Flat-top Solitons Research Articles

Solitary waves in electro-mechanical lattices

We introduce and study both analytically and numerically a class of microelectromechanical chains aiming to turn them into transmission lines of solitons. Mathematically, their analysis reduces to the study of a spatially one-dimensional nonlinear Klein–Gordon equation with a model dependent onsite nonlinearity induced by the electrical forces. Since the basic solitons appear to be unstable for most of the force regimes, we introduce a stabilizing algorithm and demonstrate that it enables a stable and persisting propagation of solitons. Among other fascinating nonlinear formations induced by the presented models, we mention the “meson”: a stable square shaped pulse with sharp fronts that expands with a sonic speed, and “flatons”: flat-top solitons of arbitrary width.

Unidirectional flow of flat-top solitons

We numerically demonstrate the unidirectional flow of flat-top solitons when interacting with two reflectionless potential wells with slightly different depths. The system is described by a nonlinear Schrödinger equation with dual nonlinearity. The results show that for shallow potential wells, the velocity window for unidirectional flow is larger than for deeper potential wells. A wider flat-top solitons also have a narrow velocity window for unidirectional flow than those for thinner flat-top solitons.

Coexistence of positive and negative polarity solitons, double layers and supersolitons in electron-positron multi-ion plasmas

In order to analyze the nonlinear propagation of arbitrary amplitude ion-acoustic waves, the Sagdeev pseudopotential formalism is used to a four-component multi-ion plasma made up of inertial light positive ions, heavy negative ions, inertialess nonextensive electrons, and positrons. At first, we derived the expression of Sagdeev Pseudopotential for our plasma model and then numerically investigated the condition for the existence of several nonlinear structures such as solitons, double layers, supersolitons, etc. Based on numerical simulation, it is observed that within a certain range of parameters, our plasma model allows compressive as well as a rarefactive soliton, flat-top soliton, double layer, and supersoliton solutions. Great emphasis is placed on the determination of the existence regions for both solitons and supersolitons in the parameter space. Also, various nonlinear structures are explored to examine the effects of various plasma properties, such as the nonextensivity parameter, Mach number, etc. The findings of the current investigations may be helpful for comprehending significant features of ion-acoustic wave structures in space plasmas such as earth's ionosphere, cometary tails, etc., and laboratory plasmas like laser-plasma interaction research.

On multi-hump solutions of reverse space-time nonlocal nonlinear Schrödinger equation

In this article multi-soliton solutions of reverse space-time nonlocal nonlinear Schr ödinger (NLS) equation have been constructed. Darboux transformation is applied to the associated linear eigenvalue problem for the generalized NLS equation and we obtain a determinant formula for multi-soliton solutions. Under suitable reduction conditions and appropriate choice of spectral parameters, the generalized expression of first-order nontrivial solution gives some novel solutions such as double-hump and flat-top soliton solutions for reverse space-time nonlocal NLS equation. The dynamics and interaction of double-hump soliton solutions are studied in detail and it is indicated that these solutions undergo collisions without any energy redistribution. For higher-order double-hump solutions, the relative velocities of solitons play a crucial role to have humps and also induce nonlinear interference in the collision zone. The dynamics of individual decaying and growing unstable and stable double-humps as well as their interactions are explained and illustrated.

Discrete solitons in competitive zigzag waveguide arrays with cubic-quintic nonlinearity

In this paper, we study one-dimensional discrete solitons in zigzag waveguide arrays with competitive cubic-quintic nonlinearity and competitive linear mixing between the nearest-neighbor (NN) and next-nearest-neighbor (NNN) couplings. The competitive nonlinearity features a cubic self-focusing associated with a quintic self-defocusing nonlinearities. The competitive linear mixing between the NN and NNN couplings is induced by making the two coefficients opposite of each other, which is assumed to be induced by the embedding synthetic gauge phase within the coupling constants. The combination of these two types of competition, linear mixing and nonlinearity can create four types of soliton: multipeak bell-shaped solitons, multipeak flat-top solitons, staggered bell-shaped solitons, and staggered flat-top solitons. The stability and dynamics of these types of solitons are verified systematically through the paper. The total power and the types of competition between the linear mixing play important roles in tuning these solitons.

Flat-Top Soliton Frequency Comb Generation Through Intra-Cavity Dispersion Engineering in a Brillouin Laser Cavity

Dissipative Kerr cavity soliton and optical frequency comb (OFC) can be generated in fiber cavities or microresonators with anomalous group velocity dispersion (GVD), which has achieved great breakthroughs and found significant applications in many fields. Despite recent progress in this field, it remains challenging to obtain OFCs with a flat-top spectrum and a high conversion efficiency in an optical cavity with anomalous GVD. In this paper, intra-cavity dispersion engineering is proposed to reshape the bright dissipative Kerr solitons in a Brillouin laser cavity with anomalous GVD for generating flat-top OFCs with a high conversion efficiency. By introducing a piece of optical fiber with normal GVD into the above Brillouin fiber laser cavity for making a hybrid fiber cavity with near zero net cavity GVD, the intra-cavity waveform is broadened in time-domain due to the existence of normal GVD, which conquers the pulse compression caused by four-wave mixing and anomalous GVD, and leads to the formation of bright dissipative Kerr solitons and OFCs with a flat-top profile and a high conversion efficiency, via interlocking of up-switching waves. Furthermore, flat-top OFCs with step tunable mode spacing from GHz to THz are achieved via intra-cavity dispersion engineering.

Reflectionless potentials and resonant scattering of flat-top and thin-top solitons.

We identify a class of potentials for which the scattering of flat-top solitons and thin-top solitons of the nonlinear Schrödinger equationwith dual nonlinearity can be reflectionless. The scattering is characterized by sharp resonances between regimes of full transmission and full quantum reflection. Perturbative expansion in terms of the magnitude of radiation losses leads to the general form of reflectionless potentials. Simulating the scattering of flat-top solitons and thin-top solitons confirms the reflectionless feature of these potentials.

Generation of vector flat-top solitons and hybrid bright–flat-top soliton complexes in optical microresonators via modulated pump

We demonstrate numerically the possibility of generation of vector platicons (or vector flat-top solitons) having orthogonally polarized components by the modulated pump at normal group velocity dispersion (GVD). Dynamics of this process can be controlled by pump polarization and depends on both the spectral interval between pumped modes and the difference of their free spectral ranges. We also show the possibility of the excitation of the hybrid soliton-platicon complexes (or hybrid complexes of bright and flat-top solitons) if pumped modes have alternating signs of the GVD coefficient.

Fast soliton interactions in cubic-quintic nonlinear media with weak dissipation

We derive the expressions for the collision-induced amplitude dynamics of two flat-top solitons in spatial dimensions of 1, 2, and 3 caused by the generic weak nonlinear loss under the framework of coupled cubic-quintic (n+1)D nonlinear Schrödinger equations for n=1,2,3. We develop a new perturbative technique which is mainly based on the calculations for the fast collision-induced changes in the soliton envelope and the use of the single perturbed soliton solution for the calculations. These results quantify the energy dropdown due to a fast collision of two pulses (n=1), or two optical beams (n=2), or two light bullets (n=3) in cubic-quintic nonlinear media with dissipation. The theoretical calculations are then confirmed by numerical simulations with the corresponding coupled nonlinear Schrödinger equations. Our approach can be used for studying the effects of dissipation on colliding solitons described by the coupled nonlinear Schrödinger models, where the unperturbed equation is nonintegrable.

Stability Analysis and Transient Attraction Dynamics of Flat-Top Solitons via Modulated Driving Fields in Normal Dispersion Microcavity

We theoretically and numerically investigate the stability and transient attraction dynamics of flat-top solitons (or platicons) in normal group velocity dispersion (GVD) microcavity via modulation-assisted four wave mixing instead of mode interaction. The spatial bifurcation structure of the stationary states depending on modulation power, pump power and cavity detuning is mapped, indicating a significantly extended existence range by the modulation-assisted approach. We explore the transformation between discrete energy spectrum of dark solitons and a quasi-continuous spectrum of platicons with and without a third-order dispersion and identify the connections between these two states. Robust trapping and manipulation of temporal platicons is also verified by implementation of phase modulation, and explained by transient attraction dynamics. One may manipulate the duration of the pulse by varying modulation power, pump power, or cavity detuning. Specifically, the effect of high-order dispersion on Kerr frequency comb generation is studied, and the results show that different orders of GVD term affect the 3dB comb bandwidth, repetition rate, spectral position of dispersive waves, as well as temporal profile and pulse peak power.

Solitons of the generalized nonlinear Schrödinger equation

Exact solutions in a form of bright solitons are obtained for the nonlinear Schrödinger equations with general quartic or quintic nonlinearities. The solutions, expressed in terms of the elliptic integrals, are implicit. An existence of flat-top solitons is shown, and the corresponding condition is obtained. The stability of solitons is analyzed for different system parameters. Numerical simulations show three different scenarios of the soliton instability, namely the soliton decay, oscillations and collapse.

Flatons: Flat-top solitons in extended Gardner-like equations

We present and study an extended Gardner-like family of equations, G(m,n;k):ut+(c+um−c−un)x+(uk)xxx=0,c±>0,n > m > 1, k ≥ 1, endowed with a non-convex convection which may be due to two opposing mechanisms that bound the range of velocities of both solitons and compactons beyond which they dissolve, and kink and/or antikink form. Close to solitons and compactons barrier, there is a narrow strip of velocities where the wave shape undergoes a structural change and rather than grow with velocity, their top flattens and they widen rapidly with minute changes in velocity. These waves, referred to as flatons, may be viewed as an approximate amalgam of a kink and anti kink placed at any distance from each other. Typical of solitons, once flatons form they are very robust with their domain of attraction being sensitive to the amplitude at which convection reverses its direction. A multi-dimensional extension of these equations unfolds a plethora of flatons which, unless m is even and n is odd, for every admissible velocity may span an entire sequence of multi-nodal radially symmetric flatons.

Electrostatic flat-top solitons near double layers and triple root structures in multispecies plasmas: How realistic are they?

Electrostatic flat-top solitons are a new acoustic-type nonlinear mode and found to be a generic feature accompanying the occurrence of double layers and/or triple root structures, in multispecies plasmas admitting the latter. Their existence domains can be parameterized by the difference between their velocities and the double layer or triple root velocities, but these velocity differences turn out to be extremely small, of the order 10−5 or less. The onset of their flat top character in the electrostatic potential is clearly seen in the corresponding electric field or charge density profiles. However, even at the limit of the numerical accuracy for vanishing velocity differences, their profiles are still soliton-like, very unlike those of double layers or triple root structures. So although the Sagdeev potential varies continuously as the structure velocity approaches that of the double layer or triple root structure, the character of the nonlinear modes changes in a discontinuous manner. For sufficiently wide flat-top solitons, the electric field signature looks very much like two unipolar signals with opposite polarities, where unipolar electric fields typically characterize double layers or triple root structures. We are not aware of flat-top solitons having been reported to date, and their extremely limited existence range raises the question of whether they may be observable at all, unless helped by a fortunate stroke of serendipity. This topic requires suitable numerical simulations to ascertain their stability and interaction properties.

Generation of stable and breathing flat-top solitons via Raman assisted four wave mixing in microresonators

Flat-top-soliton (or platicon) dynamics in coherently pumped normal dispersion microresonators is important for both fundamental nonlinear physics and microcomb generation in the visible band. Here we numerically investigate the platicon generation that is initiated via Raman assisted four wave mixing instead of mode interaction. To show the possibility of generating coherent combs in the visible band, we design an aluminum nitride (AlN) microresonator with normal dispersion and investigate the comb generation dynamics in simulations. Stable platicon Kerr combs can be generated in this AlN microresonator using a 780-nm pump. Moreover, we also observe a breather platicon dynamics induced by the narrow Raman gain spectrum of crystalline AlN, which shows distinct dynamics from the dark soliton breathers reported in previous work that are dominated by Kerr effect. A phase diagram bearing the influence of the pump detuning and pump power on the breathing dynamics of the breather platicon is also computed. Furthermore, a transition to chaotic breathing is numerically observed.

Determinant solutions and asymptotic state analysis for an integrable model of transient stimulated Raman scattering

In this paper, the transient stimulated Raman scattering equations, which describe the stimulated Raman scattering in the transient limit, are investigated by Darboux transformation. Considering the trivial initial solutions, the explicit N-soliton solutions in determinant form for the transient stimulated Raman scattering equations are presented. Based on the explicit solutions formula, the one- and two-soliton solutions are given explicitly. The asymptotic state analysis of the two-soliton solutions is conducted. And different soliton propagation phenomena are discussed and illustrated including but not limited to the flat-top soliton, the two-hump soliton and bound two soliton with periodic structure, which is good for understanding the interaction properties under different wave parameters.

Stable flat-top solitons and peakons in the PT-symmetric δ-signum potentials and nonlinear media.

We discover that the physically interesting PT-symmetric Dirac delta-function potentials can not only make sure that the non-Hermitian Hamiltonians admit fully-real linear spectra but also support stable peakons (nonlinear modes) in the Kerr nonlinear Schrödinger equation. For a specific form of the delta-function PT-symmetric potentials, the nonlinear model investigated in this paper is exactly solvable. However, for a class of PT-symmetric signum-function double-well potentials, a novel type of exact flat-top bright solitons can exist stably within a broad range of potential parameters. Intriguingly, the flat-top solitons can be characterized by the finite-order differentiable waveforms and admit the novel features differing from the usual solitons. The excitation features and the direction of transverse power flow of flat-top bright solitons are also explored in detail. These results are useful for the related experimental designs and applications in nonlinear optics and other related fields.

Gaussian-like and flat-top solitons of atoms with spatially modulated repulsive interactions

Solitons, nonlinear particle-like excitations with inalterable properties (amplitude, shape, and velocity) as they propagate, are omnipresent in many branches of science---and in physics in particular. Flat-top solitons are a novel type of bright solitons that have not been well explored in pure nonlinear media. Here, a model of nonlinear Kerr (cubic) media of ultracold atoms with spatially modulated repulsive interactions is proposed and shown to support a vast variety of stable flat-top matter-wave solitons, including one-dimensional (1D) flat-top fundamental and multipole solitons, two-dimensional (2D) flat-top fundamental and vortex solitons. We demonstrate that by varying the relevant physical parameters (nonlinearity coefficient and chemical potential) the ordinary bright (gaussian) solitons can transform into the novel flat-top solitons. The (in-)stability domains of the flat-top soliton families are checked by means of linear stability analysis and reconfirmed by direct numerical simulations. This model is generic in the contexts of nonlinear optics and Bose-Einstein condensates, which provide direct experimental access to observe the predicted solutions.

Purely Kerr nonlinear model admitting flat-top solitons.

We elaborate one- and two-dimensional (1D and 2D) models of media with self-repulsive cubic nonlinearity, whose local strength is subject to spatial modulation that admits the existence of flat-top solitons of various types, including fundamental ones, 1D multipoles, and 2D vortices. Previously, solitons of this type were only produced by models with competing nonlinearities. The present setting may be implemented in optics and Bose-Einstein condensates. The 1D version gives rise to an exact analytical solution for stable flat-top solitons, and generic families may be predicted by means of the Thomas-Fermi approximation. Stability of the obtained flat-top solitons is analyzed by means of the linear-stability analysis and direct simulations. Fundamental solitons and 1D multipoles with k=1 and 2 nodes, as well as vortices with winding number m=1, are completely stable. For multipoles with k≥3 and vortices with m≥2, alternating stripes of stability and instability are identified in their parameter spaces.

Integrability conditions and solitonic solutions of the nonlinear Schrödinger equation with generalized dual-power nonlinearities, PT-symmetric potentials, and space- and time-dependent coefficients

We consider a generalized nonlinear Schrödinger equation with dual power law nonlinearities, complex potential, and position- and time-dependent strengths of dispersion and nonlinearities. Using a standard similarity transformation, we obtain the integrability conditions and solitonic solutions of this equation by mapping it to its homogeneous version. Using a modified similarity transformation, where a solution of the homogeneous equation, which we denote as a seed, enters also in the transformation operator, a wider range of exact solutions is obtained including cases with complex potentials. We apply these two transformations to obtain two exact solitonic solutions of the homogeneous nonlinear Schrödinger equation, which are derived here for the first time for a general power of the nonlinearities, namely the flat-top soliton and tanh solution. We discuss and derive explicit solutions to the experimentally relevant cases associated with parabolic and PT-symmetric potentials.

Combined effects of frequency and higher-order effects on soliton conversion in an erbium fiber with inhomogeneous broadening

In this paper, soliton propagation in an erbium-doped fiber with inhomogeneous broadening is considered which can be described by the inhomogeneous Hirota– Maxwell–Bloch equations. The Darboux transformation method is employed to generate the soliton solution through a linear eigenvalue problem. In particular, our results demonstrate explicitly that a soliton can be converted into various nonlinear waves such as periodic wave, anti-kink soliton and flat-top soliton in the presence of higher-order effects with inhomogeneous broadening. Additionally, we found that higher-order coefficients have strong influence on the soliton transition while frequency is only responsible for phase shift. The results might be of certain value for the study of the soliton management and soliton conversion.