Soliton Trajectories Research Articles

New Soliton Solutions and Trajectory Equations of Soliton Collision to a (2+1)-Dimensional Shallow Water Wave Model

New Soliton Solutions and Trajectory Equations of Soliton Collision to a (2+1)-Dimensional Shallow Water Wave Model

Soliton interaction in a two-temperature electron plasma with trapping and superthermality effects

Head-on collision of the two small-amplitude electron-acoustic (EA) solitons is studied in an unmagnetized collisionless plasma in the presence of superthermal (hot) trapped electrons. For this purpose, using a well-known extended Poincare–Lighthill–Kuo (PLK) method, a pair of the trapped Korteweg–de Vries (tKdV) equations is derived to investigate the soliton trajectories and phase shifts. The latter are found dependent on amplitudes of the interacting solitons, effectively altering with hot-electron superthermality and plasma parameters. Typical parameters for the electron diffusion region (EDR) and day-side auroral zone have been selected to examine the impact of hot-electron superthermality, trapping parameter, hot-to-cold electron number density ratio, and cold-to-hot electron temperature ratio on the profiles of potential excitations and phase shifts of interacting solitons. It is found that phase speed of the EA waves becomes altered by varying the κ–parameter, strongly modifying the nonlinearity and dispersive coefficients in a superthermal trapped plasma. However, particle trapping phenomenon does not affect the linear phase speed but introduces a fractional nonlinearity in the tKdV equations of two interacting solitons. The impact of the adiabatic and isothermal pressures is also highlighted to show new modifications in the propagation characteristics of two interacting solitons.

Light-regulated soliton dynamics in liquid crystals

Electrically powered solitons are particle-like field configurations in out-of-equilibrium nematics that have garnered significant interest. However, their random generation and lack of controllable motion have limited their application. Here, we present a reconfigurable optoelectronic approach capable of regulating the entire lifecycle of solitons by utilizing multi-strategy digital light projection to construct delicate patterning of virtual electrode. We demonstrate that optically actuated domains with diverse geometry enable the generation of multiple solitons and further allow in-situ formation of individual soliton by matching the light pattern to its dimension. Exquisitely engineered light intensity of patterns facilitates modulation of soliton velocity and transformation of propagating direction. The utilization of a light-guided channel enables the on-demand control of soliton trajectories along customized paths. Furthermore, dynamic light patterns that vary in space and time allow for collective motion such as migration, mimicking phototaxis in biological systems. This reconfigurable manipulation strategy, grounded in the photoconductive effect, proves highly versatile and effective in directing soliton dynamics, heralding the potential for their programmable control and offering a significant advantage in multitasking scenarios.

Controlling intracavity dual-comb soliton motion in a single-fiber laser.

Ultrafast science builds on dynamic compositions of precisely timed light pulses, and evolving groups of pulses are observed in almost every mode-locked laser. However, the underlying physics has rarely been controlled or used until now. Here, we demonstrate a general approach to control soliton motion inside a dual-comb laser and the programmable synthesis of ultrashort pulse patterns. Introducing single-pulse modulation inside an Er:fiber laser, we rapidly shift the timing between two temporally separated soliton combs. Their superposition outside the cavity yields ultrashort soliton sequences. On the basis of real-time spectral interferometry, we observe the deterministic switching of intersoliton separation arising from the interplay of attracting and repulsing forces via ultrafast nonlinearity and laser gain dynamics. Harnessing these insights, we demonstrate the high-speed all-optical synthesis of nano- to picosecond pump-probe delays and programmable free-form soliton trajectories. This concept may pave the way to a new class of all-optical delay generators for ultrafast measurements at unprecedented high tuning, cycling, and acquisition speeds.

Multi-elliptic-dark soliton solutions of the defocusing nonlinear Schrödinger equation

We derive multi-elliptic-dark soliton solutions for the defocusing nonlinear Schrödinger (NLS) equation by using the Darboux-Bäcklund transformation. Additionally, we investigate the asymptotic dynamic behaviors of these solutions along the trajectories of dark solitons, which reveals that the N-elliptic-dark soliton solution can be decomposed into N single-elliptic-dark soliton solutions, as t→±∞.

Non-collisional dynamics of nonautonomous three solitons through tailoring of modulated coefficients and modulation instability gain spectra

In this paper, the generalized nonautonomous nonlinear Schrödinger equation with non-isospectral parameter is investigated from the point of view of soliton management through modulation of dispersion and nonlinearity. Based on the obtained analytic soliton solutions, the effect of modulated coefficients such as dispersion and nonlinearity on the control of soliton transmission is studied. Especially, controllable soliton interactions are investigated through suitably tailoring the inhomogeneous profiles. Additionally, modulation instability (MI) has been analyzed graphically through linear stability analysis (LSA) and MI gain spectrum is analyzed for various forms of modulated coefficients. Through this study, we confirm that the modulated coefficients in the theoretical model under investigation affect the amplitudes, widths and trajectories of the nonautonomous solitons. Thus, one can obtain the optical control system by choosing the different form of modulated coefficients for the specific problem. Results obtained in this paper would be worthy in the field of soliton management for improving the capacity of optical fiber communication system.

Interaction between double solitons in anti-PT symmetric synthetic photonic lattices

An anti-parity-time symmetry (anti-PT symmetric) double fiber loop synthetic photonic lattice is constructed theoretically by projecting the intensity of two loops and adjusting the phase and gain/loss distribution. Numerical studies on pulse dynamics show that the trajectories of double-pulse solitons may be manipulated by tuning the initial relative phases, pulse separation, nonlinearity, and non-Hermitian characteristics of the system. Our research may provide a new way to realize logic gate devices and optical switches in optical interconnection and optical communications.

Intracavity Raman scattering couples soliton molecules with terahertz phonons

Ultrafast atomic vibrations mediate heat transport, serve as fingerprints for chemical bonds and drive phase transitions in condensed matter systems. Light pulses shorter than the atomic oscillation period can not only probe, but even stimulate and control collective excitations. In general, such interactions are performed with free-propagating pulses. Here, we demonstrate intra-cavity excitation and time-domain sampling of coherent optical phonons inside an active laser oscillator. Employing real-time spectral interferometry, we reveal that Terahertz beats of Raman-active optical phonons are the origin of soliton bound-states – also termed “Soliton molecules” – and we resolve a coherent coupling mechanism of phonon and intra-cavity soliton motion. Concurring electronic and nuclear refractive nonlinearities generate distinct soliton trajectories and, effectively, enhance the time-domain Raman signal. We utilize the intrinsic soliton motion to automatically perform highspeed Raman spectroscopy of the intra-cavity crystal. Our results pinpoint the impact of Raman-induced soliton interactions in crystalline laser media and microresonators, and offer unique perspectives toward ultrafast nonlinear phononics by exploiting the coupling of atomic motion and solitons inside a cavity.

Programming Solitons in Liquid Crystals Using Surface Chemistry.

AC electric fields cause three-dimensional orientational fluctuations (solitons) to form and rapidly propagate in confined films of liquid crystals (LCs), offering the basis of a new class of active soft matter (e.g., for accelerating mixing and transport processes in microscale chemical systems). How surface chemistry impacts the formation and trajectories of solitons, however, is not understood. Here, we show that self-assembled monolayers (SAMs) formed from alkanethiols on gold, which permit precise control over surface chemistry, are electrochemically stable over voltage and frequency windows (<100 V; 1 kHz) that lead to soliton formation in achiral nematic films of 4'-butyl-4-heptyl-bicyclohexyl-4-carbonitrile (CCN-47). By comparing soliton formation in LC films confined by SAMs formed from hexadecanethiol (C16SH) or pentadecanethiol (C15SH), we reveal that the electric field required for soliton formation increases with the LC anchoring energy: surfaces patterned with regions of C16SH and C15SH SAMs thus permit spatially controlled creation and annihilation of solitons necessary to generate a net flux of solitons. We also show that solitons propagate in orthogonal directions when confined by obliquely deposited gold films decorated with SAMs formed from C16SH or C15SH and that the azimuthal direction of propagation of solitons within achiral LC films possessing surface-induced twists is not unique but reflects variation in the spatial location of the solitons across the thickness of the twisted LC film. Finally, discontinuous changes in LC orientation induced by patterned surface anchoring lead to a range of new soliton behaviors including refraction, reflection, and splitting of solitons at the domain boundaries. Overall, our results provide new approaches for the controlled generation and programming of solitons with complex and precise trajectories, principles that inform new designs of chemical soft matter.

Controllable propagation paths of gap solitons.

This paper numerically investigates the evolution of solitons in an optical lattice with gradual longitudinal manipulation. We find that the stationary solutions (with added noise to the amplitude) keep their width, profile, and intensity very well, although the propagation path is continuously changing during the modulated propagation. Discontinuities in the modulation functions cause the scattering of the beam that may end the stable propagation. Our results reveal a method to control the trajectory of solitons by designed variation of the optical lattice waveguides. Interesting examples presented include the snakelike and spiraling solitons that both can be adaptively induced in sinusoidally and helically shaped optical lattices. The controlled propagation paths provide an excellent opportunity for various applications, including optical switches and signal transmission, among others.

Asymmetric multi-vortex solitons in nonlocal nonlinear media

This paper introduces a series of asymmetric multi-vortex solitons, which made up of Gaussian beam and embedded multi off-axis shift vortex, in nonlocal nonlinear media. The analytical expression of the soliton trajectory is derived, and the other propagation properties, such as the intensity distribution, the rotating period, and the orbital angular momentum (OAM) are all investigated. It is found that the initial displacement of the soliton mass center, which can be controlled by selecting the appropriate shift values of the off-axis vortexes, can determine the trajectory of the optics beam. The OAM also depending on the shift values of the vortexes. Obviously, the controllable propagation of the asymmetric multi-vortex beam has important academic value and application prospect in optical switches, or be used as optical tweezers and wrenches.

Dark soliton collisions and method of lines approach for modeling freak waves in a positron beam plasma having superthermal electrons

There are two main goals in this research, the first one is presenting an extended Poincaré–Lighthill–Kuo method (EPLKM) for studying the collisions between two dark envelope solitons in a non-Maxwellian plasma permeated by a positron beam, drawing inspiration from recent laboratory experiments on positron beams. The second goal in the present model explains the method of lines (MOLs) approach devoted for analyzing and modeling freak waves (FWs) . In order to achieve the goals of this study, a nonlinear Schrödinger equation (NLSE) has been derived using a reductive perturbation technique (RPT). The regions of (un)stable structures (dark and bright solitons and FWs) have been precisely defined based on the studying of modulational instability (MI) of the NLSE. In the stable regions, we focus our attention in examining and investigating dark soliton collisions. To do that a counterpart pair of two dark envelope solitons is assumed as initial condition, and EPLKM is employed to get expressions for the analytical phase shifts and determine the soliton trajectories after collisions. A parametric investigation is also presented by the effect of the superthermality (kappa) parameter and the positron beam characteristics (concentration, streaming velocity) on the colliding soliton properties. On the other side the MOLs approach is introduced for studying the rogue wave (RW) solution of the NLSE in the unstable regions. The numerical results of the comparison between the numerical and analytical solutions showed the accuracy of the MOLs. What prompted us to use this method is its high accuracy and ease of use rather than more complex numerical methods. These results will be helpful in understanding the dynamics of modulated structures (envelope dark soliton collisions and FWs) in purpose designed experiments.

Two-color ‘dancing’ light bullets in the graded-index waveguide

The propagation of a spatiotemporal soliton in a focusing quadratic-nonlinear waveguide is theoretically investigated. This soliton is a bound state of localized bunches of light energy at the fundamental frequency and at its second harmonics. It is shown that with anisotropic spatial distribution of the refractive index in the cross section of the waveguide, the soliton trajectory can be a spatial Lissajous figure. Such spatiotemporal solitons are called two-color ‘dancing’ light bullets. Unlike dancing light bullets in a waveguide with Kerr nonlinearity, the stability of the dancing bullets discussed here does not require additional restrictions on their apertures, powers, and temporal durations. In addition, a two-color dancing light bullet can be formed both under anomalous and normal dispersion of group velocity. It also distinguishes a two-color dancing light bullet in a quadratic nonlinear waveguide from a dancing light bullet in a Kerr nonlinear waveguide.

Random walks of trains of dissipative solitons

The propagation of light pulses in dual-core nonlinear optical fibers is studied using a model proposed by Sakaguchi and Malomed. The system consists of a supercritical complex Ginzburg-Landau equation coupled to a linear equation. Our analysis includes single standing and walking solitons as well as walking trains of 3, 5, 6, and 12 solitons. For the characterization of the different scenarios, we used ensemble-averaged square displacement of the soliton trajectories and time-averaged power spectrum of the background waves. Power law spectra, indicative of turbulence, were found to be associated with random walks. The number of solitons (or their separations) can trigger anomalous random walks or totally suppress the background waves.

Kink-antikink scattering in the ϕ4 model without static intersoliton forces

Kink-antikink scattering in the $\phi^4$ model is investigated in the limit when the static inter-soliton force vanishes. We observe the formation of spectral walls and, further, identify a new phenomenon, the vacuum wall, whose existence gives rise to a bouncing structure for the annihilating solitons. Furthermore, we discover higher order spectral walls, i.e., spectral walls which form when higher harmonics enter the continuous spectrum. These higher order spectral walls not only deform the soliton trajectories, they also can be observed easily as very intense radiation bursts.

Dynamic dissipative solitons in nematics with positive anisotropies.

Electric field induced instabilities of nematic molecules are of importance for both fundamental science and practical applications. Complex electro-hydrodynamic (EHD) effects such as electro-convection, fingerprint textures, spatiotemporal chaos, and solitons in nematics have been broadly investigated and generated much attention. In this work, dissipative solitons as a novel EHD phenomenon are realized in nematics with positive anisotropies, presumably for the first time. Unlike the ones reported recently in nematics with negative anisotropies whose formation and dynamics are mainly attributed to the flexoelectric and electro-convection effects, the solitons discussed here arise from the nonlinear coupling between the director field and the isotropic flow induced by ion motion. The structure and dynamics of the solitons are demonstrated and the influences of chirality, azimuthal anchoring and ion concentration are also investigated. Finally, we show that the propagation trajectory of solitons can be manipulated by patterned photoalignment and micro-particles can be trapped by them as vehicles for micro-cargo transport.

A simple approach to study the boundary-induced trajectory evolution of spatial nonlocal quadratic solitons: Based on the Green’s function method

Based on the Green’s function method, we propose a simple and time saving approach to study the boundary-induced trajectory of the bell-like bright quadratic solitons. The results obtained from the proposed approach approximate well to those obtained from the direct numerical simulation of the full model equations, which are precise but suffer from time consuming. Based on the proposed approach, it is revealed that the boundary-induced zigzag trajectory of the soliton is affected by several aspects, including the degree of nonlocality, the sample width, the initial displacement of the soliton from the boundary, and the oblique degree of the incident field.

On the Higher-Order Phase Shift Contributions in Opposite Polarities Dust Plasmas

Abstract The collision of dressed dust acoustic solitons (DDASs) and the analytical higher-order phase shift are studied in a dusty plasma system that contains cold negative and positive dusty fluids and Maxwellian distributed for ions as well as electrons. The extended Poincaré–Lighthill–Kuo method is applied in order to obtain the nonlinear Korteweg–de Vries and phase shift equations, which admit the variation in soliton profiles and trajectories, respectively. Influences of the higher-order correction and the plasma fluid parameters such as charged dust grains concentration, negative-to-positive dust grain mass ratio, ion-to-negative dust grain number density ratio, and ion-to-electron temperature ratio on the characteristics of DDASs and their phase shifts are discussed. The comparisons between first- and higher-order contributions in rarefactive and compressive profiles are also taken into account. Furthermore, the present consideration may be utilised to mesosphere and magnetosphere.

Head-On Collision of Electron-Acoustic Solitons in a Magnetized Plasma

The dynamics of electron-acoustic waves (EAWs) profile and its quasi-elastic head-on collision (i.e., it causes shifts of the postcollision soliton trajectories only) is investigated in a plasma consisting of cold magnetized electrons, background hot electrons obeying nonthermal distribution, and stationary ions. Applying an extended Poincare-Lighthill-Kuo perturbation method, the coupled Korteweg-de Vries equations are derived and solved analytically. The effects of the nonthermal parameter, the angle between the magnetic field direction with the collision line, hot-to-cold electrons number density ratio, hot-to-cold electrons temperature ratio, and electron cyclotron frequency on both the solitary EAW profile and the two solitons collision are investigated. The present model is applied to the plasma in the Earth auroral zone.

Soliton-generating τ-functions revisited

In the inverse-scattering formalism and the Hirota algorithm, soliton solutions of evolution equations are images of τ-functions, which are, often, unbounded in space and time. Hence, whereas solitons are bounded, localized structures, they are generated from unbounded, spatially extended entities that lack physically intuitive significance. The goal of this paper is to provide a representation of soliton solutions as images of bounded and localized sources. To this end, appropriate equivalent τ-functions, which are different in form from the traditionally used ones, are generated. In terms of the latter, the single-soliton solutions of the Korteweg-deVries (KdV) and Kadomtsev-Petviashvili II (KP II) equations are images of single solitons (NOT solutions of theses equations), whereas multi-soliton solutions are images of positive humps that are localized in the soliton interaction region. The structure of the equivalent τ-functions has another desirable attribute: It elucidates in a simple manner the role of the arbitrary shifts in the positions of soliton trajectories in controlling the behavior of multi-soliton solutions: Whether an N-soliton solution is reduced to a solution with (N − 1) or (N − 2) solitons when two wave numbers are made to coincide. Examples are discussed in the cases of the KdV and KP II equations. The modified KdV equation provides a unique observation. Depending on the shifts in the positions of soliton trajectories, when two wave numbers are made to coincide, the two-soliton solution may tend either to a single soliton or to a δ-function in the x-t plane.