82 research outputs found

    Solitons in nonlinear lattices

    Full text link
    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

    Nonlinear edge waves in mechanical topological insulators

    Get PDF
    We show theoretically that the classical 1D nonlinear Schrödinger (NLS) and coupled nonlinear Schrödinger (CNLS) equations govern the envelope(s) of localised and unidirectional nonlinear travelling edge waves in a 2D mechanical topological insulator (MTI). The MTI consists of a collection of pendula with weak Duffing nonlinearity connected by linear springs that forms a mechanical analogue of the quantum spin Hall effect (QSHE). It is found, through asymptotic analysis and dimension reduction, that the NLS and CNLS respectively describe the unimodal and bimodal properties of the nonlinear system. The governing bimodal CNLS is found to be non-integrable by nature and as such we discover new solutions by exploring the spatial dynamics of the reduced travelling wave ODE with general parameters. Such solutions include travelling fronts and, by numerically continuing these fronts, one can find vector soliton (VS) in non integrable CNLS equations. The equilibria can also undergo both pitchfork and Turing bifurcation in the reversible spatial dynamical system and we discuss relevant conditions for the existence and consequences of such critical values. We briefly discuss the necessity of the developed front condition in forming such structures and present an analytical framework for front-grey soliton collisions by utilising conserved quantities of the non-integrable CNLS. The existence/stability of front and VS solutions can be inferred by spatial hyperbolicity and linear stability of the background fields, with the criteria presented here. VS solutions are considered in the form of bright-bright, bright-dark, and dark-dark solitons and their collision dynamics are explored qualitatively in the non-integrable regime. The Turing analysis presents the existence of periodic and localised patterned states in the CNLS, and we compare these solutions to those found in the analysis of the Swift-Hohenberg equation. Theoretical predictions from the 1D (C)NLS are confirmed by numerical simulations of the original 2D MTI for various types of travelling waves and rogue waves. As a result of topological protection the edge solitons persist over long time intervals and through irregular boundaries. Due to the robustness of topologically protected edge solitons (TPES) it is suggested that their existence may have significant implications on the design of acoustic devices. Spacetime simulations show a clear possibility of utilising MTIs in acoustical cloaking with TPES a vital player in such processes

    Roadmap on optical rogue waves and extreme events

    Get PDF
    The pioneering paper 'Optical rogue waves' by Solli et al (2007 Nature 450 1054) started the new subfield in optics. This work launched a great deal of activity on this novel subject. As a result, the initial concept has expanded and has been enriched by new ideas. Various approaches have been suggested since then. A fresh look at the older results and new discoveries has been undertaken, stimulated by the concept of 'optical rogue waves'. Presently, there may not by a unique view on how this new scientific term should be used and developed. There is nothing surprising when the opinion of the experts diverge in any new field of research. After all, rogue waves may appear for a multiplicity of reasons and not necessarily only in optical fibers and not only in the process of supercontinuum generation. We know by now that rogue waves may be generated by lasers, appear in wide aperture cavities, in plasmas and in a variety of other optical systems. Theorists, in turn, have suggested many other situations when rogue waves may be observed. The strict definition of a rogue wave is still an open question. For example, it has been suggested that it is defined as 'an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses'. This definition (as suggested by a peer reviewer) is clear at the intuitive level and can be easily extended to the case of spatial beams although additional clarifications are still needed. An extended definition has been presented earlier by N Akhmediev and E Pelinovsky (2010 Eur. Phys. J. Spec. Top. 185 1-4). Discussions along these lines are always useful and all new approaches stimulate research and encourage discoveries of new phenomena. Despite the potentially existing disagreements, the scientific terms 'optical rogue waves' and 'extreme events' do exist. Therefore coordination of our efforts in either unifying the concept or in introducing alternative definitions must be continued. From this point of view, a number of the scientists who work in this area of research have come together to present their research in a single review article that will greatly benefit all interested parties of this research direction. Whether the authors of this 'roadmap' have similar views or different from the original concept, the potential reader of the review will enrich their knowledge by encountering most of the existing views on the subject. Previously, a special issue on optical rogue waves (2013 J. Opt. 15 060201) was successful in achieving this goal but over two years have passed and more material has been published in this quickly emerging subject. Thus, it is time for a roadmap that may stimulate and encourage further research.Peer ReviewedPostprint (author's final draft

    Analysis of localized solutions in coupled Gross-Pitavskii equations

    Get PDF
    Bose-Einstein condensates (BECs) have been one of the most active areas of research since their experimental birth in 1995. The complicated nature of the experiments on BECs suggests to observe them in reduced dimensions. The dependence of the collective excitations of the systems on the spatial degrees of freedom allows the study in lower dimensions. In this thesis, we first study two effectively one-dimensional parallel linearly coupled BECs in the presence of external potentials. The system is modelled by linearly coupled Gross-Pitaevskii (GP) equations. In particular, we discuss the dark solitary waves and the grey-soliton-like solutions representing analogues of superconducting Josephson fluxons which we refer to as the fluxon analogue (FA) solutions. We analyze the existence, stability and time dynamics of FA solutions and coupled dark solitons in the presence of a harmonic trap. We observe that the presence of the harmonic trap destabilizes the FA solutions. However, stabilization is possible by controlling the effective linear coupling between the condensates. We also derive theoretical approximations based on variational formulations to study the dynamics of the solutions semi-analytically. We then study multiple FA solutions and coupled dark solitons in the same settings. We examine the effects of trapping strength on the existence and stability of the localized solutions. We also consider the interactions of multiple FA solutions as well as coupled dark solitons. In addition, we determine the oscillation frequencies of the prototypical structures of two and three FA solutions using a variational approach. Finally, we consider two effectively two-dimensional parallel coupled BECs enclosed in a double well potential. The system is modelled by two GP equations coupled by linear and nonlinear cross-phase-modulations. We study a large set of radially symmetric nonlinear solutions of the system in the focusing and defocusing cases. The relevant three principal branches, i.e. the ground state and the first two excited states, are continued as a function of either linear or nonlinear couplings. We investigate the linear stability and time evolution of these solutions in the absence and presence of a topological charge. We notice that only the chargeless or charged ground states can be stabilized by adjusting the linear or nonlinear coupling between the condensates

    Impact of modulation instability on distributed optical fiber sensors

    Get PDF
    Modulation instability (MI) as the main limit to the sensing distance of distributed fiber sensors is thoroughly investigated in this thesis in order to obtain a model for predicting its characteristics and alleviating its effects. Starting from Maxwell's equations in optical fibers, the nonlinear Schrödinger equation (NLSE) describing the propagation of wave envelope in nonlinear dispersive media is derived. As the main tool for analyzing modulation instability, the NLSE is numerically evaluated using the split-step Fourier method and its analytical closed-form solutions such as solitons are utilized to validate the numerical algorithms. As the direct consequence of the NLSE, self-phase modulation is utilized to measure the nonlinear coefficient of optical fibers via a self-aligned interferometer. The modulation instability gain is obtained by applying a linear stability analysis to the NLSE assuming a white background noise as the seeding for the nonlinear interaction. The MI gain spectrum is expressed by hyperbolic functions for lossless fibers and by Bessel functions with complex orders for fibers with attenuation. An approximate gain spectrum is presented for lossy fibers based on the gain in lossless optical fibers. The accuracy of the analytical results and approximate formulas is evaluated by performing Monte Carlo simulations on the NLSE. The impact of background noise on the onset and evolution of modulation instability is analytically investigated and experimentally demonstrated. Power depletion due to the nonlinear process of modulation instability is modeled by integrating its gain spectrum using Laplace's method. Based on that, a critical power for MI is proposed by introducing the notion of depletion ratio. The model is verified by numerical simulation and experimental measurement. An optimal input power for distributed fiber sensors is proposed to maximize the output optical power and thus, the far end signal-to-noise ratio. Furthermore, the recurrence phenomenon of Fermi-Pasta-Ulam is experimentally observed and numerically simulated, validating the utilized numerical techniques. A standard Brillouin optical time-domain analyser serves as the experimental test bench for the proposed model. As the physical phenomenon behind the experiment, stimulated Brillouin scattering is described based on a pump-probe interaction mechanism through an acoustic wave. A 25 km single-mode fiber is employed as a nonlinear medium with anomalous dispersion at the pump wavelength 1550 nm. The evolution of pump power propagating along the fiber is mapped using the Brillouin interaction with the probe lightwave. The measured longitudinal power traces are processed to extract the impact of MI on the pump power. It is experimentally demonstrated that distributed fiber sensors with orthogonally-polarized pumps suffer less from modulation instability. As the scalar modulation instability of each pump reduces, vector modulation instability occurs because of interaction between the pumps; however, the overall performance improves. A version of the coupled nonlinear Schrödinger equations known as the Manakov system is shown to describe the behavior of two-pump distributed fiber sensors employing optical fibers with random birefringence. The excellent agreement between the experimental and numerical results indicates that the performance limit of two-pump distributed fiber sensors is determined by polarization modulation instability

    Analysis of localized solutions in coupled Gross-Pitavskii equations

    Get PDF
    Bose-Einstein condensates (BECs) have been one of the most active areas of research since their experimental birth in 1995. The complicated nature of the experiments on BECs suggests to observe them in reduced dimensions. The dependence of the collective excitations of the systems on the spatial degrees of freedom allows the study in lower dimensions. In this thesis, we first study two effectively one-dimensional parallel linearly coupled BECs in the presence of external potentials. The system is modelled by linearly coupled Gross-Pitaevskii (GP) equations. In particular, we discuss the dark solitary waves and the grey-soliton-like solutions representing analogues of superconducting Josephson fluxons which we refer to as the fluxon analogue (FA) solutions. We analyze the existence, stability and time dynamics of FA solutions and coupled dark solitons in the presence of a harmonic trap. We observe that the presence of the harmonic trap destabilizes the FA solutions. However, stabilization is possible by controlling the effective linear coupling between the condensates. We also derive theoretical approximations based on variational formulations to study the dynamics of the solutions semi-analytically. We then study multiple FA solutions and coupled dark solitons in the same settings. We examine the effects of trapping strength on the existence and stability of the localized solutions. We also consider the interactions of multiple FA solutions as well as coupled dark solitons. In addition, we determine the oscillation frequencies of the prototypical structures of two and three FA solutions using a variational approach. Finally, we consider two effectively two-dimensional parallel coupled BECs enclosed in a double well potential. The system is modelled by two GP equations coupled by linear and nonlinear cross-phase-modulations. We study a large set of radially symmetric nonlinear solutions of the system in the focusing and defocusing cases. The relevant three principal branches, i.e. the ground state and the first two excited states, are continued as a function of either linear or nonlinear couplings. We investigate the linear stability and time evolution of these solutions in the absence and presence of a topological charge. We notice that only the chargeless or charged ground states can be stabilized by adjusting the linear or nonlinear coupling between the condensates

    Non-equilibrium dynamics of Bose Einstein condensates

    Get PDF

    Many body effects in one-dimensional attractive Bose gases

    Get PDF
    In this thesis we investigate the properties of ultra-cold quantum gases in reduced dimension and the effects of harmonic confinement on soliton-like properties. We study regimes of agreement between mean-field and many-body theories the generation of entanglement between initially independent finite sized atomic systems. Classical solitons are non-dispersing waves which occur in integrable systems, such as atomic Bose-Einstein condensates in one dimension. Bright and dark solitons are possible, which exist as peaks or dips in density. Quantum solitons are the bound-state solutions to a system satisfying quantum integrability, given via the Bethe Ansatz. Such integrability is broken by the introduction of harmonic confinement. We investigate the equivalence of the classical field and many-body solutions in the limit of large numbers of atoms and derive numerical and variational approaches to examine the ground state energy in harmonic confinement and the fidelity between a Hartree-product solution and a quantum soliton solution. Soliton collisions produce no entanglement between either state and result only in an asymptotic position and phase shift, however external potentials break integrability and thus give the possibility of entangling solitons. We investigate the dynamical entanglement generation between two atomic dimers in harmonic confinement via exact diagonalisation in a basis of Harmonic oscillator functions, making use of the separability of the centre-of-mass component of the Hamiltonian. We show repulsive states show complex dynamics, but with an overall tendency towards states of larger invariant correlation entropy, whereas attractive states resist entanglement unless a phase matching condition is satisfied. This phase matching condition could in theory be used to generate states with highly non-Poissonian number superpositions in atomic systems with controlled number
    corecore