Initial Soliton Research Articles

Transfer of solitons and half-vortex solitons via adiabatic passage

We show that the transfer of matter-wave solitons and half-vortex solitons in a spin-orbit coupled Bose-Einstein condensate between two (or more) arbitrarily chosen sites of an optical lattice can be implemented using the adiabatic passage. The underlying linear Hamiltonian has a flat band in its spectrum, so that even sufficiently weak interatomic interactions can sustain well-localized Wannier solitons which are involved in the transfer process. The adiabatic passage is assisted by properly chosen spatial and temporal modulations of the Rabi frequency. Within the framework of a few-mode approximation, the mechanism is enabled by a dark state created by coupling the initial and target low-energy solitons with a high-energy extended Bloch state, like in the conventional stimulated Raman adiabatic passage used for the coherent control of quantum states. In real space, however, the atomic transfer between initial and target states is sustained by the current carried by the extended Bloch state, which remains populated during the whole process. The full description of the transfer is provided by the Gross-Pitaevskii equation. Protocols for the adiabatic passage are described for one- and two-dimensional optical lattices, as well as for splitting and subsequent transfer of an initial wave packet simultaneously to two different target locations. Published by the American Physical Society 2024

Generation of a vector conventional soliton via a graphene oxide saturable absorber.

We have experimentally observed an ultrashort conventional vector soliton in an erbium-doped fiber laser. The few-layered graphene oxide (GO) is used as a saturable absorber (SA). It is found that the saturable absorption characteristic of GO is polarization independent. Therefore, vector solitons can be obtained without polarization control by using such SA. By using a polarization beam splitter to split the mode-locked pulse obtained in the oscillator, two orthogonal polarization vector solitons with equal intensity and consistent characteristics can be obtained. It demonstrates that the initial soliton consists of two orthogonal polarization components. It is worth noting that these two orthogonal polarization component solitons improve the signal-to-noise ratio (SNR) of 3dB compared with the initial soliton. The improvement in SNR is very significant and cannot be neglected. This phenomenon has not been reported before, to our knowledge. In addition, the conventional soliton generated by this mode-locked laser has a central wavelength of 1559nm with 1.1ps pulse duration. The mode-locking state of this laser can be self-started. After mode locking, the environmental stability is excellent. The experimental results indicate that GO as a broadband SA has great potential and application prospects in the field of vector soliton generation.

Quantum decay of an optical soliton

Optical solitons are known to be classically stable objects which are robust to perturbations. In this work, we show that due to quantum mechanical effects, an optical soliton that is initially in a classical soliton coherent state will shed photons into the continuum and hence decay. The standard formulation of the quantized soliton uses the linearized version of the quantum nonlinear Schrodinger equation in the background of the classical soliton, and the quantized soliton remains stable in this approximation. We show that if higher-order interaction terms are taken into account, the soliton is no longer stable, and its photon number decreases quadratically as a function of the number of soliton cycles. We compute the power spectrum for the continuum radiation and find a narrow band that is localized about the initial soliton momentum with a cut-off that is inversely proportional to the initial soliton width.

Temporal behavior of bright and dark spatial solitons in photorefractive crystals having both the linear and quadratic electro-optic effects based on low amplitude approximations

We investigate the formation or time evolution characteristics of bright and dark optical spatial solitons in novel photorefractive media with linear and quadratic electro-optic effects. This study was carried out at low amplitude. By the application of low amplitude approximation to the time-dependent wave (soliton) propagation equation, bright soliton solution (BSS) and dark soliton solution (DSS) were each obtained analytically, and both are exact solutions. The change in soliton width or the intensity full width at half maximum (FWHM) of two soliton types can also be calculated directly from the BSS and the DSS, providing a suitable description of the bright and dark solitons time evolution in novel photorefractive media. Only steady state solitons are formed in low amplitude, and the quasi-steady state soliton does not exist as reported at high saturation values. We compare analytical BSS with numerical calculation results of previous studies and find good agreement between analytics and numerics on the amplitude limit. In contrast to bright solitons, which can arise in the dominant linear or quadratic electro-optic effect, dark solitons in novel photorefractive media always exist under the dominant quadratic electro-optic effects only. We found that the time evolution of dark solitons from the DSS corresponds with time-independent optical spatial solitons studies in a similar media. Lastly, BSS and DSS found at low amplitude limits are fundamental soliton solutions or initial state soliton (at low peak intensity ratio regime) that can play an important role in further investigations of optical spatial solitons formation in novel photorefractive media.

Control of matter-wave solitons using an accelerating atomic mirror

We present a possible way to control matter-wave solitons, which is through the collision between solitons and an accelerating atomic mirror. The acceleration of the mirror has a nontrivial effect on the dynamical characters of the reflected solitons. In the one-dimensional Bose–Einstein condensates, when the acceleration of the mirror has the identical direction with the initial soliton’s velocity, the soliton will diffuse after collision; in the contrasting case, the soliton will shrink and then diffuse. We quantitatively explain the above dynamical phenomena by analyzing the atoms’ movement in the soliton, and demonstrate that the method can generate a similar effect to the phase imprinting technology. Moreover, considering the dipolar effect between atoms, this approach can be used for the generation and control of breathing solitons.

Quantum reflection of dark solitons scattered by reflectionless potential barrier and position-dependent dispersion.

We investigate theoretically and numerically quantum reflection of dark solitons propagating through an external reflectionless potential barrier or in the presence of a position-dependent dispersion. We confirm that quantum reflection occurs in both cases with a sharp transition between complete reflection and complete transmission at a critical initial soliton speed. The critical speed is calculated numerically and analytically in terms of the soliton and potential parameters. Analytical expressions for the critical speed were derived using the exact trapped mode, time-independent, and time-dependent variational calculations. It is then shown that resonant scattering occurs at a critical speed, where the energy of the incoming soliton is resonant with that of a trapped mode. Reasonable agreement between analytical and numerical values for the critical speed is obtained as long as a periodic multisoliton ejection regime is avoided.

A visual approach to the ultradiscrete KdV equation with non-integer site values

Simple visual approaches are used to examine the ultradiscrete KdV equation with arbitrary real values for the dependent variable. The number and sizes of the solitons in the system can be obtained straightforwardly using these visualizations. By identifying the initial soliton sizes and positions, the initial profile can be decomposed into its constituent parts, which can then be evolved separately, and then the evolved profile can be reconstructed.

Dark topological valley Hall edge solitons

Abstract Topological edge solitons propagating along the edge of a photonic topological insulator are localized self-sustained hybrid states that are immune to defects/disorders due to the protection of the edge states stemming from the nontrivial topology of the system. Here, we predict that exceptionally robust dark valley Hall edge solitons may form at the domain walls between two honeycomb lattices with broken inversion symmetry. The underlying structure can be created with femtosecond laser inscription, it possesses a large bandgap where well-localized dark edge solitons form, and in contrast to systems with broken time-reversal symmetry, it does not require external magnetic fields or complex longitudinal waveguide modulations for the realization of the topological phase. We present the envelope equation allowing constructing dark valley Hall edge solitons analytically. Such solitons propagate without radiation into the bulk of the lattice and can circumvent sharp corners, which allows observing their persistent circulation along the closed triangular domain wall boundary. They survive over huge distances even in the presence of disorder in the underlying lattice. We also investigate interactions of closely located dark topological valley Hall edge solitons and show that they are repulsive and lead to the formation of two gray edge solitons, moving with different group velocities departing from group velocity of the linear edge state on which initial dark solitons were constructed. Our results illustrate that nonlinear valley Hall systems can support a rich variety of new self-sustained topological states and may inspire their investigation in other nonlinear systems, such as atomic vapors and polariton condensates.

Vector Soliton in Coupled Nonlinear Schrödinger Equation

Researchers are currently interested in studying the dynamics of the wave field in a nonlinear and dispersive medium. The Nonlinear Schrödinger Equation (NLSE), which is the fundamental equation that explains the phenomenon, has paved the way for research in a variety of fields, including soliton scattering. However, if the fields have a large number of components, the Coupled NLSE should be considered. We used orthogonally polarised and equal-amplitude vector solitons with two polarization directions to model the interactions. The effect of vector soliton scattering by external Delta potential in Coupled NLSE was studied in this paper. The scattering process is primarily determined by the initial velocity, amplitude of the soliton and potential strength. The variational approximation and direct numerical methods of Coupled NLSE were used to investigate the scattering process. The variational approximation (VA) method was used to analyse the dynamics of soliton’s width and center of mass position. The soliton may thus be reflected, transmitted or trapped within the potential. Uncoupled solitons may initially create a coupled state if their kinetic energy is less than the attractive interaction potential between solitons, but once their velocity surpasses the critical velocity, the soliton will easily pass through each other. To validate the approximation, a direct numerical simulation of CNLSE was performed. The results of the VA method and direct numerical simulation of Coupled NLSE are in good agreement when the parameters for both solutions are set to the same value. The initial velocity, potential strength and soliton amplitude play a role in the scattering of the vector soliton with Delta potential.

Single-mode spatiotemporal soliton attractor in multimode GRIN fibers

Experimental and numerical studies of spatiotemporal femtosecond soliton propagation over up to 1 km spans of parabolic graded-index fibers reveal that initial multimode soliton pulses naturally and irreversibly evolve into a single-mode soliton. This is carried by the fundamental mode of the fiber, which acts as a dynamical attractor of the multimode system for up to the record value (for multimode fibers) of 5600 chromatic dispersion distances. This experimental evidence invalidates the use of variational approaches, which intrinsically require that the initial multimode propagation of a self-imaging soliton is indefinitely maintained.

Stable Legendre–Lorentzian solitons in localized optical potentials

We derive a family of optical solitons from a non-linear Schrödinger’s equation with an external potential in a (1 + 1)D system. The soliton solutions can be expressed in a closed form by using a Lorentzian apodization and associated Legendre functions, in contrast to the more common solitons using a Gaussian apodization. Similarly, an analytical and bounded form for the external optical potential is also found, and furthermore, a general formula for the power is obtained. Remarkably, we found that these solitons are stable in their propagation for certain values of power and width of the solitons. Finally, we report several interesting propagation dynamics for the unstable scenario: from loss of the beam’s inner structure by splitting of the initial soliton profile to breathing decaying solitons.

Bogoliubov–de Gennes theory of the snake instability of gray solitons in higher dimensions

Gray solitons are a one-parameter family of solutions to the one-dimensional non-linear Schr\"odinger equation (NLSE) with positive cubic nonlinearity, as found in repulsively interacting dilute Bose-Einstein condensates or electromagnetic waves in the visible spectrum in waveguides described by Gross-Pitaevskii mean field theory. In two dimensions these solutions to the NLSE appear as a line or plane of depressed condensate density or light intensity, but numerical solutions show that this line is dynamically unstable to `snaking': the initially straight line of density or intensity minimum undulates with exponentially growing amplitude. To assist future studies of quantum mechanical instability beyond mean field theory, we here pursue an approximate analytical description of the snake instability within Bogoliubov-de Gennes perturbation theory. Within this linear approximation the two-dimensional result applies trivially to three dimensions as well, describing buckling modes of the low-density plane. We extend the analytical results of Kuznetsov and Turitsyn [Sov. Phys. JETP \textbf{67}, 1583 (1988)] to shorter wavelengths of the `snake' modulation and show to what extent the snake mode can be described accurately as a parametric instability, in which the position and grayness parameter of the initial soliton simply become dependent on the transverse dimension(s). We find that the parametric picture remains accurate up to second order in the snaking wave number, if the snaking soliton is also dressed by an outward-propagating sound wave, but that beyond second order in the snaking wave number the parametric description breaks down.

Influence of dispersion slope on soliton spectral tunneling in photonic crystal fiber

We report a numerical investigation of how the dispersion slope affects the soliton spectral tunneling (SST) in a photonic crystal fiber with three zero dispersion wavelengths. It is discovered that a larger dispersion slope makes group-velocity mismatch between the initial soliton and the transferred wave thereby suppressing the SST effect, while a proper decrease of the dispersion slope enhances the SST effect to widen a supercontinuum range. Besides, we find a soliton-like leaking dispersion wave, which can sustain information and energy for a short time within a particular spectral range.

Quenched solitons and shock waves in Bose-Einstein condensates

The Bose-Einstein condensate (BEC) formed by ultracold atomic gases provides an ideal platform for studying various quantum phenomena. In this platform, researchers have explored in depth the important equilibrium and steady phenomena including superfluidity, vortices, and solitons, and recently started to study nonequilibrium problems. In a classical system, nonequilibrium problems, such as explosion, usually occur together with shock waves, which is presented when the explosion speed is larger than the local sound speed. For BEC systems which possess quantum properties, how to produce and understand the shock waves becomes a hot research topic. In this work, we systematically discuss the possibility of quantum shock wave and its essential mechanism in a one-dimensional BEC initially containing dark solitons through quenching interactions. When the system is quenched to the limit of non-interaction, we analytically obtain the post-quench dynamics of initially immobile dark solitons, and find the existence of shock wave, which can be explained through the quantum interference effect. When the system is quenched to finite interaction, we find similar phenomena through numerically solving the Gross-Pitaevskii equation, and analyze different situations. When the system is quenched to a finite weaker interaction, the situation is similar to a non-interaction case; when the system is quenched to a stronger interaction, the shock wave is accompanied by the splitting of the initial soliton, and the two objects can synchronously change; specifically when the quenched ratio of strength is an integer squared, the shock wave disappears, and the soliton is split perfectly. We further explore the properties of the shock wave including its amplitude and speed, and obtain the full scenario as the quenched interaction varies. This work provides theoretical guidance for realizing and measuring shock wave in experiment.

Interaction of a Spatial Soliton on an Interface between Two Nonlinear Media

Spatial solitons are the solutions of nonlinear partial differential equations describing the propagation of optical beams in Kerr type nonlinear medium. This paper studies the scattering of a spatial solitons of the Cubic-Quintic Nonlinear Schrodinger Equation (C-Q NLSE) on an interface between two nonlinear media. The scattering process will be investigated by variational approximation method and by direct numerical solution of C-Q NLSE. This variational approximation method has been used to analyse the dynamic of the width and center of mass position of a soliton during the scattering process. Meanwhile, a direct numerical simulation of C-Q NLSE is run to check the accuracy of the approximation by using the same range of parameters and initial condition. The results for direct numerical simulation of CQNLSE for soliton parameters are quite similar with the variational equation. The studies showed that soliton can be reflected by or transmitted through the interface, also the nonlinear surface wave can be formed depending on the parameters of interface and initial soliton.

Emergence of Envelope Solitary Waves from Initial Localized Pulses within the Ostrovsky Equation

We study the emergence of steady wave packets in the form of envelope solitary waves (envelope solitons) which evolve from localized pulse-type initial conditions in the long-term evolution within the framework of the Ostrovsky equation. Ostrovsky equation derived in 1978 describes weakly nonlinear oceanic waves affected by the Earth’s rotation. Subsequently it became clear that this equation is universal, and is widely used for the description of waves in various settings. It is well-known that in applications to media with a negative small-scale dispersion this equation does not possess steady solitary wave solutions. In particular, the evolution of an initial Korteweg-de Vries (KdV) soliton results in the emergence of envelope solitons which can be described by the nonlinear Schr¨odinger equation (NLS) when the amplitude of the initial KdV soliton is sufficiently small. However, the derivation of the corresponding NLS still has not been sufficiently comprehensive, which resulted in contradiction with some numerical simulations. Here, we revisit this problem and suggest a new approach to the construction of the emerging envelope solitons.

Numerical solutions of the equal width equation by trigonometric cubic B‐spline collocation method based on Rubin–Graves type linearization

AbstractIn this article, the equal width (EW) equation is going to be solved numerically. In order to show the accuracy of the presented method, six test problems namely single solitary wave, interaction of two solitary waves, interaction of three solitary waves, Maxwellian initial condition, undular bore, and soliton collision are going to be solved. For the first test problem, since it has exact solution, the error norms L2 and L∞ are going to be calculated and compared with some of the earlier studies existing in the literature. Moreover, the three invariants I1, I2, and I3 of the given problems during the simulations are calculated and tabulated. Besides those comparisons, the relative changes of the invariants are given. Finally, a comparison of those error norms and invariants has clearly shown that the present approach obtained compatible and better results than most of the earlier works by using the same parameters.

Cascaded single-mode fibers for higher-order soliton compression at 2 µm.

We present the design of single-mode fibers for two-stage higher-order soliton compression at 2 µm wavelength and achieve high-degree pulse compression in cascaded single-mode fibers. The compression performance for the initial input pulse width from 1 to 50 ps is also investigated. For the initial third-order soliton of 10 ps, a compression factor of 75.7 has been achieved, and the pedestal energy is only 46.66%.

Stable multi-peak vector solitons in spin–orbit coupled spin-1 polar condensates

We demonstrate the formation of multi-peak three-component stationary stripe vector solitons in a quasi-one-dimensional spin-orbit-coupled hyper-fine spin $F = 1$ polar Bose-Einstein condensate. The present investigation is carried out through a numerical solution by imaginary-time propagation and an analytic variational approximation of the underlying mean-field Gross-Pitaevskii equation. Simple analytic results for energy and component densities were found to be in excellent agreement with the numerical results for solitons with more than 100 pronounced maxima and minima. The vector solitons are one of the two types: dark-bright-dark or bright-dark-bright. In the former a maximum density in component $ F_z = 0 $ at the center is accompanied by a zero in components $F_z = \pm 1$. The opposite happens in the latter case. The vector solitons are demonstrated to be mobile and dynamically stable. The collision between two such vector solitons is found to be quasi elastic at large velocities with the conservation of total density of each vector soliton. However, at very small velocity, the collision is inelastic with a destruction of the initial vector solitons. It is possible to observe and study the predicted SO-coupled vector solitons in a laboratory.

Temporal behaviour of bright solitons in photorefractive crystals having both the linear and quadratic electro-optic effect

Abstract We investigate the characteristics of formation or time evolution of optical spatial solitons in novel photorefractive media having both the linear and quadratic electro-optic effect. We have derived a one dimensional time dependant wave propagation equation in biased photorefractive media having both the linear and quadratic electro-optic effect. The variation of soliton width with time is investigated for different values of peak intensity ratio and it is shown that these bright solitons can exist both in the quasi steady state and steady state regimes for high saturation values. Two distinct regimes of intensity ratio are identified where the soliton formation characteristics are different. Only steady state solitons are formed if the intensity ratio is small. Soliton characteristics like the initial soliton width, steady state soliton width, and quasi steady state soliton width are investigated for both regimes and found to depend on as varying functions of the intensity ratio. We also find that the individual magnitudes of the linear and quadratic electro-optic coefficients affect the temporal characteristics of formation of the solitons. Appropriate example has been taken for an illustration of our results.