Multi-soliton Complexes Research Articles

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.

Ultrafast temporal-spectral analysis probes isomeric dynamics in a dissipative soliton resonator

Self-assembly of dissipative solitons arouses versatile configurations of molecular complexes, enriching intriguing dynamics in mode-locked lasers. The ongoing studies fuel the analogy between matter physics and optical solitons, and stimulate frontier developments of ultrafast optics. However, the behaviors of multiple constituents within soliton molecules still remain challenging to be precisely unveiled, regarding both the intramolecular and intermolecular motions. Here, we introduce the concept of “soliton isomer” to elucidate the molecular dynamics of multisoliton complexes. The time-lens and time-stretch techniques assisted temporal-spectral analysis reveals the diversity of assembly patterns, reminiscent of the “isomeric molecule”. Particularly, we study the fine energy exchange during the intramolecular motions, therefore gaining insights into the degrees of freedom of isomeric dynamics beyond temporal molecular patterns. All these findings further answer the question of how far the matter-soliton analogy reaches and pave an efficient route for assisting the artificial manipulation of multisoliton structures.

A special phenomenon of wave interactions: An application of nonlinear evolution equation in (3+1)-dimension

The Lie group transformation approach can be used in the reduction of the generalized (3+1)-dimensional nonlinear evolution equation to ordinary differential equations (ODEs). In this work, infinitesimal generators, commutator table of Lie algebra, adjoint representation of subalgebras, symmetry group, and similarity reduction for (3+1)-dimensional shallow-water-like (gSWL) equation are obtained. Interactions of soliton solutions of a gSWL equation are evaluated that depict how near-inertial waves affect the dynamics of barotropic and baroclinic balanced flows. It is shown that the temporo-spatial localization of multi-soliton complexes (MSCs), gradually escalating incoherent interactivity amidst multi parameter group of MSCs along with spatial inhomogeneity alter the shape of quasi-periodic MSCs. The interactions among bright and dark solitons are shown. The increased number of solitons and increased self-interactions among solitary waves contribute to an unstable wave field. Breathers have a well-separating and initial localized energy domain. A limiting case of breather solutions is the peregrine soliton. Thus, interactions among these soliton structures show that multiple breathers retain their wave characteristic.

Multisoliton complex systems with explicit superpotential interactions

We consider scalar field theories in dimensions with N fields which interact through a potential , which is defined in terms of an explicit superpotential W. We construct W for any N in terms of a known superpotential w for a single-scalar model, such as that for the sine-Gordon equation or the ϕ 4 model, leading to an expression for V which has multiple minima that supports solitons. Static solitons which minimize the total energy in each soliton sector appear as solutions of first-order Bogomolny equations, which have a gradient structure. These are identical in form to equations which arise in the context of synchronization phenomena in complex systems, with the space and time variables interchanged. The sine-Gordon superpotential, for example, leads to an explicit periodic superpotential W for N scalar fields, with associated Bogomolny equations that are equivalent to the well-known Kuramoto equations which describe the synchronization of identical phase oscillators on the unit circle. The known asymptotic properties of the Kuramoto system, for both positive and negative coupling constants, ensure that finite-energy solitons exist for any given set of intermediate values imposed at the origin. Besides the models derived from the sine-Gordon equation, we investigate ϕ 4 and ϕ 6 models with N scalar fields and show numerically that solitons again exist over a wide range of parameters. We also derive general properties of the elementary meson excitations of the system, in particular we show that meson-soliton bound states exist over a restricted range of mass parameters with respect to an exact solution of the ϕ 6 system for N = 3.

Saturable absorber based on tapered graded-index multimode fiber with ZnO coating for bound solitons generation

A passively mode-locked erbium-doped fiber laser with a saturable absorber (SA) based on a tapered graded-index multimode fiber with zinc oxide coating is proposed and demonstrated in this paper. The device is also a platform supporting various types of solitons outputs. The conventional solitons with center wavelength of 1558.86 nm, pulse width of 537 fs, and signal-to-noise ratio of 61.27 dB are generated at the repetition frequency of 18.37 MHz. By adjusting the polarization controller and increasing pump power, tightly and loosely bound solitons and two-fold solitons can be generated. The pulse trains of tightly bound solitons exhibit the separation of 1.91 ps, the pulse width of ∼707 fs, and centered at 1560 nm, whereas the pulse separation of loose bound solitons is 18.06 ps. Such a SA is easy in fabrication and highly stable in operation and can enrich the exploration of multi-soliton complexes.

Real-time observation of dissipative multi-soliton complexes in a net-normal dispersion fiber laser

Recent progress in real-time measurement technology makes it possible to spectrally characterize multi-soliton complexes. Here, we report on real-time observation to the multi-pulse structures in a net-normal dispersion mode-locked fiber laser. With careful adjustment of cavity parameters, several kinds of multi-soliton complexes, such as the coexistence of soliton singlet and soliton molecule, as well as the coexistence of different types of soliton molecules, are realized in the experiment. Real-time spectral information of each pulse under multi-soliton states can be well resolved with the dispersive Fourier transformation technique, which is markedly different from the results measured by conventional measurement instruments. Note that this is the first reported the simultaneous generation of different types of soliton molecules in a fiber laser to our best knowledge. The findings help to understand the complicated dynamics of the coexisting pattern between different kinds of dissipative solitons.

Impact of solitons on the progression of initial lesion in aortic dissection

Aortic dissection (AD) is the most common catastrophic disease reported at cardiovascular emergency in hospitals. Herein, a tear in the tunica intima results into separation of layers of aortic wall leading to rupture and torrential bleed. Hypoxia and oxidative stress are associated with AD. The release of hypoxia inducible factor (HIF)-[Formula: see text] from the initial flap lesion in the tunica intima is the basis for aneurysmal prone factors. We framed a boundary value problem (BVP) to evaluate homeostatic saturation for oxygen dynamics using steady-state analysis. We prove uniqueness and existence of the solution of the BVP for gas exchange at capillary–tissue interface as a normal physiological function. Failure of homeostatic mechanism establishes hypoxia, a new quasi-steady-state in AD. We model permeation of two-layer fluid comprised of blood and HIF-[Formula: see text] through tunica media as a generalized [Formula: see text]-dimensional nonlinear evolution equation and solve it using Lie group of transformations method. We note that the two-layer fluid permeates the tunica media as solitary wave including solitons such as bright soliton, dark soliton, peregrine soliton, topological soliton, kink soliton, breather soliton and multi-soliton complex. Also, we introduce the main result and discuss the implications of soliton solution, using graphic interpretation, to describe the early stage of progression of AD.

Experimental Observation of Ultrashort Hyperchaotic Dark Multisoliton Complexes in a Magnonic Active Ring Resonator.

We report on the self-generation of ultrashort hyperchaotic dark multisoliton sequences with two positive large Lyapunov exponents in an active ring resonator consisting of a multifunctional L-shaped magnonic waveguide and a saturable amplifier. The irregular magnonic waveguide supports the converting of backward volume magnetostatic spin waves with negative dispersion to magnetostatic surface spin waves with positive dispersion that is accompanied by a transition from four-wave to three-wave nonlinear spin-wave interactions. Each multisoliton complex consists of four dark parametric pulses containing the soliton trains of four dark incoherent spin-wave envelope solitons possessing a subnanosecond duration. Such patterns are formed due to the dispersion and nonlinearity management, the nonlinear transformation of the pulse signal in the saturable amplifier and the partial chaotic synchronization of both the ring eigenmodes and the spin wave automodulation frequencies. We also demonstrate a new intermittency type of "hyperchaotic multisoliton complexes-hyperchaotic multisoliton gas" with the increase of the signal power level.

Experimental observation of shaking soliton molecules in a dispersion-managed fiber laser.

Recent progress in real-time spectral interferometry enables access to the internal dynamics of optical multisoliton complexes. Here, we report on the first, to the best of our knowledge, experimental observation of shaking soliton molecules by means of the dispersive Fourier transform technique. Beyond the simplex vibrating soliton pairs, multiple oscillatory motions can jointly involve in the internal dynamics, reminiscent of the shaking soliton pairs. Both quasi-periodically and chaotically evolving phase oscillations are approached in the sense of different oscillatory frequencies. In addition, the shaking soliton pair combined with sliding phase dynamics is also observed, and is interpreted as the superposition of two different internal motions. All of these results shed new light on the internal dynamics of soliton molecules with higher degrees of freedom, as well as enrich the framework toward multisoliton complexes.

Generation of Dark Multisoliton Complexes in a Magnonic Ring Resonator with Dispersion Management and Competing Nonlinear Spin-Wave Interactions

Multisoliton complexes have been generated in an active ring resonator with an L-shaped magnonic microwaveguide and a saturated amplifier. It has been shown that an irregular microwaveguide simultaneously supports both the propagation of magnetostatic spin waves with different (normal and anomalous) dispersion and competition between three- and four-wave spin-wave interactions. Generated multisoliton complexes consist of quasiperiodic sequences of two dark parametric pulses containing “soliton trains” consisting of three dark ultrashort four-wave envelope solitons. It has been found that quasiperiodic sequences of multisoliton complexes generated by the active ring resonator with a regular magnonic microwaveguide consist of one dark parametric pulse, which also contains three dark four-wave solitons. However, in contrast to the preceding case, four-wave solitons, having a longer duration, do not form soliton trains inside a dark parametric pulse.

Breather Wave Molecules.

We present both a theoretical description and experimental observation of the nonlinear mutual interactions between a pair of copropagative breathers in the framework of the focusing one-dimensional nonlinear Schrödinger equation. As a general case, we show that the resulting bound state of breathers exhibits moleculelike behavior with quasiperiodic oscillatory dynamics (i.e., internal coherent interactions and pulsations), while for commensurate conditions the molecule oscillations become exactly periodic. Our theoretical model is confirmed by an experimental observation of shaped moleculelike breather light waves propagating in a nearly conservative optical fiber system. Our work sheds new light on the existence of localized wave structures and recurrence dynamics beyond the multisoliton complexes.

Self-generation of chaotic dissipative multisoliton complexes supported by competing nonlinear spin-wave interactions.

A self-generation of chaotic dissipative spin-wave multisoliton complexes has been observed experimentally. Localized in time, these patterns are formed in a passively Q-switched and mode-locked magnetic film feedback ring due to the competing three- and four-wave nonlinear spin-wave interactions. Such competition induces a modulation instability that leads to the formation of incoherent one-color four-wave bound solitons embedded in chaotic three-wave solitonlike pulses. The development of a symmetry-breaking instability causes a transition from incoherent one-color four-wave bound solitons to chaotic multicolor ones.

Soliton molecules in a fiber laser based on optic evanescent field interaction with WS2

Passively mode-locked fiber laser serves as an ideal playground for exploring the dynamics of dissipative solitons. Recently, two-dimensional materials have attracted growing interests for their excellent optical properties in the research field of ultrafast optics. Here, we report an observation of soliton molecules in a passively mode-locked fiber laser based on optic evanescent field interaction with tungsten disulfide (WS2), which contributes to the study of multi-soliton complexes. Particularly, the WS2 saturable absorber (SA) is fabricated by optically depositing the few-layer WS2 nanosheets onto microfiber. In the experiment, stable mode-locking operations are realized, which manifest the effective performance of the WS2 SA. Harmonic mode-locking phenomena are also observed, and these separate solitons facilitate the generation of multi-soliton complexes. Furthermore, due to soliton–soliton interaction, wandering particle-like solitons can be bound together to produce soliton molecules and bunch of soliton molecules. These results enrich both the exploration of multi-soliton complexes and potential industrial applications.

Group-velocity-locked vector soliton molecules in fiber lasers

Physics phenomena of multi-soliton complexes have enriched the life of dissipative solitons in fiber lasers. By developing a birefringence-enhanced fiber laser, we report the first experimental observation of group-velocity-locked vector soliton (GVLVS) molecules. The birefringence-enhanced fiber laser facilitates the generation of GVLVSs, where the two orthogonally polarized components are coupled together to form a multi-soliton complex. Moreover, the interaction of repulsive and attractive forces between multiple pulses binds the particle-like GVLVSs together in time domain to further form compound multi-soliton complexes, namely GVLVS molecules. By adopting the polarization-resolved measurement, we show that the two orthogonally polarized components of the GVLVS molecules are both soliton molecules supported by the strongly modulated spectral fringes and the double-humped intensity profiles. Additionally, GVLVS molecules with various soliton separations are also observed by adjusting the pump power and the polarization controller.

Double-Wall Carbon Nanotube Hybrid Mode-Locker in Tm-doped Fibre Laser: A Novel Mechanism for Robust Bound-State Solitons Generation

The complex nonlinear dynamics of mode-locked fibre lasers, including a broad variety of dissipative structures and self-organization effects, have drawn significant research interest. Around the 2 μm band, conventional saturable absorbers (SAs) possess small modulation depth and slow relaxation time and, therefore, are incapable of ensuring complex inter-pulse dynamics and bound-state soliton generation. We present observation of multi-soliton complex generation in mode-locked thulium (Tm)-doped fibre laser, using double-wall carbon nanotubes (DWNT-SA) and nonlinear polarisation evolution (NPE). The rigid structure of DWNTs ensures high modulation depth (64%), fast relaxation (1.25 ps) and high thermal damage threshold. This enables formation of 560-fs soliton pulses; two-soliton bound-state with 560 fs pulse duration and 1.37 ps separation; and singlet+doublet soliton structures with 1.8 ps duration and 6 ps separation. Numerical simulations based on the vectorial nonlinear Schr¨odinger equation demonstrate a transition from single-pulse to two-soliton bound-states generation. The results imply that DWNTs are an excellent SA for the formation of steady single- and multi-soliton structures around 2 μm region, which could not be supported by single-wall carbon nanotubes (SWNTs). The combination of the potential bandwidth resource around 2 μm with the soliton molecule concept for encoding two bits of data per clock period opens exciting opportunities for data-carrying capacity enhancement.

Analytical multisoliton solutions for multicomponent Bose–Einstein condensate systems with time- and space-controlled potentials

Abstract We construct explicit multisoliton complex solutions for multicomponent Bose–Einstein condensate systems with time- and spatial-coordinate-dependent atomic potentials and interactions. The exact solutions are used to analyze the important solitary matter wave properties such as the profiles of temporal and spatial multimode beams as well as focusing effects. Results demonstrate that soliton complexes can be controlled nonlinearly during the interaction by modulating the external potentials and nonlinearities.

Adiabatic approximation and aperiodic dynamics of an elliptically polarized light wave in an isotropic gyrotropic nonlinear medium

Adiabatic approximation is used to find an analytical solution to the nonintegrable propagation problem for a plane elliptically polarized light wave in an isotropic gyrotropic medium with local and nonlocal components of Kerr-type nonlinearity and second-order dispersion of group velocity. Aperiodic consistent dynamics of bound (attributable to the nonlinearity) states of two light field components with orthogonal circular polarizations is described. It is shown that this dynamics corresponds to generalization of consistent propagation of ‘soliton–multisoliton complex’ pairs to the cases with nonlinear interaction of periodic solutions—cnoidal waves.

Approximate solutions to a nonintegrable problem of propagation of elliptically polarised waves in an isotropic gyrotropic nonlinear medium, and periodic analogues of multisoliton complexes

Using the linearization method, we obtain approximate solutions to a one-dimensional nonintegrable problem of propagation of elliptically polarised light waves in an isotropic gyrotropic medium with local and nonlocal components of the Kerr nonlinearity and group-velocity dispersion. The consistent evolution of two orthogonal circularly polarised components of the field is described analytically in the case when their phases vary linearly during propagation. The conditions are determined for the excitation of waves with a regular and ‘chaotic’ change in the polarisation state. The character of the corresponding nonlinear solutions, i.e., periodic analogues of multisoliton complexes, is analysed.

Multisoliton complexes of Bose–Einstein condensates in nonlinear optical lattices

We theoretically study the role of the mode structure of a multicomponent Bose–Einstein condensate (BEC) in the potential created by a nonlinear optical lattice. We describe a multisoliton complex (MSC) as a superposition of different fundamental soliton modes in the matter-wave system. Using a similarity transformation, we solve the nonlinear evolution equation of the multimode coupled matter-wave field and construct a set of analytical bright soliton solutions. A perturbation method is used to examine the linear stability of the constructed solitons. Based on these particular solutions, we numerically analyze the mode structure of a MSC. The results show that the periodicity causes a Bloch modulation in the envelopes of the density distribution. When different fundamental modes collide with each other in the nonlinear lattice, the collision-induced shifts, and the space-dependent modulation of external potentials change the density profile of the multimode soliton complex. Therefore, the mode structure, which is absent in a one-mode BEC, provides the possible multiscale modeling of the matter-wave field with extra degrees of freedom.

Travelling solitons in the externally driven nonlinear Schrödinger equation

We consider the undamped nonlinear Schrödinger equation driven by a periodic external force. In the absence of damping, solitons do not have undulations on their tails; yet, we show that they can bind into stationary multisoliton complexes. Using two previously known stationary solitons and two newly found stationary complexes as starting points, we obtain classes of localized travelling waves by the numerical continuation in the parameter space. Two families of stable solitons are identified: one family is stable for sufficiently low velocities while solitons from the second family stabilize when travelling faster than a certain critical speed. The stable solitons of the former family can also form stably travelling bound states.