Soliton Velocity Research Articles

Dark solitons on elliptic function background for the defocusing Hirota equation

We investigate dark solitons lying on elliptic function background in the defocusing Hirota equation with third-order dispersion and self-steepening terms. By means of the modified squared wavefunction method, we obtain the Jacobi’s elliptic solution of the defocusing Hirota equation, and solve the related linear matrix eigenvalue problem on elliptic function background. The elliptic N-dark soliton solution in terms of theta functions is constructed by the Darboux transformation and limit technique. The asymptotic dynamical behaviors for the elliptic N-dark soliton solution as t → ± ∞ are studied. Through numerical plots of the elliptic one-, two- and three-dark solitons, the amplification effect on the velocity of elliptic dark solitons, and the compression effect on the soliton spatiotemporal distributions produced by the third-order dispersion and self-steepening terms are discussed.

Asymptotic analysis on bright solitons and breather solutions of a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide

Abstract We study a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide. We obtain the Lax pair and N-fold Darboux transformation (DT) with N being a positive integer. Based on Lax pair obtained by us, we derive the infinitely-many conservation laws. We give the bright one-, two-, and N-soliton solutions, and the first-, second-, and Nth-order breather solutions based on the N-fold DT. We conclude that the velocities of the bright solitons are influenced by the distributed gain function, g(z), and variable coefficients in equation, h 1(z), p 1(z), r 1(z), and s 1(z) via the asymptotic analysis, where z represents the propagation variable or spatial coordinate. We also graphically observe that: the velocities of the first- and second-order breathers will be affected by h 1(z), p 1(z), r 1(z), and s 1(z), and the background wave depends on g(z).

Dynamics of the Davydov’s soliton in external oscillating magnetic field

The influence of an external oscillating in time magnetic field on the dynamics of the Davydov’s soliton is studied. It is shown that soliton parameters depend on the amplitude and frequency of the magnetic field, as well as on the field orientation with respect to the molecular chain axis. Electron dynamics in the parallel magnetic field is a composition of the “free” soliton coherent propagation along the molecular chain and oscillatory movement in the transverse direction. The latter is described by the functions of the harmonic oscillator with the cyclotron frequency depending on the field frequency. In a perpendicular field the soliton dynamics is a superposition of (i) the free electron plane wave in the plane perpendicular to the molecular chain, and (ii) propagation along the chain which is described by the modified Nonlinear Schrödinger equation with an extra term depending on the field. This equation is solved using the nonlinear perturbation theory. It is shown that the soliton velocity and phase are given by expressions which include the terms oscillating in time with the frequency of the main harmonic, given by the external field frequency, and higher multiple harmonics. Such complex effects of external time-depending magnetic fields on the dynamics of solitons modify the charge transport in low-dimensional molecular systems, which in its turn, can affect the functioning of the devices based on such compounds. These results suggest also the physical mechanism of therapeutic effects of oscillating magnetic fields, based on the field influence on the dynamics of solitons which provide charge transport through biological macromolecules in the redox processes.

Hamiltonian mechanics of “magnetic” solitons in two‐component Bose–Einstein condensates

AbstractWe consider the motion of a “magnetic” soliton in two‐component condensates along a nonuniform and time‐dependent background in the framework of Hamiltonian mechanics. Our approach is based on generalization of Stokes' remark that soliton's velocity is related to its inverse half‐width by the dispersion law for linear waves continued to the region of complex wave numbers. We obtain expressions for the canonical momentum and the Hamiltonian as functions of soliton's velocity and transform the Hamilton equations to a Newton‐like equation. The theory is illustrated by several examples of concrete soliton's dynamics.

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.

Effective regulation of the interaction process among three optical solitons

Abstract The interaction between three optical solitons is a complex and valuable research direction, which is of practical application for promoting the development of optical communication and all-optical information processing technology. In this paper, we start from the study of the variable-coefficient coupled higher-order nonlinear Schrödinger equation (VCHNLSE), and obtain an analytical three-soliton solution of this equation. Based on the obtained solution, the interaction of the three optical solitons is explored when they are incident from different initial velocities and phases. When the higher-order dispersion and nonlinear functions are sinusoidal, hyperbolic secant, and hyperbolic tangent functions, the transmission properties of three optical solitons before and after interactions are discussed. Besides, this paper achieves effective regulation of amplitude and velocity of optical solitons as well as of the local state of interaction process, and interaction-free transmission of the three optical solitons is obtained with a small spacing. The relevant conclusions of the paper are of great significance in promoting the development of high-speed and large-capacity optical communication, optical signal processing, and optical computing.

Controlling dark solitons on the healing length scale

While usually the optical diffraction limit is setting a limit for the lengthscales on which a typical alkali Bose–Einstein condensate can be controlled, we show that in certain situations control via matter waves can achieve smaller resolutions. For this we consider a small number of impurity atoms which are trapped inside the density dip of a dark soliton state and show that any grey soliton state can be obtained by selectively driving the impurity atoms. This allows to fully control the position and velocity of the dark soliton, and also study controlled collisions between these non-linear objects.

Riemann–Hilbert approach, dark solitons and double‐pole solutions for Lakshmanan–Porsezian–Daniel equation in an optical fiber, a ferromagnetic spin or a protein

AbstractInverse scattering transform for the defocusing Lakshmanan–Porsezian–Daniel equation with nonzero boundary condition is constructed via the Riemann–Hilbert approach. Since poles of the associated reflection coefficient are simple, ‐dark soliton solutions corresponding to simple poles are constructed. For ‐dark soliton solutions, results show that the soliton amplitude and width are not affected by the strength of the higher‐order linear and nonlinear effects , but soliton velocity has a linear correlation with ; the interactions between the two‐dark solitons and among the three‐dark solitons are elastic and experience phase and position shifts. Besides, asymptotic analysis of the double‐pole solutions for the focusing Lakshmanan–Porsezian–Daniel equation with nonzero boundary condition is presented. Different from ‐dark soliton solutions which locate in the straight lines and experience position shift after the interaction, the double‐pole solutions diverge from each other logarithmically and experience no position shift after the interaction.

Moving Bragg Solitons in a Dual-Core System Composed of a Linear Bragg Grating with Dispersive Reflectivity and a Uniform Nonlinear Core

The existence and stability of moving Bragg grating solitons are systematically investigated in a dual-core system, where one core is uniform and has Kerr nonlinearity, and the other is linear with Bragg grating and dispersive reflectivity. It is found that moving soliton solutions exist throughout the upper and lower bandgaps, whereas no soliton solutions exist in the central bandgap. Similar to the quiescent solitons in the system, it is found that when dispersive reflectivity is nonzero, for certain values of parameters, sidelobes appear in the solitons’ profiles. The stability of the moving solitons is characterized using systematic numerical stability analysis. Additionally, the impact and interplay of dispersive reflectivity, soliton velocity, and group velocity on the stability border are analyzed.

Interaction of soliton gases in deep-water surface gravity waves.

Soliton gases represent large random soliton ensembles in physical systems that display integrable dynamics at leading order. We report hydrodynamic experiments in which we investigate the interaction between two beams or jets of soliton gases having nearly identical amplitudes but opposite velocities of the same magnitude. The space-time evolution of the two interacting soliton gas jets is recorded in a 140-m-long water tank where the dynamics is described at leading order by the focusing one-dimensional nonlinear Schrödinger equation. Varying the relative initial velocity of the two species of soliton gas, we change their interaction strength and we measure the macroscopic soliton gas density and velocity changes due to the interaction. Our experimental results are found to be in good quantitative agreement with predictions of the spectral kinetic theory of soliton gas despite the presence of perturbative higher-order effects that break the integrability of the wave dynamics.

Atomic soliton transmission and induced collapse in scattering from a narrow barrier

We report systematic numerical simulations of the collision of a bright matter-wave soliton made of Bose-condensed alkali-metal atoms through a narrow potential barrier by using the three-dimensional Gross–Pitaevskii equation. In this way, we determine how the transmission coefficient depends on the soliton impact velocity and the barrier height. Quite remarkably, we also obtain the regions of parameters where there is the collapse of the bright soliton induced by the collision. We compare these three-dimensional results with the ones obtained by three different one-dimensional nonlinear Schrödinger equations. We find that a specifically modified nonpolynomial Schrödinger equation is able to accurately assess the transmission coefficient even in a region in which the usual nonpolynomial Schrödinger equation collapses. In particular, this simplified but very effective one-dimensional model takes into account the transverse width dynamics of the soliton with an ordinary differential equation coupled to the partial differential equation of the axial wave function of the Bose–Einstein condensate.

Trapping and guiding soliton through Airy pulse

We show that Airy soliton interaction in anomalous dispersion and focussing nonlinear regime provides an effective control on the co-propagating sech soliton dynamics. Cross phase modulation (XPM) induced nonlinear phase results varying group velocity of the sech soliton which leads to its temporal shift towards leading edge. The soliton could be made either advanced or delayed by choosing an appropriate initial time delay between the two co-propagating pulses. The effect of the initial time delay on trapping efficiency is also disclosed. Additionally, we have studied the effect of temporal chirp in Airy pulse on sech soliton dynamics. It is found that introduction of positive chirp inhibits the XPM-induced temporal shift of the soliton, while negative chirp boosts the XPM-induced acceleration. Further, the effect of group delay on the trapping and guiding of the soliton has been investigated. Moreover, the role of relative dispersion and nonlinear length on the soliton pulse acceleration are investigated. These results are expected to provide a better understanding on manipulation of light by light that could be useful in all-optical switching and signal control.

Domain walls and vector solitons in the coupled nonlinear Schrödinger equation

We outline a program to classify domain walls (DWs) and vector solitons in the 1D two-component coupled nonlinear Schrödinger (CNLS) equation without restricting the signs or magnitudes of any coefficients. The CNLS equation is reduced first to a complex ordinary differential equation (ODE), and then to a real ODE after imposing a restriction. In the real ODE, we identify four possible equilibria including ZZ, ZN, NZ, and NN, with Z(N) denoting a zero (nonzero) value in a component, and analyze their spatial stability. We identify two types of DWs including asymmetric DWs between ZZ and NN and symmetric DWs between ZN and NZ. We identify three codimension-1 mechanisms for generating vector solitons in the real ODE including heteroclinic cycles, local bifurcations, and exact solutions. Heteroclinic cycles are formed by assembling two DWs back-to-back and generate extended bright-bright (BB), dark-dark (DD), and dark-bright (DB) solitons. Local bifurcations include the Turing (Hamiltonian–Hopf) bifurcation that generates Turing solitons with oscillatory tails and the pitchfork bifurcation that generates DB, bright-antidark, DD, and dark-antidark solitons with monotonic tails. Exact solutions include scalar bright and dark solitons with vector amplitudes. Any codimension-1 real vector soliton can be numerically continued into a codimension-0 family. Complex vector solitons have two more parameters: a dark or antidark component can be numerically continued in the wavenumber, while a bright component can be multiplied by a constant phase factor. We introduce a numerical continuation method to find real and complex vector solitons and show that DWs and DB solitons in the immiscible regime can be related by varying bifurcation parameters. We show that collisions between two DB solitons with a nonzero phase difference in their bright components typically feature a mass exchange that changes the frequencies and phases of the two bright components and the two soliton velocities.

Theoretical Analysis of Microcavity Simultons Reinforced by χ^{(2)} and χ^{(3)} Nonlinearities.

High-Q microcavities with quadratic and cubic nonlinearities add lots of versatility in controlling microcombs. Here, we study microcavity simulton and soliton dynamics reinforced by both χ^{(2)} and χ^{(3)} nonlinearities in a continuously pumped microcavity. Theoretical analysis based on the Lagrangian approach reveals the soliton peak power and gain-loss balance are impacted by the flat part of the intracavity pump, while the dark-pulse part of the pump leads to a nearly constant soliton group velocity change. We also derived a soliton conversion efficiency upper limit that is fully determined by the coupling condition and the quantum-limited soliton timing jitter in the χ^{(2,3)} system. Numerical simulations confirm the analytical results. Our theory is particularly useful for investigating AlN microcombs and sheds light on the interplay between χ^{(2)} and χ^{(3)} nonlinearities within microcavity simultons.

A study on solitary wave solutions for the Zoomeron equation supported by two-dimensional dynamics

This study emphasizes the importance of understanding natural phenomena through various observations and relating them to scientific studies. Nonlinear partial differential equations serve as fundamental tools for modeling these phenomena, with a focus on nonlinear evolution equations when involving time. This paper investigates the dynamics of the Zoomeron equation, highlighting not only a result derived from the KdV and Schrödinger equations but also its contribution to the modeling of Boomeron and Trappon solitons. Different analytical techniques yield various solutions for the Zoomeron equation, each offering unique insights. The behavior of the solutions generated in kink and singular soliton forms in various dynamics using the -expansion method is investigated and compared under restrictive conditions, and the distribution of energy density is explained with the assistance of the gradient function. It is aimed at gaining a physically unique perspective by revealing the connections between the velocities of solitons produced for this equation and the effects of gradient flow directions. In addition, stability analysis for some of the solutions generated in the kink form is investigated. The structure of the paper includes method introduction, application, stability property, results and discussion, resulting in a unique perspective on understanding the physical dynamics of the Zoomeron model.

Dynamical behavior of fractional nonlinear dispersive equation in Murnaghan’s rod materials

The primary objective of this study aims to carry out a more thorough investigation into a fractional nonlinear double dispersive equation that is used to represent wave propagation in an elastic, inhomogeneous Murnaghan’s rod. By Murnaghan’s rod, we mean the materials, which include the constitutive constant, Poisson ratio, and Lame’́s coefficient, are considered to be compressible in nature forming up the elastic rod. To solve the fractional version of Murnaghan’s rod problem, we employed β-fractional and M-Truncated fractional derivative. Regarding the extraction of polynomial and rational function solutions of the Murnaghan’s rod problem, which degenerate into several wave solutions including solitary, soliton (dromions), as well as periodic wave solutions. We employ the well-known unified and new auxiliary equation methods of nonlinear sciences. A finite series of certain functions satisfying an ordinary differential equation of first order, second degree is used to represent the projected solution. Based on the given approach, numerous types of solutions for exponential, hyperbolic, and trigonometric functions are generated. In this research study, the behavior of a dynamical planer system has been examined by giving various values to parameters and by depicting every possible situation as a phase portrait. The sensitivity analysis, where the soliton wave velocity and wave number parameters influence the water wave singularity, is demonstrated using the wave profiles of the constructed dynamical structural system. With the use of graphs, we have simulated the solitons to determine their kinds. All of solutions found in this manuscript is been confirmed through back substituting them into the original model using computational software.

Variable-coefficient polynomial function method for finding the lump-type solutions of integrable system with variable coefficients

In this work, we study a ([Formula: see text])-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation in fluid mechanics. The new lump and lump-soliton solutions are obtained by the variable-coefficient polynomial function method. We used 3D graphs, contour plots and density graphs to show that the amplitude and velocity of solitons are affected by some variable coefficients. It is proved that the polynomial function method with variable coefficients is very direct and effective for solving lump-type solutions in variable coefficient integrable systems, and more new conclusions can be obtained.

Propagation of generalized Korteweg-de Vries solitons along large-scale waves.

We consider propagation of solitons along large-scale background waves in the generalized Korteweg-de Vries (gKdV) equationtheory when the width of the soliton is much smaller than the characteristic size of the background wave. Due to this difference in scales, the soliton's motion does not affect the dispersionless evolution of the background wave. We obtained the Hamilton equationsfor soliton's motion and derived simple relationships which express the soliton's velocity in terms of a local value of the background wave. Solitons' paths obtained by integration of these relationships agree very well with the exact numerical solutions of the gKdV equation.

Conservation laws, bound-state solitons and modulation instability for a variable-coefficient higher-order nonlinear Schrödinger equation in an optical fiber

Under investigation in this paper is a variable-coefficient sixth-order nonlinear Schrödinger equation, which describes the propagation of attosecond pulses in an optical fiber. Based on Lax pair, infinitely-many conservation laws are constructed. With the aid of auxiliary functions, bilinear forms are derived. In addition, the one- and two-soliton solutions are obtained via the Hirota method. The influences of variable coefficients for soliton velocity and profile are discussed. Particularly, the interaction periods and soliton separation factor of bound-state solitons are analyzed. Finally, modulation instability is investigated. The reported results could be used to understand related soliton molecule and optical instability phenomena in nonlinear optics.

Voltage-controlled magnetic solitons motion in an anisotropic ferromagnetic nanowire

The precise manipulation of magnetic solitons remains a challenge and is considered a crucial process in magnetic storage. In this paper, we investigate the control of velocity and spatial manipulation of magnetic solitons using the voltage-controlled magnetic anisotropy effect. A long-wave model, known as the generalized derivative nonlinear Schrödinger (GDNLS) equation, is developed to describe the dynamics of magnetic solitons in an anisotropic ferromagnetic nanowire. By constructing the Lax pair for the GDNLS equation, we obtain the exact solutions including magnetic dark solitons, anti-dark solitons, and periodic solutions. Moreover, we propose two approaches to manipulate magnetic solitons: direct voltage application and inhomogeneous insulation layer design. Numerically results show the direct modulation of soliton velocity by a constant voltage, while time-varying voltage induces periodic oscillations. Investigation of Gaussian-type defects reveals soliton being trapped beyond a critical defect depth. These results provide a theoretical basis for future applications in magnetic soliton-based memory devices.