Solitons in nonlinear lattices

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by purely nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are surveyed too, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation (BEC). The solitons are considered in one, two, and three dimensions (1D, 2D, and 3D). Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of 1D solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for the theoretical and experimental studies alike. In both the 1D and 2D cases, the mechanism which creates solitons in NLs is principally different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.Comment: 169 pages, 35 figures, a comprehensive survey of results on solitons in purely nonlinear and mixed lattices, to appear in Reviews of Modern Physic

Soliton generation and control in engineered materials

Optical solitons provide unique opportunities for the control of light‐bylight. Today, the field of soliton formation in natural materials is mature, as the main properties of the possible soliton states are well understood. In particular, optical solitons have been observed experimentally in a variety of materials and physical settings, including media with cubic, quadratic, photorefractive, saturable, nonlocal and thermal nonlinearities. New opportunities for soliton generation, stability and control may become accessible in complex engineered, artificial materials, whose properties can be modified at will by, e.g., modulations of the material parameters or the application gain and absorption landscapes. In this way one may construct different types of linear and nonlinear optical lattices by transverse shallow modulations of the linear refractive index and the nonlinearity coefficient or complex amplifying structures in dissipative nonlinear media. The exploration of the existence, stability and dynamical properties of conservative and dissipative solitons in settings with spatially inhomogeneous linear refractive index, nonlinearity, gain or absorption, is the subject of this PhD Thesis. We address stable conservative fundamental and multipole solitons in complex engineered materials with an inhomogeneous linear refractive index and nonlinearity. We show that stable two‐dimensional solitons may exist in nonlinear lattices with transversally alternating domains with cubic and saturable nonlinearities. We consider multicomponent solitons in engineered materials, where one field component feels the modulation of the refractive index or nonlinearity while the other component propagates as in a uniform nonlinear medium. We study whether the cross‐phase‐modulation between two components allows the stabilization of the whole soliton state. Media with defocusing nonlinearity growing rapidly from the center to the periphery is another example of a complex engineered material. We study such systems and, in contrast to the common belief, we have found that stable bright solitons do exist when defocusing nonlinearity grows towards the periphery rapidly enough. We consider different nonlinearity landscapes and analyze the types of soliton solution available in each case. Nonlinear materials with complex spatial distributions of gain and losses also provide important opportunities for the generation of stable one‐ and multidimensional fundamental, multipole, and vortex solitons. We study onedimensional solitons in focusing and defocusing nonlinear dissipative materials with single‐ and double‐well absorption landscapes. In two‐dimensional geometries, stable vortex solitons and complexes of vortices could be observed. We not only address stationary vortex structures, but also steadily rotating vortex solitons with azimuthally modulated intensity distributions in radially symmetric gain landscapes. Finally, we study the possibility of forming stable topological light bullets in focusing nonlinear media with inhomogeneous gain landscapes and uniform twophoton absorption

The existence and stability of solitons in discrete nonlinear Schrödinger equations

In this thesis, we investigate analytically and numerically the existence and stability of discrete solitons governed by discrete nonlinear Schrödinger (DNLS) equations with two types of nonlinearity, i.e., cubic and saturable nonlinearities. In the cubic-type model we consider stationary discrete solitons under the effect of parametric driving and combined parametric driving and damping, while in the saturable-type model we examine travelling lattice solitons. First, we study fundamental bright and dark discrete solitons in the driven cubic DNLS equation. Analytical calculations of the solitons and their stability are carried out for small coupling constant through a perturbation expansion. We observe that the driving can not only destabilise onsite bright and dark solitons, but also stabilise intersite bright and dark solitons. In addition, we also discuss a particular application of our DNLS model in describing microdevices and nanodevices with integrated electrical and mechanical functionality. By following the idea of the work above, we then consider the cubic DNLS equation with the inclusion of parametric driving and damping. We show that this model admits a number of types of onsite and intersite bright discrete solitons of which some experience saddle-node and pitchfork bifurcations. Most interestingly, we also observe that some solutions undergo Hopf bifurcations from which periodic solitons (limit cycles) emerge. By using the numerical continuation software Matcont, we perform the continuation of the limit cycles and determine the stability of the periodic solitons. Finally, we investigate travelling discrete solitons in the saturable DNLS equation. A numerical scheme based on the discretization of the equation in the moving coordinate frame is derived and implemented using the Newton-Raphson method to find traveling solitons with non-oscillatory tails, i.e., embedded solitons. A variational approximation (VA) is also applied to examine analytically the travelling solitons and their stability, as well as to predict the location of the embedded solitons

The existence and stability of solitons in discrete nonlinear Schrödinger equations

In this thesis, we investigate analytically and numerically the existence and stability of discrete solitons governed by discrete nonlinear Schrödinger (DNLS) equations with two types of nonlinearity, i.e., cubic and saturable nonlinearities. In the cubic-type model we consider stationary discrete solitons under the effect of parametric driving and combined parametric driving and damping, while in the saturable-type model we examine travelling lattice solitons. First, we study fundamental bright and dark discrete solitons in the driven cubic DNLS equation. Analytical calculations of the solitons and their stability are carried out for small coupling constant through a perturbation expansion. We observe that the driving can not only destabilise onsite bright and dark solitons, but also stabilise intersite bright and dark solitons. In addition, we also discuss a particular application of our DNLS model in describing microdevices and nanodevices with integrated electrical and mechanical functionality. By following the idea of the work above, we then consider the cubic DNLS equation with the inclusion of parametric driving and damping. We show that this model admits a number of types of onsite and intersite bright discrete solitons of which some experience saddle-node and pitchfork bifurcations. Most interestingly, we also observe that some solutions undergo Hopf bifurcations from which periodic solitons (limit cycles) emerge. By using the numerical continuation software Matcont, we perform the continuation of the limit cycles and determine the stability of the periodic solitons. Finally, we investigate travelling discrete solitons in the saturable DNLS equation. A numerical scheme based on the discretization of the equation in the moving coordinate frame is derived and implemented using the Newton-Raphson method to find traveling solitons with non-oscillatory tails, i.e., embedded solitons. A variational approximation (VA) is also applied to examine analytically the travelling solitons and their stability, as well as to predict the location of the embedded solitons

Asymmetric Wave Propagation Through Saturable Nonlinear Oligomers

In the present paper we consider nonlinear dimers and trimers (more generally, oligomers) embedded within a linear Schrödinger lattice where the nonlinear sites are of saturable type. We examine the stationary states of such chains in the form of plane waves, and analytically compute their reflection and transmission coefficients through the nonlinear oligomer, as well as the corresponding rectification factors which clearly illustrate the asymmetry between left and right propagation in such systems. We examine not only the existence but also the dynamical stability of the plane wave states. Lastly, we generalize our numerical considerations to the more physically relevant case of Gaussian initial wavepackets and confirm that the asymmetry in the transmission properties also persists in the case of such wavepackets

Swinging of two-dimensional solitons in harmonic and Bessel optical lattices

We consider parametric amplification of two-dimensional spatial soliton swinging in longitudinally modulated harmonic and Bessel lattices in Kerr-type saturable medium. We show that soliton center oscillations along different axes in two-dimensional lattices are coupled, which give rise to a number of interesting propagation scenarios including periodic damping and excitation of soliton oscillations along perpendicular axes, selective amplification of soliton swinging along one of transverse axes and enhancement of soliton spiraling.Comment: 15 pages, 4 figures, to appear in Physical Review