Embedded solitons in a three-wave system

2D in-band solitons in PT-symmetric waveguide arrays

We address two types of two-dimensional (2D) localized solitons in Kerr media with an imprinted quasi-one-dimensional lattice featuring a parity-time (PT) symmetry. Solitary waves originating from the edges of the Bloch bands are stable in their entire existence domains. Purely nonlinear multipeaked states propagate stably in wide parameter windows. Both types of nonlinear waves exist in the finite bandgaps of the corresponding linear system and, cross-continuously, the Bloch band (continuous spectrum) sandwiched between (or neighboring) them. To the best of our knowledge, our findings thus provide the first example of "embedded solitons" in 2D PT periodic systems.

Solitary waves in diatomic chains

We consider the mechanism of formation of isolated localized wave structures in the diatomic Fermi-Pasta-Ulam (FPU) model. Using a singular multiscale asymptotic analysis in the limit of high mass mismatch between the alternating elements, we obtain the typical slow-fast time scale separation and formulate the Fredholm orthogonality condition approximating a sequence of mass ratios supporting the formation of solitary waves in the general type of diatomic FPU models. This condition is made explicit in the case of a diatomic Toda lattice. Results of numerical integration of the full diatomic Toda lattice equations confirm the formation of these genuinely localized wave structures at special values of the mass ratio that are close to the analytical predictions when the ratio is sufficiently small.

Diversity of solitons in a generalized nonlinear Schrödinger equation with self-steepening and higher-order dispersive and nonlinear terms

In this article, we show that if the nonlinear Schrödinger (NLS) equation is generalized by simultaneously taking into account higher-order dispersion, a quintic nonlinearity, and self-steepening terms, the resulting equation is interesting as it has exact soliton solutions which may be (depending on the values of the coefficients) stable or unstable, standard or "embedded," fixed or "moving" (i.e., solitons which advance along the retarded-time axis). We investigate the stability of these solitons by means of a modified version of the Vakhitov-Kolokolov criterion, and numerical tests are carried out to corroborate that these solitons respond differently to perturbations. It is shown that this generalized NLS equation can be derived from a Lagrangian density which contains an auxiliary variable, and Noether's theorem is then used to show that the invariance of the action integral under infinitesimal gauge transformations generates a whole family of conserved quantities. Finally, we study if this equation has the Painlevé property.

Supertransmission channel for an intrinsic localized mode in a one-dimensional nonlinear physical lattice

It is well known that a moving intrinsic localized mode (ILM) in a nonlinear physical lattice looses energy because of the resonance between it and the underlying small amplitude plane wave spectrum. By exploring the Fourier transform (FT) properties of the nonlinear force of a running ILM in a driven and damped 1D nonlinear lattice, as described by a 2D wavenumber and frequency map, we quantify the magnitude of the resonance where the small amplitude normal mode dispersion curve and the FT amplitude components of the ILM intersect. We show that for a traveling ILM characterized by a specific frequency and wavenumber, either inside or outside the plane wave spectrum, and for situations where both onsite and intersite nonlinearity occur, either of the hard or soft type, the strength of this resonance depends on the specific mix of the two nonlinearities. Examples are presented demonstrating that by engineering this mix the resonance can be greatly reduced. The end result is a supertransmission channel for either a driven or undriven ILM in a nonintegrable, nonlinear yet physical lattice.

Moving embedded lattice solitons

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 differential-difference version of a higher-order nonlinear Schrodinger equation were found [Gonzalez-Perez-Sandi, Fujioka, and Malomed, Physica D 197, 86 (2004)]. The discovery of these ELS gives rise to relevant questions such as the following: (1) Are there continuous families of ELS? (2) Can ELS be stable? (3) Is it possible for ELS to move along the lattice? (4) How do ELS interact? The present work addresses these questions by showing that a novel equation (a discrete version of a complex modified Korteweg-de Vries equation that 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.

Stability analysis of embedded solitons in the generalized third-order nonlinear Schrodinger equation

We study the generalized third-order nonlinear Schrodinger (NLS) equation which admits a one-parameter family of single-hump embedded solitons. Analyzing the spectrum of the linearization operator near the embedded soliton, we show that there exists a resonance pole in the left half-plane of the spectral parameter, which explains linear stability, rather than nonlinear semistability, of embedded solitons. Using exponentially weighted spaces, we approximate the resonance pole both analytically and numerically. We confirm in a near-integrable asymptotic limit that the resonance pole gives precisely the linear decay rate of parameters of the embedded soliton. Using conserved quantities, we qualitatively characterize the stable dynamics of embedded solitons.

Accumulation of embedded solitons in systems with quadratic nonlinearity

Previous numerical studies have revealed the existence of embedded solitons (ESs) in a class of multiwave systems with quadratic nonlinearity, families of which seem to emerge from a critical point in the parameter space, where the zero solution has a fourfold zero eigenvalue. In this paper, the existence of such solutions is studied in a three-wave model. An appropriate rescaling casts the system in a normal form, which is universal for models supporting ESs through quadratic nonlinearities. The normal-form system contains a single irreducible parameter epsilon, and is tantamount to the basic model of type-I second-harmonic generation. An analytical approximation of Wentzel-Kramers-Brillouin type yields an asymptotic formula for the distribution of discrete values of epsilon at which the ESs exist. Comparison with numerical results shows that the asymptotic formula yields an exact value of the scaling index, -65, and a fairly good approximation for the numerical factor in front of the scaling term.

Stability of double-peaked solitons in Bragg gratings with the quadratic nonlinearity

We report systematic results for the existence and stability of double-peaked (DP) solitons in the known model including the Bragg grating, which acts on the fundamental- and second-harmonic waves, and the quadratic (chi(2)) nonlinearity, which accounts for the parametric interaction between the harmonics. We identify existence and stability regions for the DP solitons in the plane of relevant parameters (the relative Bragg reflectivity at the two harmonics, and phase mismatch q between them). We conclude that the existence region considerably expands with the soliton's velocity v, while the stability area remains nearly constant up to a critical value of v. The stability region quickly vanishes as one crosses the critical value, while the region of the existence of unstable DP solitons does not disappear. The stability is confined to negative soliton frequencies, and almost entirely to q<0. Collisions between stable moving solitons are investigated too, with a conclusion that they are always destructive.

Standard and embedded solitons in nematic optical fibers

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 wave packets of transverse magnetic modes propagating through the fiber. This procedure allows us to generate different hierarchies of nonlinear partial differential equations 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 (CM KdV) which governs the dynamics for the amplitude of the wave packet. In this derivation the dispersion, self-focussing, and diffraction in the nematic fiber are taken into account. It is shown that this CM KdV 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 wave numbers 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.

Solitary waves in systems with separated Bragg grating and nonlinearity

The existence and stability of solitons in a dual-core optical waveguide, in which one core has Kerr nonlinearity while the other one is linear with a Bragg grating written on it, are investigated. The system's spectrum for the frequency omega of linear waves always contains a gap. If the group velocity c in the linear core is zero, it also has two other, upper and lower (in terms of omega) gaps. If c not equal to 0, the upper and lower gaps do not exist in the rigorous sense, as each overlaps with one branch of the continuous spectrum. When c=0, a family of zero-velocity soliton solutions, filling all the three gaps, is found analytically. Their stability is tested numerically, leading to a conclusion that only solitons in an upper section of the upper gap are stable. For c not equal to 0, soliton solutions are sought for numerically. Stationary solutions are only found in the upper gap, in the form of unusual solitons, which exist as a continuous family in the former upper gap, despite its overlapping with one branch of the continuous spectrum. A region in the parameter plane (c,omega) is identified where these solitons are stable; it is again an upper section of the upper gap. Stable moving solitons are found too. A feasible explanation for the (virtual) existence of these solitons, based on an analytical estimate of their radiative-decay rate (if the decay takes place), is presented.

Three-dimensional walking spatiotemporal solitons in quadratic media

Two-parameter families of chirped stationary three-dimensional spatiotemporal solitons in dispersive quadratically nonlinear optical media featuring type-I second-harmonic generation are constructed in the presence of temporal walk-off. Basic features of these walking spatiotemporal solitons, including their dynamical stability, are investigated in the general case of unequal group-velocity dispersions at the fundamental and second-harmonic frequencies. In the cases when the solitons are unstable, the growth rate of a dominant perturbation eigenmode is found as a function of the soliton wave number shift. The findings are in full agreement with the stability predictions made on the basis of a marginal linear-stability curve. It is found that the walking three-dimensional spatiotemporal solitons are dynamically stable in most cases; hence in principle they may be experimentally generated in quadratically nonlinear media.

From parametric gap solitons to chaos by means of second-harmonic generation in Bragg gratings

We review the theory of light localization due to the combined action of single or double Bragg coupling between dichromatic counterpropagating envelopes and parametric mixing nonlinearities. We discuss existence, stability, and excitation of such localized envelopes. We also investigate the link between stationary gap solitons and input-output response of nonlinear quadratic Bragg gratings. Frustrated transmission and multistable switching is expected to occur under suitable integrable (cascading) limits. Substantial deviations from these conditions lead to the onset of spatial chaos. (c) 2000 American Institute of Physics.