254 research outputs found
Comment on "localized vortices with a semi-integer charge in nonlinear dynamical lattices"
No full textIn a recent paper by Kevrekidis, Malomed, Bishop, and Frantzeskakis [Phys. Rev. E 65, 016605 (2001)] the existence of localized vortices with semi-integer topological charge as exact stationary solutions in a two-dimensional discrete nonlinear Schrödinger model is claimed, as well as the existence of an analog solution in the one-dimensional model. We point out that the existence of such exact stationary solutions would violate fundamental conservation laws, and therefore these claims are erroneous and appear as a consequence of inaccurate numerics. We illustrate the origin of these errors by performing similar numerical calculations using more accurate numerics.</p
Soliton dynamics in linearly coupled discrete nonlinear Schrodinger equations
No full textWe study soliton dynamics in a system of two linearly coupled discrete nonlinear Schrodinger equations, which describe the dynamics of a two-component Bose gas, coupled by an electromagnetic field, and confined in a strong optical lattice. When the nonlinear coupling strengths are equal, we use a unitary transformation to remove the linear coupling terms, and show that the existing soliton solutions oscillate from one species to the other. When the nonlinear coupling strengths are different, the soliton dynamics is numerically investigated and the findings are compared to the results of an effective two-mode model. The case of two linearly coupled Ablowitz-Ladik equations is also briefly discussed. (C) 2009 IMACS. Published by Elsevier B.V. All rights reserved
Patterns ofwater in light
No full textThe intricate patterns emerging from the interactions between soliton stripes of a two-dimensional defocusing nonlinear Schrödinger (NLS) model with a non-local nonlinearity are considered. We show that, for sufficiently strong non-locality, the model is asymptotically reduced to a Kadomtsev-Petviashvilli- II (KPII) equation, which is a common model arising in the description of shallow water waves, as such patterns of water may indeed exist in light (this non-local NLS finds applications in nonlinear optics, modelling beam propagation in media featuring thermal nonlinearities, in plasmas, and in nematic liquid crystals). This way, approximate antidark soliton solutions of the NLS model are constructed from the stable KPII line solitons. By means of direct numerical simulations, we demonstrate that non-resonant and resonant two- and three-antidark NLS stripe soliton interactions give rise to wave configurations that are found in the context of the KPII equation. Thus, our study indicates that patterns which are usually observed in water can also be found in optics. © 2019 The Author(s) Published by the Royal Society. All rights reserved
On quadratic eigenvalue problems arising in stability of discrete vortices
No full textAbstractWe develop a count of unstable eigenvalues in a finite-dimensional quadratic eigenvalue problem arising in the context of stability of discrete vortices in a multi-dimensional discrete nonlinear Schrödinger equation [D.E. Pelinovsky, P.G. Kevrekidis, D.J. Frantzeskakis, Persistence and stability of discrete vortices in nonlinear Schrödinger lattices, Physica D 212 (2005) 20–53]. The count is based on the Pontryagin Invariant Subspace Theorem and the parameter continuation arguments. Another application of the method is given in the context of front–pulse solutions of neuron networks with piecewise constant nonlinear functions [D.E. Pelinovsky, V.G.Yakhno, Generation of collective–activity structures in a homogeneous neuron-like medium, Int. J. Bifurcation and Chaos 6 (1996) 81–87]
Dark solitons in atomic Bose-Einstein condensates: From theory to experiments
No full textThis review paper presents an overview of the theoretical and experimental progress on the study of matter-wave dark solitons in atomic Bose-Einstein condensates. Upon introducing the general framework, we discuss the statics and dynamics of single and multiple matter-wave dark solitons in the quasi one-dimensional setting, in higher dimensional settings, as well as in the dimensionality crossover regime. Special attention is paid to the connection between theoretical results, obtained by various analytical approaches, and relevant experimental observations. © 2010 IOP Publishing Ltd
Vector solitons supported by the third-order dispersion
No full textNovel vector solitons, composed by a bright soliton and a dark or an antidark soliton are derived for a system of two incoherently coupled nonlinear Schrödinger equations with the third-order dispersion. It is demonstrated that in the small-amplitude limit these solitons are described by the completely integrable Mel'nikov system. © 2001 Elsevier Science B.V. All rights reserved
Solitons in coupled nonlinear Schrödinger models: A survey of recent developments
No full textAbstractIn this review we try to capture some of the recent excitement induced by experimental developments, but also by a large volume of theoretical and computational studies addressing multi-component nonlinear Schrödinger models and the localized structures that they support. We focus on some prototypical structures, namely the dark-bright and dark-dark solitons. Although our focus will be on one-dimensional, two-component Hamiltonian models, we also discuss variants, including three (or more)-component models, higher-dimensional states, as well as dissipative settings. We also offer an outlook on interesting possibilities for future work on this theme
On the properties of a nonlocal nonlinear schrödinger model and its soliton solutions
No full textNonlinear waves are normally described by means of certain nonlinear evolution equations. However, finding physically relevant exact solutions of these equations is, in general, particularly difficult. One of the most known nonlinear evolution equation is the nonlinear Schrödinger (NLS), a universal equation appearing in optics, Bose-Einstein condensates, water waves, plasmas, and many other disciplines. In optics, the NLS system is used to model a unique balance between the critical effects that govern propagation in dispersive nonlinear media, namely dispersion/diffraction and nonlinearity. This balance leads to the formation of solitons, namely robust localized waveforms that maintain their shape even when they interact. However, for several physically relevant contexts the standard NLS equation turns out to be an oversimplified description. This occurs in the case of nonlocal media, such as nematic liquid crystals, plasmas, and optical media exhibiting thermal nonlinearities. Here, we study the properties and soliton solutions of such a nonlocal NLS system, composed by a paraxial wave equation for the electric field envelope and a diffusion-type equation for the medium’s refractive index. The study of this problem is particularly interesting since remarkable properties of the traditional NLS—associated with complete integrability—are lost in the nonlocal case. Nevertheless, we identify cases where derivation of exact solutions is possible while, in other cases, we resort to multiscale expansions methods. The latter, allows us to reduce this systems to a known integrable equation with known solutions, which in turn, can be used to approximate the solutions of the initial system. By doing so, a plethora of solutions can be found; solitary waves solutions, planar or ring-shaped, and of dark or anti-dark type, are predicted to occur. © Springer International Publishing AG, part of Springer Nature 2018
Nonlinear self-phase modulation in optical soliton systems with lumped amplifiers
No full textThe nonlinear Schrödinger equation, governing pulse propagation in lossy optical fibers with periodic amplification, is analyzed under the average soliton concept. Bright and dark solitons solutions, corresponding to separatrices of one-dimensional hamiltonian dynamical systems, have been derived. The shift of the soliton velocity, relative to the group velocity, as well as the effect of nonlinear self-phase modulation on the angular frequency and the wavenumber are investigated. The obtained results are connected with the initial pulse amplitude and its spatial and temporal derivatives. © 1993
Light Meets Water in Nonlocal Media: Surface Tension Analogue in Optics
No full textShallow water wave phenomena find their analogue in optics through a nonlocal nonlinear Schrödinger (NLS) model in 2+1 dimensions. We identify an analogue of surface tension in optics, namely, a single parameter depending on the degree of nonlocality, which changes the sign of dispersion, much like surface tension does in the shallow water wave problem. Using multiscale expansions, we reduce the NLS model to a Kadomtsev-Petviashvili (KP) equation, which is of the KPII (KPI) type, for strong (weak) nonlocality. We demonstrate the emergence of robust optical antidark solitons forming Y-, X-, and H-shaped wave patterns, which are approximated by colliding KPII line solitons, similar to those observed in shallow waters. © 2017 American Physical Society
- …
