Helmholtz bright and boundary solitons

Abstract

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...

Helmholtz Solitons in Optical Materials With a Dual Power-Law Refractive Index

“…The case n σ > 0, n 2σ < 0 supports bright solitons and a class of nonlinear boundary wave; 23 when n σ < 0, n 2σ > 0, one finds coexisting bright hyperbolic solitons, algebraic (bright and dark) solitons, and also class of nonlinear periodic wave. 24 The stability properties of those various solutions have been characterized by semi-analytical and computational investigations.…”

“…3) are distinct from their competitive-focusing counterparts. 23,24 Previously, the VK criterion has been successfully deployed in the analysis of Helmholtz solitons. 23,24,38,39 The validity of this approach lies in spatial symmetry (see Fig.…”

“…We classify these solitons as stable limitcycle attractors of the system. 23,24,38,39 We now analyse solitons in the conditionally-stable regime (where 2 < σ < 4) in terms of the power of the input beam, denoted by P in . Typical curves in the (β, P) plane for solution (20) are shown in Fig.…”

Nonlinear interfaces: intrinsically nonparaxial regimes and effects

“…This nonparaxiality is well described by the scalar nonlinear Helmholtz (NLH) equation [23,24] which has been proposed to overcome the limitations of the NLS, for instance, by arresting soliton collapse in a focusing Kerr-type medium [23] and for which exact analytical soliton solutions have been found [24,25,26]. Substantial differences with paraxial theory are not only revealed by the exact bright Kerr soliton solutions of the NLH equation but are also found in dark Kerr [27], two-component [28], boundary [29] and bistable [30] Helmholtz soliton solutions. When the full Helmholtz approach is used, significant differences with the predictions of NLS theory are also found at a fundamental level, for example, when analysing soliton collisions [31].…”

“…Backscattered waves are filtered out, thus avoiding an evanescent backward field, that can appear to grow in the forward direction and hence masks the contribution of the forward propagating field. This scheme has been applied to the phenomena studied in [27,28,29,30,31].…”