, interest in nonlinear wave propagation continues to grow, stimulated by new applications, such as fiber-optic communication systems, as well as the many classical unresolved issues of fluid dynamics. What is arguably the turning point for the modern perspective of nonlinear systems took place at Los Alamos over 40 years ago with the pioneering numerical simulations of Fermi, Pasta, and Ulam. A decade later, this research initiated the next major advance of Zabusky and Kruskal that motivated the revolution in completely integrable systems. With this in mind, the conference on Nonlinear Waves in Solitions in Physical Systems (NWSPS) was organized by the Center for Nonlinear Studies (CNLS) at Los Alamos National Laboratory in May of 1997, to assess the current state-of-the-art in this very active field. Papers from the conference attendees as well as from researchers unable to attend the conference were collected in this special volume of Physica D . In this paper, the contributions to the conference and to this special issue are reviewed, with an emphasis on the many unifying principles that all these works share.

Adaptive control methods for ultrafast pulse propagation in optical fibers

Adaptive control in combination with ultrafast pulse shaping provides a compelling approach to harness events that occur on the fastest timescale available. This paper illustrates the application of adaptive pulse shaping to femtosecond pulse propagation at /spl lambda/=1550 nm in single-mode optical fibers. The approach is illustrated first by through a numerical simulation of the technique. An experimental demonstration is described. The propagation of /spl sim/200-fs pulses is successfully achieved in the nonlinear regime by suitably preshaping the input pulse using an adaptive feedback loop.

Low-noise picosecond soliton transmission by use of concatenated nonlinear amplifying loop mirrors

We compare two systems that are specially configured with loop mirrors to provide stable picosecond soliton transmission. One configuration, created by Smith Doran [J. Opt. Soc. Am. B12, 1117 (1995)], uses nonlinear optical loop mirrors. This configuration is compared with another that uses nonlinear amplifying loop mirrors, the configuration of which we systematically determine from the amplitude and the pulse-width switching characteristics of the nonlinear optical loop mirror. It is found that the nonlinear optical loop mirror configuration allows stable pulse transmission with much less dispersive background waves than does the nonlinear optical loop mirror configuration. This clean performance is obtained at the cost of slightly lessening the parameter region of global stability of pulses. A technique for accurately estimating the region of global stability for amplitude and pulse-width perturbations of the stable pulse is also given.

Recovery of solitons with nonlinear amplifying loop mirrors

We study the use of nonlinear amplifying loop mirrors to recover soliton pulses nonadiabatically deformed by losses. We approach this problem as a mapping problem of input pulse to output pulse, for segments of fiber followed by a combination of linear and nonlinear amplification. For a wide range of amplifier spacings, we find numerically that a single optimal input pulse of soliton shape exists for each amplifier spacing, which is well recovered at output. The recovered output pulses contain only ~3% continuous radiation.

Power enhancement with super-Gaussian sliding-frequency guiding filters.

I show with numerical simulations that higher-order sliding filters, especially super-Gaussian filters, can produce dramatic power enhancement of optical solitons, similar to that of dispersion-managed solitons. This approach should allow smaller timing jitter without sacrificing the signal-to-noise ratio.

On the relationship of periodic wavetrains and solitary waves of complex Ginzburg-Landau type equations

Abstract The relationship between periodic wavetrains and solitary waves in complex Ginzburg-Landau type equations, such as those that model optical fiber amplifiers, is studied in the nonlinear Schrodinger limit with a Melnikov method. An important example, the cubic complex Ginzburg-Landau equation, is studied in detail. For this equation it is found that in the NLS limit particular families of periodic wavetrains all deform asymptotically to a single, persisting, stationary, nonlinear Schrodinger soliton as their periods tend to infinity.