Black and gray Helmholtz Kerr soliton refraction

efraction of black and gray solitons at boundaries separating different defocusing Kerr media is analyzed within a Helmholtz framework. A universal nonlinear Snell’s law is derived that describes gray soliton refraction, in addition to capturing the behavior of bright and black Kerr solitons at interfaces. Key regimes, defined by beam and interface characteristics, are identified and predictions are verified by full numerical simulations. The existence of a unique total non-refraction angle for gray solitons is reported; both internal and external refraction at a single interface is shown possible (dependent only on incidence angle). This, in turn, leads to the proposal of positive or negative lensing operations on soliton arrays at planar boundaries

Paraxial light in a Cole-Cole nonlocal medium: integrable regimes and singularities

Nonlocal nonlinear Schroedinger-type equation is derived as a model to describe paraxial light propagation in nonlinear media with different `degrees' of nonlocality. High frequency limit of this equation is studied under specific assumptions of Cole-Cole dispersion law and a slow dependence along propagating direction. Phase equations are integrable and they correspond to dispersionless limit of Veselov-Novikov hierarchy. Analysis of compatibility among intensity law (dependence of intensity on the refractive index) and high frequency limit of Poynting vector conservation law reveals the existence of singular wavefronts. It is shown that beams features depend critically on the orientation properties of quasiconformal mappings of the plane. Another class of wavefronts, whatever is intensity law, is provided by harmonic minimal surfaces. Illustrative example is given by helicoid surface. Compatibility with first and third degree nonlocal perturbations and explicit solutions are also discussed.Comment: 12 pages, 2 figures; eq. (36) corrected, minor change

Stationary Black Holes: Uniqueness and Beyond

The spectrum of known black-hole solutions to the stationary Einstein equations has been steadily increasing, sometimes in unexpected ways. In particular, it has turned out that not all black-hole-equilibrium configurations are characterized by their mass, angular momentum and global charges. Moreover, the high degree of symmetry displayed by vacuum and electro-vacuum black-hole spacetimes ceases to exist in self-gravitating non-linear field theories. This text aims to review some developments in the subject and to discuss them in light of the uniqueness theorem for the Einstein-Maxwell system.Comment: Major update of the original version by Markus Heusler from 1998. Piotr T. Chru\'sciel and Jo\~ao Lopes Costa succeeded to this review's authorship. Significantly restructured and updated all sections; changes are too numerous to be usefully described here. The number of references increased from 186 to 32

Dynamics of ring dark solitons in Bose-Einstein condensates and nonlinear optics

Quasiparticle approach to dynamics of dark solitons is applied to the case of ring solitons. It is shown that the energy conservation law provides the effective equations of motion of ring dark solitons for general form of the nonlinear term in the generalized nonlinear Schroedinger or Gross-Pitaevskii equation. Analytical theory is illustrated by examples of dynamics of ring solitons in light beams propagating through a photorefractive medium and in non-uniform condensates confined in axially symmetric traps. Analytical results agree very well with the results of our numerical simulations.Comment: 10 pages, 4 figure

Korteweg-de Vries description of Helmholtz-Kerr dark solitons

A wide variety of different physical systems can be described by a relatively small set of universal equations. For example, small-amplitude nonlinear Schrödinger dark solitons can be described by a Korteweg-de Vries (KdV) equation. Reductive perturbation theory, based on linear boosts and Gallilean transformations, is often employed to establish connections to and between such universal equations. Here, a novel analytical approach reveals that the evolution of small-amplitude Helmholtz–Kerr dark solitons is also governed by a KdV equation. This broadens the class of nonlinear systems that are known to possess KdV soliton solutions, and provides a framework for perturbative analyses when propagation angles are not negligibly small. The derivation of this KdV equation involves an element that appears new to weakly nonlinear analyses, since transformations are required to preserve the rotational symmetry inherent to Helmholtz-type equations

Off-diagonal cosmological solutions in emergent gravity theories and Grigory Perelman entropy for geometric flows

We develop an approach to the theory of relativistic geometric flows and emergent gravity defined by entropy functionals and related statistical thermodynamics models. Nonholonomic deformations of G. Perelman's functionals and related entropic values are used for deriving relativistic geometric evolution flow equations. For self-similar configurations, such equations describe generalized Ricci solitons defining modified Einstein equations. We analyze possible connections between relativistic models of nonholonomic Ricci flows and emergent modified gravity theories. We prove that corresponding systems of nonlinear partial differential equations, PDEs, for entropic flows and modified gravity possess certain general decoupling and integration properties. There are constructed new classes of exact and parametric solutions for nonstationary configurations and locally anisotropic cosmological metrics in modified gravity theories and general relativity. Such solutions describe scenarios of nonlinear geometric evolution and gravitational and matter field dynamics with pattern-forming and quasiperiodic structure and various space quasicrystal and deformed spacetime crystal models. We analyze new classes of generic off-diagonal solutions for entropic gravity theories and show how such solutions can be used for explaining structure formation in modern cosmology. Finally, we speculate why the approaches with Perelman-Lyapunov type functionals are more general or complementary to the constructions elaborated using the concept of Bekenstein-Hawking entropy.Comment: accepted to EPJC; latex2e 11pt, 35 pages with a table of contents; v3 is substantially modified with a new title and a new co-autho

Stability of Spatial Optical Solitons

We present a brief overview of the basic concepts of the soliton stability theory and discuss some characteristic examples of the instability-induced soliton dynamics, in application to spatial optical solitons described by the NLS-type nonlinear models and their generalizations. In particular, we demonstrate that the soliton internal modes are responsible for the appearance of the soliton instability, and outline an analytical approach based on a multi-scale asymptotic technique that allows to analyze the soliton dynamics near the marginal stability point. We also discuss some results of the rigorous linear stability analysis of fundamental solitary waves and nonlinear impurity modes. Finally, we demonstrate that multi-hump vector solitary waves may become stable in some nonlinear models, and discuss the examples of stable (1+1)-dimensional composite solitons and (2+1)-dimensional dipole-mode solitons in a model of two incoherently interacting optical beams.Comment: 34 pages, 9 figures; to be published in: "Spatial Optical Solitons", Eds. W. Torruellas and S. Trillo (Springer, New York

High frequency integrable regimes in nonlocal nonlinear optics

We consider an integrable model which describes light beams propagating in nonlocal nonlinear media of Cole-Cole type. The model is derived as high frequency limit of both Maxwell equations and the nonlocal nonlinear Schroedinger equation. We demonstrate that for a general form of nonlinearity there exist selfguided light beams. In high frequency limit nonlocal perturbations can be seen as a class of phase deformation along one direction. We study in detail nonlocal perturbations described by the dispersionless Veselov-Novikov (dVN) hierarchy. The dVN hierarchy is analyzed by the reduction method based on symmetry constraints and by the quasiclassical Dbar-dressing method. Quasiclassical Dbar-dressing method reveals a connection between nonlocal nonlinear geometric optics and the theory of quasiconformal mappings of the plane.Comment: 45 pages, 4 figure

An accurate envelope equation for light propagation in photonic nanowires: new nonlinear effects

We derive a new unidirectional evolution equation for photonic nanowires made of silica. Contrary to previous approaches, our formulation simultaneously takes into account both the vector nature of the electromagnetic field and the full variations of the effective modal profiles with wavelength. This leads to the discovery of new, previously unexplored nonlinear effects which have the potential to affect soliton propagation considerably. We specialize our theoretical considerations to the case of perfectly circular silica strands in air, and we support our analysis with detailed numerical simulations.Comment: 5 figures. The normalization of the fields is now more appropriate (orthonormal). Figure concerning dispersion of gamma0 has been eliminated. New figures for nonlinear coefficients and pulse propagation for the corrected envelope functio