Modification of solitary waves by third-harmonic generation.

The effect of phase-matched third-harmonic generation on the structure and stability of spatial solitary waves is investigated. A power threshold for the existence of two-frequency spatial solitons is found, and the multistability of solitary waves in a Kerr medium owing to a higher-order nonlinear phase shift caused by cascaded third-order processes is revealed.

Solitons and collapse suppression due to parametric interaction in bulk Kerr media.

We demonstrate that weak parametric interaction of a fundamental beam with its harmonic field in a Kerr medium can drastically modify the beam dynamics, giving rise to very complex bifurcation phenomena and quasi solitons. Most importantly, we reveal a novel physical mechanism of the collapse suppression in a bulk optical Kerr medium: parametric coupling to a weakly radiating harmonic field.

Quadratic solitary waves in a counterpropagating quasi-phase-matched configuration.

We demonstrate the possibility of self-trapping of optical beams by use of quasi phase matching in a counterpropagating configuration in quadratic media. We also show the predominant stability of these beams and estimate the power level required for their observation.

Nonlinear dynamics of higher-order solitons near the oscillatory instability threshold.

Nonlinear theory describing the dynamics of solitons in the vicinity of oscillatory instability threshold with a low frequency offset is developed. The theory is tested on the example of parametric degenerate four-wave mixing. All major predictions of our theory are in agreement with the results of direct numerical modeling.

Interaction of two one-dimensional Bose-Einstein solitons: chaos and energy exchange.

We analyze the bright soliton interactions of a Bose-Einstein condensate in quasi-one-dimensional traps putting an emphasis on integrability break down due to a trapping potential. In particular, we derive a simple analytical model, which describes well all major features of soliton dynamics including chaos and energy exchange between interacting solitons induced by the trapping potential.

Stable localized vortex solitons.

We demonstrate that parametric interaction of a fundamental beam with its second harmonic in bulk media, in the presence of self-defocusing third-order nonlinearity, gives rise to the first ever examples of completely stable localized ring-shaped solitons with intrinsic vorticity n=1 and n=2. The stability is demonstrated both in direct simulations and by computing eigenvalues of the corresponding linearized equations. A potential application of the (2+1)-dimensional ring solitons in optics is a possibility to design a reconfigurable multichannel system guiding signal beams.

Long-range interaction and nonlinear localized modes in photonic crystal waveguides.

We develop the theory of nonlinear localized modes (intrinsic localized modes or discrete breathers) in two-dimensional (2D) photonic crystal waveguides. We consider different geometries of the waveguides created by an array of nonlinear dielectric rods embedded into an otherwise perfect linear 2D photonic crystal, and demonstrate that the effective interaction in such waveguides is nonlocal, being described by a nonlinear lattice model with long-range coupling and nonlocal nonlinearity. We reveal the existence of different types of nonlinear guided mode that are also localized in the waveguide direction, and describe their unique properties, including bistability.

Higher-order nonlinear modes and bifurcation phenomena due to degenerate parametric four-wave mixing.

We demonstrate that weak parametric interaction of a fundamental beam with its third harmonic field in Kerr media gives rise to a rich variety of families of nonfundamental (multihumped) solitary waves. Making a comprehensive comparison between bifurcation phenomena for these families in bulk media and planar waveguides, we discover two types of soliton bifurcations and other interesting findings. The latter includes (i) multihumped solitary waves without even or odd symmetry and (ii) multihumped solitary waves with large separation between their humps which, however, may not be viewed as bound states of several distinct one-humped solitons.

Optical vortex solitons in parametric wave mixing.

We analyze two-component spatial optical vortex solitons supported by parametric wave mixing processes in a nonlinear bulk medium. We study two distinct cases of such localized waves, namely, parametric vortex solitons due to phase-matched second-harmonic generation in an optical medium with competing quadratic and cubic nonlinear response, and vortex solitons in the presence of third-harmonic generation in a cubic medium. We find, analytically and numerically, the structure of two-component vortex solitons, and also investigate modulational instability of their plane-wave background.

Interaction of self-guided beams of different frequencies.

We show that a self-guided beam in a nonlinear planar film can be split, switched, or deflected in a controlled manner by use of another self-guided beam of a different frequency. The new phenomena are power dependent and are potentially attractive for all-optical photonic device applications.

Simple model for spatial optical solitons in planar waveguides.

We use a variational method to develop a simple model for the steady-state behavior of self-trapped beams that have recently been observed in nonlinear planar waveguides. The model allows us to study the transition from two-dimensional to three-dimensional self-trapping as power levels increase and also to investigate the effect of saturation on these beams.

Group velocity and dispersion in nonlinear-optical fibers.

Many of the nonlinear effects being studied in optical fibers depend critically on the dispersion characteristics of the fiber. We generalize the scalar form of Brown's identity for group velocity in a linear waveguide to deal with arbitrary nonlinearity and use this generalization to show that the dispersion characteristics of a step-index fiber at the high powers required for many nonlinear processes may be significantly different from their linear values.