p-star models, mean-field random networks, and the heat hierarchy.

We consider the mean-field analog of the p-star model for homogeneous random networks, and we compare its behavior with that of the p-star model and its classical mean-field approximation in the thermodynamic regime. We show that the partition function of the mean-field model satisfies a sequence of partial differential equations known as the heat hierarchy, and the models connectance is obtained as a solution of a hierarchy of nonlinear viscous PDEs. In the thermodynamic limit, the leading-order solution develops singularities in the space of parameters that evolve as classical shocks regularized by a viscous term.

Excitation of switching waves in normally dispersive Kerr cavities.

A coherently pumped, passive cavity supports, in the normal dispersion regime, the propagation of still interlocked fronts or switching waves that form invariant localized temporal structures. We address theoretically the problem of the excitation of this type of wave packet. First, we map all the dynamical behaviors of the switching waves as a function of accessible parameters, namely, the cavity detuning and input energy deficiency, using box-like excitation of the intracavity field.

Interactions of solitary waves in integrable and nonintegrable lattices.

We study how the dynamics of solitary wave (SW) interactions in integrable systems is different from that in nonintegrable systems in the context of crossing of two identical SWs in the (integrable) Toda and the (non-integrable) Hertz systems. We show that the collision process in the Toda system is perfectly symmetric about the collision point, whereas in the Hertz system, the collision process involves more complex dynamics. The symmetry in the Toda system forbids the formation of secondary SWs (SSWs), while the absence of symmetry in the Hertz system allows the generation of SSWs.

Nonlinear interactions between solitons and dispersive shocks in focusing media.

Nonlinear interactions in focusing media between traveling solitons and the dispersive shocks produced by an initial discontinuity are studied using the one-dimensional nonlinear Schrödinger equation. It is shown that, when solitons travel from a region with nonzero background toward a region with zero background, they always pass through the shock structure without generating dispersive radiation. However, their properties (such as amplitude, velocity, and shape) change in the process.

Auto-modulation versus breathers in the nonlinear stage of modulational instability.

The nonlinear stage of modulational instability in optical fibers induced by a wide and easily accessible class of localized perturbations is studied using the nonlinear Schrödinger equation. It is shown that the development of associated spatio-temporal patterns is strongly affected by the shape and the parameters of the perturbation. Different scenarios are presented that involve an auto-modulation developing in a characteristic wedge, possibly coexisting with breathers which lie inside or outside the wedge.

Solitons and rogue waves in spinor Bose-Einstein condensates.

We present a general classification of one-soliton solutions as well as families of rogue-wave solutions for F=1 spinor Bose-Einstein condensates (BECs). These solutions are obtained from the inverse scattering transform for a focusing matrix nonlinear Schrödinger equation which models condensates in the case of attractive mean-field interactions and ferromagnetic spin-exchange interactions. In particular, we show that when no background is present, all one-soliton solutions are reducible via unitary transformations to a combination of oppositely polarized solitonic solutions of single-component BECs.

Recurrence due to periodic multisoliton fission in the defocusing nonlinear Schrödinger equation.

We address the degree of universality of the Fermi-Pasta-Ulam recurrence induced by multisoliton fission from a harmonic excitation by analyzing the case of the semiclassical defocusing nonlinear Schrödinger equation, which models nonlinear wave propagation in a variety of physical settings. Using a suitable Wentzel-Kramers-Brillouin approach to the solution of the associated scattering problem we accurately predict, in a fully analytical way, the number and the features (amplitude and velocity) of solitonlike excitations emerging post-breaking, as a function of the dispersion smallness parameter. This also permits us to predict and analyze the near-recurrences, thereby inferring the universal character of the mechanism originally discovered for the Korteweg-deVries equation.

Whitham modulation theory for the two-dimensional Benjamin-Ono equation.

Whitham modulation theory for the two-dimensional Benjamin-Ono (2DBO) equation is presented. A system of five quasilinear first-order partial differential equations is derived. The system describes modulations of the traveling wave solutions of the 2DBO equation.

Whitham modulation theory for the Kadomtsev- Petviashvili equation.

The genus-1 Kadomtsev-Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.

Oscillation structure of localized perturbations in modulationally unstable media.

We characterize the properties of the asymptotic stage of modulational instability arising from localized perturbations of a constant background, including the number and location of the individual peaks in the oscillation region. We show that, for long times, the solution tends to an ensemble of classical (i.e.

Universal Nature of the Nonlinear Stage of Modulational Instability.

We characterize the nonlinear stage of modulational instability (MI) by studying the longtime asymptotics of the focusing nonlinear Schrödinger (NLS) equation on the infinite line with initial conditions tending to constant values at infinity. Asymptotically in time, the spatial domain divides into three regions: a far left and a far right field, in which the solution is approximately equal to its initial value, and a central region in which the solution has oscillatory behavior described by slow modulations of the periodic traveling wave solutions of the focusing NLS equation. These results demonstrate that the asymptotic stage of MI is universal since the behavior of a large class of perturbations characterized by a continuous spectrum is described by the same asymptotic state.

Anisotropic hinge model for polarization-mode dispersion in installed fibers.

We discuss a generalized waveplate hinge model to characterize anisotropic effects associated with the hinge model of polarization-mode dispersion in installed systems. In this model, the action of the hinges is a random time-dependent rotation about a fixed axis. We obtain the probability density function of the differential group delay and the outage probability of an individual wavelength band using a combination of importance sampling and the cross-entropy method, and we then compute the noncompliant capacity ratio by averaging over wavelength bands.

Collision-induced timing shifts in dispersion-managed soliton systems.

The frequency and timing shifts associated with dispersion-managed solitons in a wavelength-division multiplexed system are computed by the numerically efficient Poisson sum technique. Analytical formulas are attainable by use of this approach with a Gaussian approximation for the soliton. The results are favorably compared with known results for the frequency shift.

Line soliton interactions of the Kadomtsev-Petviashvili equation.

We study soliton solutions of the Kadomtsev-Petviashvili II equation (-4u(t)+6uu(x)+3u(xxx))(x)+u(yy)=0 in terms of the amplitudes and directions of the interacting solitons. In particular, we classify elastic N-soliton solutions, namely, solutions for which the number, directions, and amplitudes of the N asymptotic line solitons as y-->infinity coincide with those of the N asymptotic line solitons as y-->-infinity. We also show that the (2N-1)!! types of solutions are uniquely characterized in terms of the individual soliton parameters, and we calculate the soliton position shifts arising from the interactions.

Reduction of collision-induced timing shifts in dispersion-managed quasi-linear systems with periodic-group-delay dispersion compensation.

Periodic-group-delay (PGD) dispersion-compensation modules were recently proposed as mechanisms to alleviate collision-induced timing shifts in dispersion-managed (DM) systems. Frequency and timing shifts in quasi-linear DM systems with PGDs were obtained, and it is shown that significant reductions are achieved when even a small fraction of the total dispersion is compensated for by PGDs.

Optical solitons: Perspectives and applications.

This article serves as an introduction to the focus issue on optical solitons. After a short review of the history of solitons and the field of integrable systems, a brief overview of the development of nonlinear optics and optical solitons is provided. Next, the various contributions to this focus issue are presented, and a few separate remarks are devoted to optical communications, where solitons promise to play a decisive role in the next generation of commercial systems.

Methods for discrete solitons in nonlinear lattices.

A method to find discrete solitons in nonlinear lattices is introduced. Using nonlinear optical waveguide arrays as a prototype application, both stationary and traveling-wave solitons are investigated. In the limit of small wave velocity, a fully discrete perturbative analysis yields formulas for the mode shapes and velocity.