Electrostatic wave interaction via asymmetric vector solitons as precursor to rogue wave formation in non-Maxwellian plasmas.

An asymmetric pair of coupled nonlinear Schrödinger (CNLS) equations has been derived through a multiscale perturbation method applied to a plasma fluid model, in which two wavepackets of distinct (carrier) wavenumbers ([Formula: see text] and [Formula: see text]) and amplitudes ([Formula: see text] and [Formula: see text]) are allowed to co-propagate and interact. The original fluid model was set up for a non-magnetized plasma consisting of cold inertial ions evolving against a [Formula: see text]-distributed electron background in one dimension. The reduction procedure resulting in the CNLS equations has provided analytical expressions for the dispersion, self-modulation and cross-coupling coefficients in terms of the two carrier wavenumbers. These coefficients present no symmetry whatsoever, in the general case (of different wavenumbers). The possibility for coupled envelope (vector soliton) solutions to occur has been investigated. Although the CNLS equations are asymmetric and non-integrable, in principle, the system admits various types of vector soliton solutions, physically representing nonlinear, localized electrostatic plasma modes, whose areas of existence is calculated on the wavenumbers' parameter plane. The possibility for either bright (B) or dark (D) type excitations for either of the (2) waves provides four (4) combinations for the envelope pair (BB, BD, DB, DD), if a set of explicit criteria is satisfied. Moreover, the soliton parameters (maximum amplitude, width) are also calculated for each type of vector soliton solution, in its respective area of existence. The dependence of the vector soliton characteristics on the (two) carrier wavenumbers and on the spectral index [Formula: see text] characterizing the electron distribution has been explored. In certain cases, the (envelope) amplitude of one component may exceed its counterpart (second amplitude) by a factor 2.5 or higher, indicating that extremely asymmetric waves may be formed due to modulational interactions among copropagating wavepackets. As [Formula: see text] decreases from large values, modulational instability occurs in larger areas of the parameter plane(s) and with higher growth rates. The distribution of different types of vector solitons on the parameter plane(s) also varies significantly with decreasing [Formula: see text], and in fact dramatically for [Formula: see text] between 3 and 2. Deviation from the Maxwell-Boltzmann picture therefore seems to favor modulational instability as a precursor to the formation of bright (predominantly) type envelope excitations and freak waves.

Transverse instability and dynamics of nonlocal bright solitons.

We study the transverse instability and dynamics of bright soliton stripes in two-dimensional nonlocal nonlinear media. Using a multiscale perturbation method, we derive analytically the first-order correction to the soliton shape, which features an exponential growth in time-a signature of the transverse instability. The soliton's characteristic timescale associated with its exponential growth is found to depend on the square root of the nonlocality parameter. This, in turn, highlights the nonlocality-induced suppression of the transverse instability. Our analytical predictions are corroborated by direct numerical simulations, with the analytical results being in good agreement with the numerical ones.

Solitary and periodic waves in collisionless plasmas: The Adlam-Allen model revisited.

We consider the Adlam-Allen (AA) system of partial differential equations, which, arguably, is the first model that was introduced to describe solitary waves in the context of propagation of hydrodynamic disturbances in collisionless plasmas. Here, we identify the solitary waves of the model by implementing a dynamical systems approach. The latter suggests that the model also possesses periodic wave solutions-which reduce to the solitary wave in the limiting case of an infinite period-as well as rational solutions that are obtained herein. In addition, employing a long-wave approximation via a relevant multiscale expansion method, we establish the asymptotic reduction of the AA system to the Korteweg-de Vries equation. Such a reduction is not only another justification for the above solitary wave dynamics, but may also offer additional insights for the emergence of other possible plasma waves. Direct numerical simulations are performed for the study of multiple solitary waves and their pairwise interactions. The stability of solitary waves is discussed in terms of potentially relevant criteria, while the robustness of spatially periodic wave solutions is touched upon via numerical experiments.

Self-similar evolution in nonlocal nonlinear media.

The self-similar propagation of optical beams in a broad class of nonlocal, nonlinear optical media is studied utilizing a generic system of coupled equations with linear gain. This system describes, for instance, beam propagation in nematic liquid crystals and optical thermal media. It is found, both numerically and analytically, that the nonlocal response has a focusing effect on the beam, concentrating its power around its center during propagation. In particular, the beam narrows in width and grows in amplitude faster than in local media, with the resulting beam shape being parabolic. Finally, a general initial localized beam evolves to a common shape.

Hydrodynamics and two-dimensional dark lump solitons for polariton superfluids.

We study a two-dimensional incoherently pumped exciton-polariton condensate described by an open-dissipative Gross-Pitaevskii equation for the polariton dynamics coupled to a rate equation for the exciton density. Adopting a hydrodynamic approach, we use multiscale expansion methods to derive several models appearing in the context of shallow water waves with viscosity. In particular, we derive a Boussinesq/Benney-Luke-type equation and its far-field expansion in terms of Kadomtsev-Petviashvili-I (KP-I) equations for right- and left-going waves. From the KP-I model, we predict the existence of vorticity-free, weakly (algebraically) localized two-dimensional dark-lump solitons. We find that, in the presence of dissipation, dark lumps exhibit a lifetime three times larger than that of planar dark solitons. Direct numerical simulations show that dark lumps do exist, and their dissipative dynamics is well captured by our analytical approximation. It is also shown that lumplike and vortexlike structures can spontaneously be formed as a result of the transverse "snaking" instability of dark soliton stripes.

Three-Component Soliton States in Spinor F=1 Bose-Einstein Condensates.

Dilute-gas Bose-Einstein condensates are an exceptionally versatile test bed for the investigation of novel solitonic structures. While matter-wave solitons in one- and two-component systems have been the focus of intense research efforts, an extension to three components has never been attempted in experiments. Here, we experimentally demonstrate the existence of robust dark-bright-bright (DBB) and dark-dark-bright solitons in a multicomponent F=1 condensate. We observe lifetimes on the order of hundreds of milliseconds for these structures. Our theoretical analysis, based on a multiscale expansion method, shows that small-amplitude solitons of these types obey universal long-short wave resonant interaction models, namely, Yajima-Oikawa systems. Our experimental and analytical findings are corroborated by direct numerical simulations highlighting the persistence of, e.g., the DBB soliton states, as well as their robust oscillations in the trap.

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

Consider 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. 93, 240403 (2004)PRLTAO0031-900710.1103/PhysRevLett.93.240403], 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 or 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.

From solitons to rogue waves in nonlinear left-handed metamaterials.

In the present work, we explore soliton and roguelike wave solutions in the transmission line analog of a nonlinear left-handed metamaterial. The nonlinearity is expressed through a voltage-dependent, symmetric capacitance motivated by recently developed ferroelectric barium strontium titanate thin-film capacitor designs. We develop both the corresponding nonlinear dynamical lattice and 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 (wave number) 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 model 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.

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 nontrivial 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.

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 multiring 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 multiring complexes into stable vortex-bright 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.

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 to space-time periodic solutions, to the emergence of almost periodic and chaotic behavior. One of our main results is that third-order dispersion has a dominant role in the development of such complex dynamics, since it can be chiefly responsible (even in the absence of other higher-order effects) for the existence of periodic, quasiperiodic, 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.

Solitons and vortices in two-dimensional discrete nonlinear Schrödinger systems with spatially modulated nonlinearity.

We consider a two-dimensional (2D) generalization of a recently proposed model [Gligoric et al., Phys. Rev. E 88, 032905 (2013)], which gives rise to bright discrete solitons supported by the defocusing nonlinearity whose local strength grows from the center to the periphery. We explore the 2D model starting from the anticontinuum (AC) limit of vanishing coupling. In this limit, we can construct a wide variety of solutions including not only single-site excitations, but also dipole and quadrupole ones. Additionally, two separate families of solutions are explored: the usual "extended" unstaggered bright solitons, in which all sites are excited in the AC limit, with the same sign across the lattice (they represent the most robust states supported by the lattice, their 1D counterparts being those considered as 1D bright solitons in the above-mentioned work), and the vortex cross, which is specific to the 2D setting. For all the existing states, we explore their stability (also analytically, when possible). Typical scenarios of instability development are exhibited through direct simulations.

Acoustic solitons in waveguides with Helmholtz resonators: transmission line approach.

We report experimental results and study theoretically soliton formation and propagation in an air-filled acoustic waveguide side loaded with Helmholtz resonators. We propose a theoretical modeling of the system, which relies on a transmission-line approach, leading to a nonlinear dynamical lattice model. The latter allows for an analytical description of the various soliton solutions for the pressure, which are found by means of dynamical systems and multiscale expansion techniques. These solutions include Boussinesq-like and Korteweg-de Vries pulse-shaped solitons that are observed in the experiment, as well as nonlinear Schrödinger envelope solitons, that are predicted theoretically. The analytical predictions are in excellent agreement with direct numerical simulations and in qualitative agreement with the experimental observations.

Dark-bright solitons in coupled nonlinear Schrödinger equations with unequal dispersion coefficients.

We study a two-component nonlinear Schrödinger system with equal, repulsive cubic interactions and different dispersion coefficients in the two components. We consider states that have a dark solitary wave in one component. Treating it as a frozen one, we explore the possibility of the formation of bright-solitonic structures in the other component. We identify bifurcation points at which such states emerge in the bright component in the linear limit and explore their continuation into the nonlinear regime. An additional analytically tractable limit is found to be that of vanishing dispersion of the bright component. We numerically identify regimes of potential stability, not only of the single-peak ground state (the dark-bright soliton), but also of excited states with one or more zero crossings in the bright component. When the states are identified as unstable, direct numerical simulations are used to investigate the outcome of the instability development. Although our principal focus is on the homogeneous setting, we also briefly touch upon the counterintuitive impact of the potential presence of a parabolic trap on the states of interest.

Vector rogue waves and dark-bright boomeronic solitons in autonomous and nonautonomous settings.

In this work we consider the dynamics of vector rogue waves and dark-bright solitons in two-component nonlinear Schrödinger equations with various physically motivated time-dependent nonlinearity coefficients, as well as spatiotemporally dependent potentials. A similarity transformation is utilized to convert the system into the integrable Manakov system and subsequently the vector rogue and dark-bright boomeronlike soliton solutions of the latter are converted back into ones of the original nonautonomous model. Using direct numerical simulations we find that, in most cases, the rogue wave formation is rapidly followed by a modulational instability that leads to the emergence of an expanding soliton train. Scenarios different than this generic phenomenology are also reported.

From nodeless clouds and vortices to gray ring solitons and symmetry-broken states in two-dimensional polariton condensates.

We consider the existence, stability and dynamics of the nodeless state and fundamental nonlinear excitations, such as vortices, for a quasi-two-dimensional polariton condensate in the presence of pumping and nonlinear damping. We find a series of interesting features that can be directly contrasted to the case of the typically energy-conserving ultracold alkali-atom Bose-Einstein condensates (BECs). For sizeable parameter ranges, in line with earlier findings, the nodeless state becomes unstable towards the formation of stable nonlinear single or multi-vortex excitations. The potential instability of the single vortex is also examined and is found to possess similar characteristics to those of the nodeless cloud. We also report that, contrary to what is known, e.g., for the atomic BEC case, stable stationary gray ring solitons (that can be thought of as radial forms of Nozaki-Bekki holes) can be found for polariton condensates in suitable parametric regimes. In other regimes, however, these may also suffer symmetry-breaking instabilities. The dynamical, pattern-forming implications of the above instabilities are explored through direct numerical simulations and, in turn, give rise to waveforms with triangular or quadrupolar symmetry.

Exploring rigidly rotating vortex configurations and their bifurcations in atomic Bose-Einstein condensates.

In the present work, we consider the problem of a system of few vortices N ≤ 5 as it emerges from its experimental realization in the field of atomic Bose-Einstein condensates. Starting from the corresponding equations of motion for an axially symmetric trapped condensate, we use a two-pronged approach in order to reveal the configuration space of the system's preferred dynamical states. We use a Monte Carlo method parametrizing the vortex particles by means of hyperspherical coordinates and identifying the minimal energy ground states thereof for N=2,...,5 and different vortex particle angular momenta. We then complement this picture with a dynamical system analysis of the possible rigidly rotating states. The latter reveals a supercritical and subcritical pitchfork, as well as saddle-center bifurcations that arise, exposing the full wealth of the problem even for such low-dimensional cases. By corroborating the results of the two methods, it becomes fairly transparent which branch the Monte Carlo approach selects for different values of the angular momentum that is used as a bifurcation parameter.

Effects of parity-time symmetry in nonlinear Klein-Gordon models and their stationary kinks.

In this work, we introduce some basic principles of PT-symmetric Klein-Gordon nonlinear field theories. By formulating a particular antisymmetric gain and loss profile, we illustrate that the stationary states of the model do not change. However, the stability critically depends on the gain and loss profile. For a symmetrically placed solitary wave (in either the continuum model or a discrete analog of the nonlinear Klein-Gordon type), there is no effect on the steady state spectrum. However, for asymmetrically placed solutions, there exists a measurable effect of which a perturbative mathematical characterization is offered. It is generally found that asymmetry towards the lossy side leads towards stability, while towards the gain side produces instability. Furthermore, a host of finite size effects, which disappear in the infinite domain limit, are illustrated in connection to the continuous spectrum of the problem.

Coupled backward- and forward-propagating solitons in a composite right- and left-handed transmission line.

We study the coupling between backward- and forward-propagating wave modes, with the same group velocity, in a composite right- and left-handed nonlinear transmission line. Using an asymptotic multiscale expansion technique, we derive a system of two coupled nonlinear Schrödinger equations governing the evolution of the envelopes of these modes. We show that this system supports a variety of backward- and forward-propagating vector solitons of the bright-bright, bright-dark, and dark-bright type. Performing systematic numerical simulations in the framework of the original lattice that models the transmission line, we study the propagation properties of the derived vector soliton solutions. We show that all types of the predicted solitons exist, but differ on their robustness: Only bright-bright solitons propagate undistorted for long times, while the other types are less robust, featuring shorter lifetimes. In all cases, our analytical predictions are in very good agreement with the results of the simulations, at least up to times of the order of the solitons' lifetimes.