Collisional-inhomogeneity-induced generation of matter-wave dark solitons

We propose an experimentally relevant protocol for the controlled generation of matter-wave dark solitons in atomic Bose-Einstein condensates (BECs). In particular, using direct numerical simulations, we show that by switching-on a spatially inhomogeneous (step-like) change of the s-wave scattering length, it is possible to generate a controllable number of dark solitons in a quasi-one-dimensional BEC. A similar phenomenology is also found in the two-dimensional setting of "disk-shaped" BECs but, as the solitons are subject to the snaking instability, they decay into vortex structures. A detailed investigation of how the parameters involved affect the emergence and evolution of solitons and vortices is provided.Comment: 8 pages, 5 Figures, Physics Letters A (in press

On the properties of a nonlocal nonlinear schrödinger model and its soliton solutions

Nonlinear 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

Patterns ofwater in light

The 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

Dark solitons in the presence of higher-order effects

Dark soliton propagation is studied in the presence of higher-order effects, including third-order dispersion, selfsteepening, linear/nonlinear gain/loss, and Raman scattering. It is found that for certain values of the parameters a stable evolution can exist for both the soliton and the relative continuous-wave background. Using a newly developed perturbation theory we show that the perturbing effects give rise to a shelf that accompanies the soliton in its propagation. Although, the stable solitons are not affected by the shelf it remains an integral part of the dynamics otherwise not considered so far in studies of higher-order nonlinear Schrödinger models. © 2013 Optical Society of America

Dynamical oscillations in nonlinear optical media

The spatial dynamics of pulses in Kerr media with parabolic index profile are examined. It is found that when diffraction and graded-index have opposite signs propagating pulses exhibit an oscillatory pattern, similar to a breathing behavior. Furthermore, if the pulse and the index profile are not aligned the pulse oscillates around the index origin with frequency that depends on the values of the diffraction and index of refraction. These oscillations are not observed when diffraction and graded-index share the same sign. © 2009 Elsevier B.V. All rights reserved

Dynamics of a Higher-Order Ginzburg–Landau-Type Equation

We study possible dynamical scenarios associated with a higher-order Ginzburg–Landau-type equation. In particular, first we discuss and prove the existence of a limit set (attractor), capturing the long-time dynamics of the system. Then, we examine conditions for finite-time collapse of the solutions of the model at hand, and find that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. Finally, considering the model as a perturbed nonlinear Schrödinger equation, we employ perturbation theory for solitons to analyze the influence of gain/loss and other higher-order effects on the dynamics of bright and dark solitons. © 2021, Springer Nature Switzerland AG

Dynamics of a Higher-Order Ginzburg–Landau-Type Equation

We study possible dynamical scenarios associated with a higher-order Ginzburg–Landau-type equation. In particular, first we discuss and prove the existence of a limit set (attractor), capturing the long-time dynamics of the system. Then, we examine conditions for finite-time collapse of the solutions of the model at hand, and find that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. Finally, considering the model as a perturbed nonlinear Schrödinger equation, we employ perturbation theory for solitons to analyze the influence of gain/loss and other higher-order effects on the dynamics of bright and dark solitons. © 2021, Springer Nature Switzerland AG

Soliton pairs in two-dimensional nonlocal media

We study the interaction of optical beams of different wavelengths, described by a two-component, two-dimensional (2D) nonlocal nonlinear Schrödinger (NLS) model. Using a multiscale expansion method the NLS model is asymptotically reduced to the completely integrable 2D Mel'nikov system, the soliton solutions of which are used to construct approximate dark-bright and antidark-bright soliton solutions of the original NLS model; the latter being unique to the nonlocal NLS system with no relevant counterparts in the local case. Direct numerical simulations show that, for sufficiently small amplitudes, both these types of soliton stripes do exist and propagate undistorted, in excellent agreement with the analytical predictions. Larger amplitude of these soliton stripes, when perturbed along the transverse direction, disintegrate either to filled vortex structures (the dark-bright solitons) or to radiation (the antidark-bright solitons). © 2020 American Physical Society

Exciting extreme events in the damped and AC-driven NLS equation through plane-wave initial conditions

We investigate, by direct numerical simulations and for certain parametric regimes, the dynamics of the damped and forced nonlinear Schrödinger (NLS) equation in the presence of a time-periodic forcing. It is thus revealed that the wave number of a plane-wave initial condition dictates the number of emerged Peregrine-type rogue waves at the early stages of modulation instability. The formation of these events gives rise to the same number of transient "triangular"spatiotemporal patterns, each of which is reminiscent of the one emerging in the dynamics of the integrable NLS in its semiclassical limit, when supplemented with vanishing initial conditions. We find that the L 2-norm of the spatial derivative and the L 4-norm detect the appearance of rogue waves as local extrema in their evolution. The impact of the various parameters and noisy perturbations of the initial condition in affecting the above behavior is also discussed. The long-time behavior, in the parametric regimes where the extreme wave events are observable, is explained in terms of the global attractor possessed by the system and the asymptotic orbital stability of spatially uniform continuous wave solutions. © 2021 Author(s)

A Davey–Stewartson description of two-dimensional solitons in nonlocal media

Novel soliton solutions of a two-dimensional (2D) nonlocal nonlinear Schrödinger (NLS) system are revealed by asymptotically reducing the system to a completely integrable Davey–Stewartson (DS) set of equations. In so doing, the reductive perturbation method in addition to a multiple scales scheme are utilized to derive both the DS-I and DS-II systems, depending on the strength of the nonlocality, which in turn, may be regarded here as a measure of the surface tension. As such, two different soliton solutions are obtained: the breather and dromion solutions in the case of DS-I (weak nonlocality), as well as lump solutions in the case of DS-II (strong nonlocality). Besides their immediate mathematical importance, our results find a wide range of applications due the high applicability of the relative nonlocal NLS (optics, plasmas, liquid crystals, and thermal media in the strong nonlocality regime, etc.) and hence these structures can also be realized experimentally in various physical setting. © 2019 Wiley Periodicals, Inc., A Wiley Compan