Analytic and numerical investigations of a cavity containing a Kerr medium are reported. The mean field equation with plane-wave excitation and diffraction is assumed. Stable hexagons are dominant close to threshold for a self-focusing medium. Bistable switching frustrates pattern formation for a self-defocusing medium. Under appropriate parametric conditions that we identify, there is coexistence of a homogeneous stationary solution, of a hexagonal pattern solution and of a large (in principle infinite) number of localized structure solutions which connect the homogeneous and hexagonal state. Further above threshold, the hexagons show defects, and then break up with apparent turbulence. For Gaussian beam excitation, the different symmetry leads to polygon formation for narrow beams, but quasihexagonal structures appear for broader beams.

In this paper, we propose the use of ultranarrow soliton beams in miniaturized nonlinear optical devices. We derive a nonparaxial nonlinear Schrödinger equation and show that it has an exact non-paraxial soliton solution from which the paraxial soliton is recovered in the appropriate limit. The physical and mathematical geometry of the non-paraxial soliton is explored through the consideration of dispersion relations, rotational transformations and approximate solutions. We highlight some of the unphysical aspects of the paraxial limit and report modifications to the soliton width, the soliton area and the soliton (phase) period which result from the breakdown of the slowly varying envelope approximation

A general dark-soliton solution of the Helmholtz equation (with defocusing Kerr nonlinearity) that has on- and off-axis, gray and black, paraxial and Helmholtz solitons as particular solutions, is reported. Modifications to soliton transverse velocity, width, phase period, and existence conditions are derived and explained in geometrical terms. Simulations verify analytical predictions and also demonstrate spontaneous formation of Helmholtz solitons and transparency of their interactions.

Exact analytical soliton solutions of the nonlinear Helmholtz equation are reported. A lucid generalization of paraxial soliton theory that incorporates nonparaxial effects is found.

We consider some features of spatial solitary-wave switching in a unidirectional ring cavity that is partially filled with a fast and saturably self-focusing nonlinear medium. Large (part-beam switched) solitary arrays are considered. It is found that prescribed binary patterns may be encoded in the duration of a single cavity transit and subsequently remain stable over thousands of transits. Beam interrupt allows pixels to be switched off in fewer than ten cavity transits. Pixel instabilities on an unpixelated beam are shown to arise from spatial solitary attractive forces and intensity gradients.

Reflection and refraction of spatial solitons at dielectric interfaces, accommodating arbitrarily angles of incidence, is studied. Analysis is based on Helmholtz soliton theory, which eliminates the angular restriction associated with the paraxial approximation. A novel generalization of Snell's law is discovered that is valid for collimated light beams and the entire angular domain. Our new theoretical predictions are shown to be in excellent agreement with full numerical simulations. New qualitative features of soliton refraction and limitations of previous paraxial analyses are highlighted.

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Abstract. We report, for the first time, exact analytical boundary solitons of a generalized cubic-quintic Non-Linear Helmholtz (NLH) equation. These solutions have a linked-plateau topology that is distinct from conventional dark soliton solutions; their amplitude and intensity distributions are spatially delocalized and connect regions of finite and zero wave-field disturbances (suggesting also the classification as "edge solitons"). Extensive numerical simulations compare the stability properties of recentlyreported Helmholtz bright solitons, for this type of polynomial non-linearity, to those of the new boundary solitons. The latter are found to possess a remarkable stability characteristic, exhibiting robustness against perturbations that would otherwise lead to the destabilizing of their bright-soliton counterparts. Helmholtz bright and boundary solitons 2 IntroductionSolitons are ubiquitous entities in nature. Whenever linear effects (such as dispersion, diffraction or diffusion) are balanced exactly by non-linearity (self-phase modulation, self-focusing or reactionkinetic properties, respectively), robust self-trapped structures -solitons -can emerge as dominant modes of the system dynamics. These localized self-stabilizing non-linear waves arise widely in nature since quite different physical systems are governed by a relatively small set of universal equations, at least to first approximation. Solitons are often sech ("bell")-or tanh ("S")-shaped structures. The latter class are sometimes referred to as kink solitons, and they generally possess topologically non-trivial phase distributions. Phase-topological kink solitons appear in a range of physically diverse systems, and play the role of "fronts" and "domain walls". In classical mechanics, for example, they describe collective long-wave excitations on a line of weakly-coupled pendula. In condensed matter, kink solitons arise in simple models of one-dimensional lattice-dynamics when studying the motion of dislocations and domain walls in ferromagnetic crystals, and they also play a key role in the phenomenological understanding of phase transitions. In chemical kinetics, kink solitons appear as solutions to reaction-diffusion equations. They also occur in hydrodynamics, plasma physics, quantum field theory and cosmology. Comprehensive reviews of these systems can be found in Refs. [1][2][3][4].Our principle concern in this paper is with spatial soliton beams found in non-linear optics [5,6]. These types of soliton can arise when the tendency of a collimated light beam to diffract is opposed by the non-linear properties of the optical medium. When these two effects (diffractive broadening, and narrowing due to self-focusing) become comparable, then a stationary beam can exist whose transverse intensity distribution is invariant along the propagation direction. Spatial solitons are of theoretical interest as particular solutions to generic non-linear evolution equations, but they are also the subject of considerable experimental investigation. The robu...