Optical solitons which remain radiationless in spite of having wavenumbers immersed in the spectrum of linear waves are rather unusual. This article shows that moving solitons of this type are solutions of an extended NLS equation with third and fourth-order dispersion, and a quintic nonlinearity. The mechanism which prevents the emission of radiation in these solitons is presented. The radiation emitted when these solitons are perturbed is also studied. This radiation exhibits four propagating fronts, and the velocities of these fronts are calculated and explained.

A model for a non-Kerr cylindrical nematic fiber is presented. We use the multiple scales method to show the possibility of constructing different kinds of wavepackets of transverse magnetic (T M ) modes propagating through the fiber. This procedure allows us to generate different hierarchies of nonlinear partial differential equations (P DEs) which describe the propagation of optical pulses along the fiber. We go beyond the usual weakly nonlinear limit of a Kerr medium and derive a complex modified Korteweg-de Vries equation (cmKdV) which governs the dynamics for the amplitude of the wavepacket. In this derivation the dispersion, self-focussing and diffraction in the nematic are taken * Fellow of SNI, Mexico † Correspondence author. E-mail: zepeda@fisica.unam.mx ‡ Fellow of SNI, Mexico 1 into account. It is shown that this cmKdV equation has two-parameter families of bright and dark complex solitons. We show analytically that under certain conditions, the bright solitons are actually double embedded solitons. We explain why these solitons do not radiate at all, even though their wavenumbers are contained in the linear spectrum of the system. We study (numerically and analytically) the stability of these solitons. Our results show that these embedded solitons are stable solutions, which is an interesting property since in most systems the embedded solitons are weakly unstable solutions. Finally, we close the paper by making comments on the advantages as well as the limitations of our approach, and on further generalizations of the model and method presented.

It was recently proved that solitons embedded in the spectrum of linear waves may exist in discrete systems, and explicit solutions for isolated unstable embedded lattice solitons (ELS) of a differentialdifference version of a higher-order NLS equation were found [Physica D 197 (2004) 86]. The discovery of these ELS gives rise to relevant questions such as the following: are there continuous families of ELS?, can ELS be stable?, is it possible for ELS to move along the lattice?, how do ELS interact?. The present work addresses these questions by showing that a novel equation (a discrete version of a complex modified KdV equation which includes next-nearest-neighbor couplings) has a two-parameter continuous family of exact ELS. These solitons can move with arbitrary velocities across the lattice, and the numerical simulations demonstrate that these ELS are completely stable. Moreover, the numerical tests show that these ELS are robust enough to withstand collisions, and the result of a collision is only a shift in the positions of the solitons. The model may apply to the description of a Bose-Einstein condensate with dipole-dipole interactions between the atoms, trapped in a deep optical-lattice potential. In nonlinear systems where solitons can exist, the propagation of small-amplitude linear waves, which obey the linearized version of the nonlinear equations, is possible too. However, for a soliton to exist, it is absolutely necessary that no resonances occur between the soliton and these linear waves. Otherwise, the soliton would decay due to an energy transfer towards the linear waves. Based on this no-resonance argument, it was frequently assumed that the solitons' internal frequencies could not be contained in the linear spectrum of the system, i.e., they could not lie within the band of frequencies permitted to linear waves. However, at the end of the nineties exceptions to this rule were found, and a special type of solitons were discovered, which do not resonate with linear waves, in spite of having frequencies immersed in the spectrum of these waves. In 1999 these peculiar solitary waves were given the name of embedded solitons (ES), and in the following years a number of models supporting ES were identified. Most of these models describe continuous systems. However, some examples of discrete ES were recently found too. These embedded lattice solitons (ELS) are isolated solutions which are stable against small perturbations in the linear approximation, but are nonlinearly unstable. The discovery of these isolated unstable ELS triggered the search for models admitting continuous families of stable ELS. In this article we present a novel differential-difference equation which has a two-parameter continuous family of exact ELS. These solitons can move with arbitrary velocities across the lattice, and the numerical simulations show that they are completely stable solutions.

We consider solitons in a system of linearly coupled Korteweg-de Vries (KdV) equations, which model two-layer settings in various physical media. We demonstrate that traveling symmetric solitons with identical components are stable at velocities lower than a certain threshold value. Above the threshold, which is found exactly, the symmetric modes are unstable against spontaneous symmetry breaking, which gives rise to stable asymmetric solitons. The shape of the asymmetric solitons is found by means of a variational approximation and in the numerical form. Simulations of the evolution of an unstable symmetric soliton sometimes produce its breakup into two different asymmetric modes. Collisions between moving stable solitons, symmetric and asymmetric ones, are studied numerically, featuring noteworthy features. In particular, collisions between asymmetric solitons with identical polarities are always elastic, while in the case of opposite polarities the collision leads to a switch of the polarities of both solitons. Three-soliton collisions are studied too, featuring quite complex interaction scenarios.

Abstract:We study linear and nonlinear pulse propagation models whose linear dispersion relations present bands of forbidden frequencies or forbidden wavenumbers. These bands are due to the interplay between higher-order dispersion and one of the terms (a second-order derivative with respect to the propagation direction) which appears when we abandon the slowly varying envelope approximation. We show that as a consequence of these forbidden bands, narrow pulses radiate in a novel and peculiar way. We also show that the nonlinear equations studied in this paper have exact soliton-like solutions of different forms, some of them being embedded solitons. The solutions obtained (of the linear as well as the nonlinear equations) are interesting since several arguments suggest that the Cauchy problems for these equations are ill-posed, and therefore the specification of the initial conditions is a delicate issue. It is also shown that some of these equations are related to elliptic curves, thus suggesting that these equations might be related to other fields where these curves appear, such as the theory of modular forms and Weierstrass ℘ functions, or the design of cryptographic protocols.