Number Of Solitons Research Articles

Temporal and spatiotemporal soliton molecules in ultrafast fibre lasers

Abstract Ultrafast fibre lasers, characterized by ultrashort pulse duration and broad spectral bandwidth, have drawn significant attention due to their vast potential across a wide range of applications, from fundamental scientific to industrial processing and beyond. As dissipative nonlinear systems, ultrafast fibre lasers not only generate single solitons, but also exhibit various forms of spatiotemporal soliton bunching. Analogous to molecules composed of multiple atoms in chemistry, soliton molecules (SMs) – alias bound states – in ultrafast fibre lasers are a key concept for gaining a deeper understanding of nonlinear interaction and hold a promise for advancing high-capacity fibre-optic communications. SMs are particularly notable for their high degree of controllability, including their internal temporal separation, and relative phase differences, thereby suggesting new possibilities for manipulating multi-pulse systems. In this review, we provide a comprehensive overview of recent advancements in the studies of SMs with the multidimensional parameter space in ultrafast fibre lasers. Owing to the flexibility afforded by mode-locking techniques and dispersion management, various types of SMs – with diverse values of the soliton number, relative phase, pulse separation, carrier frequencies, and even modal dispersion – have been experimentally demonstrated. We also discuss other basic nonlinear optical phenomena observed in fibre lasers, including the formation, spatiotemporal pulsations, and interaction dynamics of SMs. Furthermore, we explore the multidimensional control of SMs through approaches such as gain modulation, polarization control, dispersion management, and photomechanical effects, along with their applications to optical data encoding. Finally, we discuss challenges and future development of multidimensional technologies for the manipulation of SMs.

All-optical azimuthal trapping of dissipative Kerr multi-solitons for relative noise suppression

Temporal cavity solitons, or dissipative Kerr solitons (DKSs) in integrated microresonators, are essential for deployable metrology technologies. Such applications favor the lowest noise state, typically the single-DKS state where one soliton is in the resonator. Other multi-DKS states can also be reached, offering better conversion efficiency and thermal stability, potentially simplifying DKS-based technologies. Yet they exhibit more noise due to relative soliton jitter and are usually not compatible with targeted applications. We demonstrate that Kerr-induced synchronization, an all-optical trapping technique, can azimuthally pin the multi-DKS state to a common reference field. This method ensures repetition rate noise is independent of the number of solitons, making a multi-DKS state indistinguishable from a single-DKS state in that regard, akin to trapped-soliton molecule behavior. Supported by theoretical analysis and experimental demonstration in an integrated microresonator, this approach provides metrological capacity regardless of the number of cavity solitons, benefiting numerous DKS-based metrology applications.

Thermal tuning behavior and robust control of a single-soliton microcomb with a modulated sideband.

A chip-scaled single-soliton microcomb source promises wide applications in various fields. We demonstrate the deterministic single-soliton generation from both pump forward and backward tunings via sideband thermal compensation. The total soliton existing range (SER) is effectively expanded due to the thermal-lock effect and remains nearly the same regardless of the soliton states. Besides, we prove the degeneration of the forward-tuning SER, accompanied by the decrease in the soliton number. This scheme remains robust against a significant pump wavelength chirp, sustaining a free-running single soliton for over 9 h with line power fluctuations below 0.6 dB.

Diverse and controllable soliton molecules in a fiber laser based on PbBi4Te7 saturable absorber

With the advancement of optical communication, optical encoding, and optical storage technologies, the demand for tunable mode-locked fiber lasers (MLFL) with soliton molecules is continuously increasing. The novel ternary two-dimensional topological insulator PbBi4Te7 provides an effective pathway for realizing tunable soliton molecules in MLFL. This study employs a combined liquid phase exfoliation (LPE) and optical deposition methods (ODMs) to prepare PbBi4Te7 saturable absorbers (SAs). In the same MLFL, conventional soliton (CS), controllable soliton molecules with adjustable soliton numbers and intervals, as well as multi-soliton complexes are obtained. Among them, a fundamental frequency CS pulse at 22.778 MHz with a signal-to-noise ratio (SNR) of up to 71 dB is achieved. By adjusting the pump power, 2nd-, 3rd-, 4th-, 9th-, and 11th-order soliton molecules were obtained. By tuning the polarization controller (PC), switching of the soliton interval for 2nd-, 3rd-, and 4th-order soliton molecules was achieved. Additionally, we obtained three unequally spaced multi-soliton complexes: 2 + 1 soliton molecule, 2 + 2 soliton molecule complex and 2 + 2 + 1 soliton molecule complex. The results indicate that PbBi4Te7-SA exhibits outstanding nonlinear optical performance.

Observation of Two-Dimensional Dam Break Flow and a Gaseous Phase of Solitons in a Photon Fluid.

We report the observation of a two-dimensional (2D) dam break flow of a photon fluid in a nonlinear optical crystal. By precisely shaping the amplitude and phase of the input wave, we investigate the transition from one-dimensional (1D) to 2D nonlinear dynamics. We observe wave breaking in both transverse spatial dimensions with characteristic timescales determined by the aspect ratio of the input box-shaped field. The interaction of dispersive shock waves propagating in orthogonal directions gives rise to a 2D ensemble of solitons. Depending on the box size, we report the evidence of a dynamic phase characterized by a constant number of solitons, resembling a 1D soliton gas in integrable systems. We measure the statistical features of this gaslike phase. Our findings pave the way to the investigation of collective solitonic phenomena in two dimensions, demonstrating that the loss of integrability does not disrupt the dominant phenomenology.

Abundant vortex dynamics in spin-1 Bose–Einstein condensates induced by Rashba spin–orbit coupling

The paper studies the dynamics of vortices evolved from the ring dark solitons (RDSs) in spin-1 Bose–Einstein condensates with Rashba spin–orbit coupling (SOC). We find that the SOC induces abundant vortex dynamic phenomena, and the numbers of vortex dipoles and lump-like solitons are all determined by the initial depths’ weighted average of the three components of RDSs. For a shallow RDS, the ring first expands and then contracts into a mini RDS, eventually evolving into late vortex dipoles. For the RDS with a moderate initial depth, during the contraction of the ring, it evolves into early vortex dipoles or lump-like solitons, and they evolve back into the mini RDS, which then evolves into late vortex dipoles during the expansion process. For the deep and black RDSs, the early vortex dipoles continue to transform between vortex dipoles and lump-like solitons without forming mini RDS. The motions of vortices mentioned above have a higher disorder degree than that in the system without SOC, because the SOC breaks the mirror symmetry of the condensate. Finally, we reveal that the SOC strength only influences the number of late vortex dipoles, while it does not affect the number of early vortex dipoles and lump-like solitons. Stronger SOC intensity results in a shallower critical depth, and vortices form only at or beyond this depth.

Dynamics of circular Airy beams with spatial and frequency modulations in a cubic-quintic nonlinear fractional Schrödinger equation: from linear to soliton control.

The transmission dynamics of a circular Airy beam (CAB) with quadratic phase modulation (QPM) and cross-phase modulation (XPM) in the cubic-quintic nonlinear fractional Schrödinger equation (FSE) optical system is investigated. In the linear case, the energy distribution of the beam is affected by XPM and the focusing position of the beam is influenced by QPM. CAB undergoes splitting and its intensity is shifted as the absolute value of the XPM coefficient (|c|) increases. When XPM coefficients are opposite to each other, CABs are transmitted in opposite states in space. The degree of interference between beams gradually enhances with the increase of the XPM coefficient, leading to the formation of interference resembling water ripples. In the nonlinear regime, different results (evolving into solitons or undergoing diffraction transmission) are observed in CABs based on cubic-quintic nonlinear combination modes. Furthermore, nonlinear combination modes that can generate solitons and changes in solitons under actions of XPM and QPM are studied in detail. The distribution of solitons can be altered by positive or negative XPM, and solitons exist when QPM coefficients are within a certain range. The spacing and number of solitons can be modified by adjusting the magnitude of the QPM coefficient. The research shows that the control for solitons (number, distribution, and propagation) can be achieved through flexible selection of cubic-quintic nonlinear combination modes and parameter optimization (XPM coefficient, QPM coefficient, Lévy index).

On the Flux Density Spectral Property of High Linearly Polarized Signal from Pulsar J0332+5434

The polarization position angles (PPA) of time samples with high linear polarization often show two parallel tracks across the pulsar profile that follow the rotating vector model (RVM). This feature supports coherent curvature radiation (CCR) as the underlying mechanism of radio emission from pulsars, where the parallel tracks of the PPA represent the orthogonal extraordinary (X) and ordinary (O) eigenmodes of strongly magnetized pair plasma. However, the frequency evolution of these high linearly polarized signals remains unexplored. In this work, we explore the flux density spectral nature of high linearly polarized signals by studying the emission from PSR J0332+5434 over a frequency range between 300 and 750 MHz, using the Giant Metrewave Radio Telescope. The pulsar average profile comprises a central core and a pair of conal components. We find the high linearly polarized time samples to be broadband in nature, and in many cases, they resemble a narrow spiky feature in the conal regions. These spiky features are localized within a narrow pulse longitude over the entire frequency range, and their spectral shapes sometimes resemble an inverted parabolic shape. In all such cases, the PPA is exclusively along one of the orthogonal RVM tracks, likely corresponding to the X-mode. The inverted spectral shape can, in principle, be explained if the high linearly polarized emission in these time samples is formed due to the incoherent addition of CCR from a large number of charged solitons (charge bunches) exciting the X-mode.

The control for multiple kinds of solitons generated in the nonlinear fractional Schrödinger optical system based on Hermite-Gaussian beams

In this paper, we investigate the control for Hermite-Gaussian (HG) solitons in the nonlinear fractional Schrödinger equation (FSE) by sequentially applying power function modulations, cosine modulations, parabolic potentials, and quadratic phase modulations (QPM). In the photorefractive media, the HG beam forms scattered breathing solitons when the fractional diffraction effect equilibrates with nonlinear effect. Under the power function modulation, the soliton maintains an equidistant linear transmission along the z-axis, and the number of solitons is equal to the mode. In the cosine modulation, the soliton distorts and its energy rapidly decreases after a certain distance of transmission. The time of the distortion varies with the Lévy index, photorefractive coefficient, modulation frequency and order. The freak spots exhibit a “flower” shape pattern. If a parabolic potential is introduced, the HG beam forms crawling soliton pairs or merges into a single bounded breathing soliton by adjusting the correlation among the Lévy index, nonlinear and parabolic coefficients. By increasing the nonlinear coefficient in the negative QPM regime, the defocusing HG beam emits several “filiform” breathing solitons during its propagation, which move in a parallel straight line to each other. The HG beam is transformed into a single fine breathing soliton after being focused under a positive QPM. The time of the formation and breathing rate varies with the Lévy index, QPM and nonlinear coefficients. Moreover, the number of solitons changes irregularly with modes. These results are significant for applications in optical communication, optical device design, and optical signal processing.

Hybrid soliton, breather waves and solution molecules of the (2+1)-dimensional generalized fifth-order KdV equation

In this paper, we take the (2+1)-dimensional generalized fifth-order KdV (fKdV) equation as an example. We obtain many wave solutions, including hybrid soliton, breather waves and solution molecules. In order to solve hybrid soliton, we give two transformation forms. For Transform one, real and complex parameters lead to different wave solutions. Soliton solutions are obtained when the parameter values are real, but some breather waves appear when the parameter values are complex. When the parameter value changes, the number of solitons and the position of the waves will also change. For Transform two, we assign the parameter values to real values and obtain the corresponding two-, three- and four-soliton solutions. The most important thing is that, on the basis of Transform one, we add different constraint conditions according to different coordinate systems, and finally get a very interesting phenomenon: soliton molecules. In addition, we give not only the three-dimensional plots but also the density plots. In general, these results will enrich the existing literature on the (2+1)-dimensional generalized fKdV equation and help to understand this kind of KdV equation.

Frequency comb enhancement via the self-crystallization of vectorial cavity solitons.

Long-range interactions between dark vectorial temporal cavity solitons are induced by the formation of patterns via spontaneous symmetry breaking of orthogonally polarized fields in ring resonators. Turing patterns of alternating polarizations form between adjacent solitons, pushing them apart so that a random distribution of solitons along the cavity length spontaneously reaches equal equilibrium distances, the soliton crystal, without any mode crossing or external modulation. Enhancement of the frequency comb is achieved through the spontaneous formation of regularly spaced soliton crystals, 'self-crystallization', with greater power and spacing of the spectral lines for increasing soliton numbers. Partial self-crystallization is also achievable in long cavities, allowing one to build crystal sections with controllable numbers of cavity solitons separated by intervals of pattern solutions of, again, controllable length.

Two-dimensional vortex dipole, tripole, and quadrupole solitons in nonlocal nonlinearity with Gaussian potential well and barrier.

In this work, we investigate the dynamics and stability of two-dimensional (2D) vortex dipole, tripole, and quadrupole solitons with fundamental topological charge (m = 1) and higher topological charge (m > 1) in nonlocal nonlinearity with Gaussian potential well and barrier. Both analytical and numerical methods are applied to explore these vortex solitons. The analytical expressions are derived by utilizing the variational approach. The numerical simulations show that nonlocality cannot stabilize the vortex dipole, tripole, and quadrupole beams with topological charge m = 1. Interestingly, it is found that these vortex solitons remain stable during propagation only when the topological charge is m = 2 and when the propagation constants are below specific thresholds, where the vortex beams can maintain their profile no matter whether the nonlocality is weak, intermediate, or strong or how the Gaussian potential barrier height (well depth) increases. Furthermore, for the solitons with higher topological charge (m = 4), another consistent pattern emerges, that is, vortex dipole, tripole, and quadrupole solitons split into stable petal solitons and fundamental solitons with the number of petal solitons corresponding to the number of vortex solitons present. The analytical results are verified by numerical simulations.

Generation and modulation of soliton molecules in an all-polarization-maintaining fiber laser mode-locked by NPE

In this paper, we report the experimental observation of multiple soliton molecule forms and modulation results in an all-polarization-maintaining (PM) linear fiber laser mode-locked by nonlinear polarization evolution (NPE). The relationship between the waveplate angle and the reflectivity profile of the structure is analyzed numerically. Experimentally, it was found that the soliton number inside the bound states decreases from 4 solitons to 1 soliton by reducing the pump power. The soliton separation and the spectral shape could be evolved by slightly rotating the angle of the half-wave plate or λ/8-wave plate respectively. In addition to the above soliton molecules with regular triangular autocorrelation envelopes, (2+1)-type and (1+2+1)-type compound bound states are also obtained, and the fine temporal structure inside these soliton complexes are obtained by numerical simulation. The investigated results enrich the nonlinear dynamics of femtosecond mode-locked fiber lasers and lay the foundation for the application of all-PM fiber lasers mode-locked by NPE.

Discovery of dissipative microwave photonic soliton molecules in dual-bandpass optoelectronic oscillator

Optoelectronic oscillators (OEOs), which have attracted extensive studies in the past decades, are high quality-factor optoelectronic feedback loops for generating various ultra-pure microwave signals. In essence, OEOs are also dissipative nonlinear systems with multiple timescale characteristics and abundant nonlinearities, which open the possibilities for exploring localized dissipative solitary waves. In this paper, we demonstrate a new-class temporal dissipative soliton, i.e., dissipative microwave photonic soliton molecule (DMPSM), in a dual-bandpass OEO. Both the numerical simulation and experiment are conducted to reveal the physical mechanism of DMPSM generation and to evaluate the characteristics of the generated DMPSM sequences. Unlike optical soliton molecules in mode-locked lasers, the formation of DMPSMs arises from the combined action of multiple timescale coupling, nonlinear bistability, and time-delayed feedback in the OEO cavity, where the soliton interval and number in a DMPSM can be well-controlled through varying the multiple timescale variables in the OEO cavity, and the repetition frequency of the DMPSMs can be tuned through changing that of the initially injected perturbation signal. Meanwhile, the generated DMPSM sequence performs with high stability and excellent coherence, which shows enormous application potentials in pulse radar detection, dense microwave comb generation, and neuromorphology.

Controllable multi-soliton spectral tunneling by airy pulses in photonic crystal fiber

A straightforward method to induce tunable multi-soliton spectral tunneling (SST) using a finite energy Airy pulse in a photonic crystal fiber (PCF) featuring three zero-dispersion wavelengths is proposed numerically in the paper. The study reveals that the distinctive multi-envelope structure of the Airy pulse enables the control of tunneling distances, properties, and soliton numbers through modulation of its attenuation factor and peak intensity. Notably, the SST process involving the Airy pulse generates multiple blue-shifted and red-shifted dispersion waves, along with associated trapping waves, contributing to the formation of an ultra-flat and ultra-wide supercontinuum. Furthermore, the velocity and efficiency of multi-SST can be manipulated to enhance the overall tunability of the phenomenon. The findings underscore the potential application of Airy pulse-driven SST in secure multi-frequency signal transmission and controlled supercontinuum generation, with the tunability of these processes being governed by the specific attenuation factor and peak power settings of the input Airy pulse. This work advances the understanding of multi-SST dynamics in PCFs and provides a theoretical framework for exploiting these dynamics in practical information security and optical spectrum management contexts.

Resonant water-waves in ducts with different geometries: Forced KdV solutions

In a remarkable paper, Cox and Mortell (1986) (A.A. Cox, M.P. Mortell 1986. J. Fluid Mech. 162, pp. 99-116) showed that for an oscillating water tank, the evolution of small-amplitude, long-wavelength, resonantly forced waves follow a forced Korteweg–de Vries (fKdV) equation. The solutions of this model agree well with experimental results by Chester and Bones (1968) (W. Chester and J.A. Bones 1968. Proc. Roy. Soc. A, 306, 23 (Part II)). We compare the fKdV solutions with a number of channel flows with different geometry that have been studied experimentally and numerically. When sweeping the selected wide parameter range, extreme cases of the fKdV equation are covered: single soliton solutions as well as multiple solitons with a rather short wavelength challenging the long-wave fKdV assumption. The transition of solutions with a different number of solitons is rather abrupt and we show that the parameter values for transitions from single soliton towards multi-soliton solutions can be predicted and follow a simple exponential relation. In particular, we compare the fKdV model with solutions from a fully nonlinear Navier–Stokes model. We further consider a case for which the 2D assumption of the fKdV equation is strictly speaking violated.

Optical Darboux Transformer.

The optical Darboux transformer for solitons is introduced as a photonic device that performs the Darboux transformation directly in the optical domain. This enables two major advances for optical signal processing based on the nonlinear Fourier transform: (i)the multiplexing of solitonic waveforms corresponding to different discrete eigenvalues of the Zakharov-Shabat system, and (ii)the selective filtering of an arbitrary number of individual solitons too. The optical Darboux transformer can be built using existing commercially available photonic technology components and constitutes a universal tool for signal processing, optical communications, optical rogue waves generation, and waveform shaping and control in the nonlinear Fourier domain.

Internal solitary waves observation and feature extraction based on wavelet transform by Sentinel-1A in Lombok Strait, Indonesia

Lombok Strait is an essential pathway in trans-oceanic water mass transport. Due to this water mass transport flowing over Indonesian waters, the current is known as Indonesian Throughflow (ITF), which plays a role in the transfer of warm water masses from the Pacific to the Indian Ocean annually. Lombok Strait has intensive characteristics of the internal waves (IWs) generation because of the strong current that passes through the complex bathymetry along the strait area. IWs with large amplitude and nonlinear properties are known as Internal Solitary Waves (ISW) that can be detected by Synthetic Aperture Radar (SAR) images. A Wavelet Transform method for ISW feature extraction was applied to SAR images by Sentinel-1A (C-band). ISW packet characteristics can be distinguished from other phenomena based on their geometrical structure and shape. ISW packet pattern consists of light and dark lines which decrease intensity from front to rear. SAR observation detected 5 parameters (phase speed, soliton numbers, wavelength, first crest length, and propagation direction). The arc-like ISW in the Lombok Strait propagated to the North of the sill with average phase speeds of about 2.13 m/s and was frequently detected during the northwest monsoon (NWM). The detected soliton number is less than 6 solitons per packet with a wavelength of about 1 – 4 km, and the first crest length varies from about 12.16 km to more than 100 km. ISW detected in Sentinel-1A images were located at the bathymetry about 800 meters around the Lombok Strait area.

Scaling limit of soliton lengths in a multicolor box-ball system

Abstract The box-ball systems are integrable cellular automata whose long-time behavior is characterized by soliton solutions, with rich connections to other integrable systems such as the Korteweg-de Vries equation. In this paper, we consider a multicolor box-ball system with two types of random initial configurations and obtain sharp scaling limits of the soliton lengths as the system size tends to infinity. We obtain a sharp scaling limit of soliton lengths that turns out to be more delicate than that in the single color case established in [LLP20]. A large part of our analysis is devoted to studying the associated carrier process, which is a multidimensional Markov chain on the orthant, whose excursions and running maxima are closely related to soliton lengths. We establish the sharp scaling of its ruin probabilities, Skorokhod decomposition, strong law of large numbers and weak diffusive scaling limit to a semimartingale reflecting Brownian motion with explicit parameters. We also establish and utilize complementary descriptions of the soliton lengths and numbers in terms of modified Greene-Kleitman invariants for the box-ball systems and associated circular exclusion processes.

Multisoliton interactions approximating the dynamics of breather solutions

AbstractNowadays, breather solutions are generally accepted models of rogue waves. However, breathers exist on a finite background and therefore are not localized, while wavefields in nature can generally be considered as localized due to the limited sizes of physical domain. Hence, the theory of rogue waves needs to be supplemented with localized solutions, which evolve locally as breathers. In this paper, we present a universal method for constructing such solutions from exact multisoliton solutions, which consists in replacing the plane wave in the dressing construction of the breathers with a specific exact N‐soliton solution converging asymptotically to the plane wave at large number of solitons N. On the example of the Peregrine, Akhmediev, Kuznetsov–Ma, and Tajiri–Watanabe breathers, we show that constructed with our method multisoliton solutions, being localized in space with characteristic width proportional to N, are practically indistinguishable from the breathers in a wide region of space and time at large N. Our method makes it possible to build solitonic models with the same dynamical properties for the higher order rational and super‐regular breathers, and can be applied to general multibreather solutions, breathers on a nontrivial background (e.g., cnoidal waves), and other integrable systems. The constructed multisoliton solutions can also be generalized to capture the spontaneous emergence of rogue waves through the spontaneous synchronization of soliton norming constants, though finding these synchronization conditions represents a challenging problem for future studies.