arXiv subjects

D. J. Frantzeskakis

Publications and source records attributed to D. J. Frantzeskakis.

At least 55 records · Page 3Linked to original sources

arXiv2017

Floquet Analysis of Kuznetsov--Ma breathers: A Path Towards Spectral Stability of Rogue Waves

In the present work, we aim at taking a step towards the spectral stability analysis of Peregrine solitons, i.e., wave structures that are used to emulate extreme wave events. Given the space-time localized nature of Peregrine solitons, this is a priori a non-trivial task. Our main tool in this effort will be the study of the spectral stability of the periodic generalization of the Peregrine soliton in the evolution variable, namely the Kuznetsov--Ma breather. Given the periodic structure of the latter, we compute the corresponding Floquet multipliers, and examine them in the limit where the period of the orbit tends to infinity. This way, we extrapolate towards the stability of the limiting structure, namely the Peregrine soliton. We find that multiple unstable modes of the background are enhanced, yet no additional unstable eigenmodes arise as the Peregrine limit is approached. We explore the instability evolution also in direct numerical simulations.

nlin.PS↗

arXiv2017

An Adiabatic Invariant Approach to Transverse Instability: Landau Dynamics of Soliton Filaments

Assume a lower-dimensional solitonic structure embedded in a higher dimensional space, e.g., a 1D dark soliton embedded in 2D space, a ring dark soliton in 2D space, a spherical shell soliton in 3D space etc. By extending the Landau dynamics approach [Phys. Rev. Lett. {\bf 93}, 240403 (2004)], we show that it is possible to capture the transverse dynamical modes (the "Kelvin modes") of the undulation of this "soliton filament" within the higher dimensional space. These are the transverse stability/instability modes and are the ones potentially responsible for the breakup of the soliton into structures such as vortices, vortex rings etc. We present the theory and case examples in 2D and 3D, corroborating the results by numerical stability and dynamical computations.

nlin.PS↗

arXiv2017

Stability of single and multiple matter-wave dark solitons in collisionally inhomogeneous Bose-Einstein condensates

We examine the spectral properties of single and multiple matter-wave dark solitons in Bose-Einstein condensates confined in parabolic traps, where the scattering length is periodically modulated. In addition to the large-density limit picture previously established for homogeneous nonlinearities, we explore a perturbative analysis in the vicinity of the linear limit, which provides good agreement with the observed spectral modes. Between these two analytically tractable limits, we use numerical computations to fill in the relevant intermediate regime. We find that the scattering length modulation can cause a variety of features absent for homogeneous nonlinearities. Among them, we note the potential oscillatory instability even of the single dark soliton, the potential absence of instabilities in the immediate vicinity of the linear limit for two dark solitons, and the existence of an exponential instability associated with the in-phase motion of three dark solitons.

nlin.PS↗

arXiv2017

Dark Soliton Scattering in Symmetric and Asymmetric Double Potential Barriers

Motivated by the recent theoretical study of (bright) soliton diode effects in systems with multiple scatterers, as well as by experimental investigations of soliton-impurity interactions, we consider some prototypical case examples of interactions of dark solitons with a pair of scatterers. In a way fundamentally opposite to the case of bright solitons (but consonant to their "anti-particle character"), we find that dark solitons accelerate as they pass the first barrier and hence cannot be trapped by a second equal-height barrier. A pair of unequal barriers may lead to reflection from the second one, however trapping in the inter-barrier region cannot occur. We also give some examples of dynamical adjusting the barriers to trap the dark soliton in the inter-barrier region, yet we show that this can only occur over finite time horizons, with the dark soliton always escaping eventually, contrary again to what is potentially the case with bright solitons.

nlin.PS↗

arXiv2017

Single and Multiple Vortex Rings in Three-Dimensional Bose-Einstein Condensates: Existence, Stability and Dynamics

In the present work, we explore the existence, stability and dynamics of single and multiple vortex ring states that can arise in Bose-Einstein condensates. Earlier works have illustrated the bifurcation of such states, in the vicinity of the linear limit, for isotropic or anisotropic three-dimensional harmonic traps. Here, we extend these states to the regime of large chemical potentials, the so-called Thomas-Fermi limit, and explore their properties such as equilibrium radii and inter-ring distance, for multi-ring states, as well as their vibrational spectra and possible instabilities. In this limit, both the existence and stability characteristics can be partially traced to a particle picture that considers the rings as individual particles oscillating within the trap and interacting pairwise with one another. Finally, we examine some representative instability scenarios of the multi-ring dynamics including breakup and reconnections, as well as the transient formation of vortex lines.

nlin.PS↗

arXiv2016

From Solitons to Rogue Waves in Nonlinear Left-Handed Metamaterials

In the present work, we explore soliton and rogue-like wave solutions in the transmission line analogue of a nonlinear left-handed metamaterial. The nonlinearity is expressed through a voltagedependent and symmetric capacitance motivated by the recently developed ferroelectric barium strontium titanate (BST) thin film capacitor designs. We develop both the corresponding nonlinear dynamical lattice, as well as its reduction via a multiple scales expansion to a nonlinear Schrödinger (NLS) model for the envelope of a given carrier wave. The reduced model can feature either a focusing or a defocusing nonlinearity depending on the frequency (wavenumber) of the carrier. We then consider the robustness of different types of solitary waves of the reduced model within the original nonlinear left-handed medium. We find that both bright and dark solitons persist in a suitable parametric regime, where the reduction to the NLS is valid. Additionally, for suitable initial conditions, we observe a rogue wave type of behavior, that differs significantly from the classic Peregrine rogue wave evolution, including most notably the breakup of a single Peregrine-like pattern into solutions with multiple wave peaks. Finally, we touch upon the behavior of generalized members of the family of the Peregrine solitons, namely Akhmediev breathers and Kuznetsov-Ma solitons, and explore how these evolve in the left-handed transmission line.

nlin.PS↗

arXiv2016

Rogue Waves in Ultracold Bosonic Seas

In this work, we numerically consider the initial value problem for nonlinear Schrödinger (NLS) type models arising in the physics of ultracold boson gases, with generic Gaussian wavepacket initial data. The corresponding Gaussian's width and, wherever relevant also its amplitude, serve as control parameters. First we explore the one-dimensional, standard NLS equation with general power law nonlinearity, in which large amplitude excitations reminiscent of Peregrine solitons or regular solitons appear to form, as the width of the relevant Gaussian is varied. Furthermore, the variation of the nonlinearity exponent aims at a first glimpse of the interplay between rogue or soliton formation and collapse features. The robustness of the main features to noise in the initial data is also confirmed. To better connect our study with the physics of atomic condensates, and explore the role of dimensionality effects, we also consider the nonpolynomial Schrödinger equation (NPSE), as well as the full three-dimensional NLS equation, and examine the degree to which relevant considerations generalize.

nlin.PS↗

arXiv2016

Vortex-soliton complexes in coupled nonlinear Schrödinger equations with unequal dispersion coefficients

We consider a two-component, two-dimensional nonlinear Schrödinger system with unequal dispersion coefficients and self-defocusing nonlinearities, chiefly with equal strengths of the self- and cross-interactions. In this setting, a natural waveform with a nonvanishing background in one component is a vortex, which induces an effective potential well in the second component, via the nonlinear coupling of the two components. We show that the potential well may support not only the fundamental bound state, but also multi-ring excited radial state complexes for suitable ranges of values of the dispersion coefficient of the second component. We systematically explore the existence, stability, and nonlinear dynamics of these states. The complexes involving the excited radial states are weakly unstable, with a growth rate depending on the dispersion of the second component. Their evolution leads to transformation of the multi-ring complexes into stable VB solitons ones with the fundamental state in the second component. The excited states may be stabilized by a harmonic-oscillator trapping potential, as well as by unequal strengths of the self- and cross-repulsive nonlinearities.

nlin.PS↗

arXiv2016

The dynamical playground of a higher-order cubic Ginzburg-Landau equation: from orbital connections and limit cycles to invariant tori and the onset of chaos

The dynamical behavior of a higher-order cubic Ginzburg-Landau equation is found to include a wide range of scenarios due to the interplay of higher-order physically relevant terms. We find that the competition between the third-order dispersion and stimulated Raman scattering effects, gives rise to rich dynamics: this extends from Poincaré-Bendixson--type scenarios, in the sense that bounded solutions may converge either to distinct equilibria via orbital connections, or space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that the third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (i.e., even in the absence of the other higher-order effects) for the existence of the periodic, quasi-periodic and chaotic spatiotemporal structures. Suitable low-dimensional phase space diagnostics are devised and used to illustrate the different possibilities and identify their respective parametric intervals over multiple parameters of the model.

nlin.PS↗

arXiv2016

$SO(2)$-induced breathing patterns in multi-component Bose-Einstein condensates

In this work, we employ the $SO(2)$-rotations of a two-component, one-, two- and three-dimensional nonlinear Schrödinger system at and near the Manakov limit, to construct vector solitons and vortex structures. This way, stable stationary dark-bright solitons and their higher-dimensional siblings are transformed into robust oscillatory dark-dark solitons (and generalizations thereof), with and without a harmonic confinement. By analogy to the one-dimensional case, vector higher-dimensional structures take the form of vortex-vortex states in two dimensions and, e.g., vortex ring-vortex ring ones in three dimensions. We consider the effects of unequal (self- and cross-) interaction strengths, where the $SO(2)$ symmetry is only approximately satisfied, showing the dark-dark soliton oscillation is generally robust. Similar features are found in higher dimensions too, although our case examples suggest that phenomena such as phase separation may contribute to the associated dynamics. These results, in connection with the experimental realization of one-dimensional variants of such states in optics and Bose-Einstein condensates (BECs), suggest the potential observation of the higher-dimensional bound states proposed herein.

nlin.PS↗

arXiv2016

Dark spherical shell solitons in three-dimensional Bose-Einstein condensates: Existence, stability and dynamics

In this work we study spherical shell dark soliton states in three-dimensional atomic Bose-Einstein condensates. Their symmetry is exploited in order to analyze their existence, as well as that of topologically charged variants of the structures, and, importantly, to identify their linear stability Bogolyubov-de Gennes spectrum. We compare our effective 1D spherical and 2D cylindrical computations with the full 3D numerics. An important conclusion is that such spherical shell solitons can be stable sufficiently close to the linear limit of the isotropic condensates considered herein. We have also identified their instabilities leading to the emergence of vortex line and vortex ring cages. In addition, we generalize effective particle pictures of lower dimensional dark solitons and ring dark solitons to the spherical shell solitons concerning their equilibrium radius and effective dynamics around it. In this case too, we favorably compare the resulting predictions such as the shell equilibrium radius, qualitatively and quantitatively, with full numerical solutions in 3D.

cond-mat.quant-gas↗

arXiv2015

Solitons in Multi-Component Nonlinear Schrodinger Models: A Survey of Recent Developments

In 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 Schrodinger 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.

cond-mat.quant-gas↗

arXiv2015

Collapse for the higher-order nonlinear Schrödinger equation

We examine conditions for finite-time collapse of the solutions of the higher-order nonlinear Schrödinger (NLS) equation incorporating third-order dispersion, self-steepening, linear and nonlinear gain and loss, and Raman scattering; this is a system that appears in many physical contexts as a more realistic generalization of the integrable NLS. By using energy arguments, it is found that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. We identify a critical value of the linear gain, separating the possible decay of solutions to the trivial zero-state, from collapse. The numerical simulations, performed for a wide class of initial data, are found to be in very good agreement with the analytical results, and reveal long-time stability properties of localized solutions. The role of the higher-order effects to the transient dynamics is also revealed in these simulations.

nlin.PS↗

arXiv2015

Robust Vortex Lines, Vortex Rings and Hopfions in 3D Bose-Einstein Condensates

Performing a systematic Bogoliubov-de Gennes spectral analysis, we illustrate that stationary vortex lines, vortex rings and more exotic states, such as hopfions, are robust in three-dimensional atomic Bose-Einstein condensates, for large parameter intervals. Importantly, we find that the hopfion can be stabilized in a simple parabolic trap, without the need for trap rotation or inhomogeneous interactions. We supplement our spectral analysis by studying the dynamics of such stationary states; we find them to be robust against significant perturbations of the initial state. In the unstable regimes, we not only identify the unstable mode, such as a quadrupolar or hexapolar mode, but we also observe the corresponding instability dynamics. Furthermore, deep in the Thomas-Fermi regime, we investigate the particle-like behavior of vortex rings and hopfions.

cond-mat.quant-gas↗

arXiv2015

Dark solitons near potential and nonlinearity steps

We study dark solitons near potential and nonlinearity steps and combinations thereof, forming rectangular barriers. This setting is relevant to the contexts of atomic Bose-Einstein condensates (where such steps can be realized by using proper external fields) and nonlinear optics (for beam propagation near interfaces separating optical media of different refractive indices). We use perturbation theory to develop an equivalent particle theory, describing the matter-wave or optical soliton dynamics as the motion of a particle in an effective potential. This Newtonian dynamical problem provides information for the soliton statics and dynamics, including scenarios of reflection, transmission, or quasi-trapping at such steps. The case of multiple such steps and its connection to barrier potentials is also touched upon. Our analytical predictions are found to be in very good agreement with the corresponding numerical results.

cond-mat.quant-gas↗

arXiv2015

Oscillons and oscillating kinks in the Abelian-Higgs model

We study the classical dynamics of the Abelian Higgs model employing an asymptotic multiscale expansion method, which uses the ratio of the Higgs to the gauge field amplitudes as a small parameter. We derive an effective nonlinear Schrödinger equation for the gauge field, and a linear equation for the scalar field containing the gauge field as a nonlinear source. This equation is used to predict the existence of oscillons and oscillating kinks for certain regimes of the ratio of the Higgs to the gauge field masses. Results of numerical simulations are found to be in very good agreement with the analytical findings, and show that the oscillons are robust, while kinks are unstable. It is also demonstrated that oscillons emerge spontaneously as a result of the onset of the modulational instability of plane wave solutions of the model. Connections of the obtained solutions with the phenomenology of superconductors is discussed.

hep-th↗

arXiv2015

On the Bifurcation and Stability of Single and Multiple Vortex Rings in Three-Dimensional Bose-Einstein Condensates

In the present work, we investigate how single- and multi-vortex-ring states can emerge from a planar dark soliton in three-dimensional (3D) Bose-Einstein condensates (confined in isotropic or anisotropic traps) through bifurcations. We characterize such bifurcations quantitatively using a Galerkin-type approach, and find good qualitative and quantitative agreement with our Bogoliubov-de Gennes (BdG) analysis. We also systematically characterize the BdG spectrum of the dark solitons, using perturbation theory, and obtain a quantitative match with our 3D BdG numerical calculations. We then turn our attention to the emergence of single- and multi-vortex-ring states. We systematically capture these as stationary states of the system and quantify their BdG spectra numerically. We find that although the vortex ring may be unstable when bifurcating, its instabilities weaken and may even eventually disappear, for sufficiently large chemical potentials and suitable trap settings. For instance, we demonstrate the stability of the vortex ring for an isotropic trap in the large chemical potential regime.

cond-mat.quant-gas↗

arXiv2015

Conservation laws, exact travelling waves and modulation instability for an extended nonlinear Schrödinger equation

We study various properties of solutions of an extended nonlinear Schrödinger (ENLS) equation, which arises in the context of geometric evolution problems -- including vortex filament dynamics -- and governs propagation of short pulses in optical fibers and nonlinear metamaterials. For the periodic initial-boundary value problem, we derive conservation laws satisfied by local in time, weak $H^2$ (distributional) solutions, and establish global existence of such weak solutions. The derivation is obtained by a regularization scheme under a balance condition on the coefficients of the linear and nonlinear terms -- namely, the Hirota limit of the considered ENLS model. Next, we investigate conditions for the existence of traveling wave solutions, focusing on the case of bright and dark solitons. The balance condition on the coefficients is found to be essential for the existence of exact analytical soliton solutions; furthermore, we obtain conditions which define parameter regimes for the existence of traveling solitons for various linear dispersion strengths. Finally, we study the modulational instability of plane waves of the ENLS equation, and identify important differences between the ENLS case and the corresponding NLS counterpart. The analytical results are corroborated by numerical simulations, which reveal notable differences between the bright and the dark soliton propagation dynamics, and are in excellent agreement with the analytical predictions of the modulation instability analysis.

nlin.PS↗