M. Desaix

Wave-breaking-free pulses in nonlinear-optical fibers

A qualitative as well as quantitative investigation is made of the conditions for avoiding wave breaking during pulse propagation in optical fibers. In particular, it is shown that pulses having a parabolic intensity variation are approximate wave-breaking-free solutions of the nonlinear Schrodinger equation in the high-intensity limit. A simple expression for the compression factor of a fiber-grating compressor based on parabolic pulses is also derived.

Wave breaking in nonlinear-optical fibers

An analytical investigation is made of the interplay between dispersion and nonlinearity in the creation of wave breaking in optical fibers. Wave breaking is found to involve two independent processes: (a) overtaking of different parts of the pulse and (b) nonlinear generation of new frequencies during overtaking. Analytical predictions for the distance of wave breaking are obtained and found to be in good agreement with numerical results.

Variational approach to collapse of optical pulses

Optical pulses, which propagate under the combined effects of nonlinearity, dispersion, and diffraction, may collapse in space and time. The standard method for analyzing these collapses is the aberrationless paraxial ray approximation. This method is known to give a quantitatively correct, although not particularly accurate, picture of most properties of the pulse dynamics. However, it is found that the predictions for some of the important pulse parameters are qualitatively wrong and could lead to incorrect conclusions. An alternative variational approach is suggested that remedies these deficiencies and gives results in good agreement with numerical results.

Dynamics of self-focusing and self-phase modulation in a parabolic index optical fiber.

The dynamics that result from the combined effects of spatial diffraction, nonlinearity, and a parabolic graded index for wave propagation in optical fibers are presented. An approximate analytical solution of the nonlinear Schrödinger equation in a graded-index fiber is obtained by using a variational approach. Particular emphasis is put on the variation of both the pulse width and the longitudinal phase delay with the distance of propagation along the fiber.

Dynamic effects of Kerr nonlinearity and spatial diffraction on self-phase modulation of optical pulses

An investigation is made of the combined effects of nonlinearity and diffraction on self-phase modulation of optical pulses propagating in dispersionless homogeneous bulk material. It is found that the presence of transverse spatial dynamics leaves the phase characteristics qualitatively the same as for conventional self-phase modulation. In particular, this implies that, contrary to what has recently been claimed, the red always leads the blue in the supercontinuum.

Solitons emerging from pulses launched in the vicinity of the zero-dispersion point in a single-mode optical fiber

It is shown that solitons emerging from pulses launched with frequencies at, or close to, the zero-dispersion point in a single-mode optical fiber are ordinary bright solitons corresponding to the nonlinear Schrödinger equation. These solitons are inherently frequency shifted into the anomalous-dispersion regime, where the third-order dispersion acts as a small perturbation. We therefore conclude that an optical-soliton-based communication system should be designed with the carrier frequency in the anomalous-dispersion regime but not too close to the zero-dispersion point, where the formation of an unwanted dispersive-wave component would degrade the system performance.

VARIATIONALLY OBTAINED APPROXIMATE EIGENVALUES OF THE ZAKHAROV-SHABAT SCATTERING PROBLEM FOR REAL POTENTIALS

Abstract A variational procedure, based on trial functions, for finding approximate eigenvalues of the Zakharov-Shabat eigenvalue problem for real potentials is presented.

Variational approach to the Thomas-Fermi equation

Using a direct variational approach, accurate approximate solutions are obtained for the famous Thomas–Fermi nonlinear differential equation describing the screening of the Coulomb potential inside heavy neutral atoms. In addition, the analysis demonstrates the flexibility and power, but also the limitations of the variational approach.

The brachistochrone problem?an introduction to variational calculus for undergraduate students

The importance of variational calculus is usually not reflected in the basic, undergraduate engineering curriculum. In this paper, we present the result of a pedagogical project aimed at introducing variational calculus at an early stage in the engineering education. This introduction is based on the classical brachistochrone problem and combines both theoretical and experimental analysis.

Diffusion of super-Gaussian profiles

The present analysis describes an analytically simple and systematic approximation procedure for modelling the free diffusive spreading of initially super-Gaussian profiles. The approach is based on a self-similar ansatz for the evolution of the diffusion profile, and the parameter functions involved in the modelling are determined by suitable moments of the diffusion equation. The resulting approximate solutions provide an accurate description of the evolution of the initial profile into the Gaussian profile, which is the asymptotically exact solution of the diffusion problem for, in fact, arbitrary initial conditions.