Breather Soliton Research Articles

Intermediate state between steady and breathing solitons in fiber lasers

Ultrafast fiber laser, a vital tool in both science and industry, exhibits two distinct pulse states: the steady soliton (SS) and the breathing soliton (BS). While these states have been extensively studied individually, understanding the complex transition between them is crucial for controlling lasing states effectively. Herein, our experimental observations reveal an intermediate state that toggles between SS and BS, enabled by the dispersive Fourier transform technique. We find that energy hop and decaying breathing processes, driven respectively by the energy quantization effect and Q-switched modulation, govern this transition. Additionally, we observe that the transition between different BS states primarily involves a pure decaying breathing process. Numerical simulations are used to generate similar transition dynamics in a model that combines equations describing the population inversion in a mode-locked laser. This study sheds light on the transition dynamics in non-equilibrium systems, offering insights for intelligently manipulating lasing states.

Spatio-temporal breather dynamics in microcomb soliton crystals

Solitons, the distinct balance between nonlinearity and dispersion, provide a route toward ultrafast electromagnetic pulse shaping, high-harmonic generation, real-time image processing, and RF photonic communications. Here we uniquely explore and observe the spatio-temporal breather dynamics of optical soliton crystals in frequency microcombs, examining spatial breathers, chaos transitions, and dynamical deterministic switching – in nonlinear measurements and theory. To understand the breather solitons, we describe their dynamical routes and two example transitional maps of the ensemble spatial breathers, with and without chaos initiation. We elucidate the physical mechanisms of the breather dynamics in the soliton crystal microcombs, in the interaction plane limit cycles and in the domain-wall understanding with parity symmetry breaking from third-order dispersion. We present maps of the accessible nonlinear regions, the breather frequency dependences on third-order dispersion and avoided-mode crossing strengths, and the transition between the collective breather spatio-temporal states. Our range of measurements matches well with our first-principles theory and nonlinear modeling. To image these soliton ensembles and their breathers, we further constructed panoramic temporal imaging for simultaneous fast- and slow-axis two-dimensional mapping of the breathers. In the phase-differential sampling, we present two-dimensional evolution maps of soliton crystal breathers, including with defects, in both stable breathers and breathers with drift. Our fundamental studies contribute to the understanding of nonlinear dynamics in soliton crystal complexes, their spatio-temporal dependences, and their stability-existence zones.

M-shaped rational, homoclinic breather, kink-cross rational, multi-wave and interactional soliton solutions to the fifth-order Sawada-Kotera equation

In this work, we develop multi-wave, homoclinic breathers, M-shaped rational, 1-kink interactions with M-shaped, periodic-cross rational and kink-cross rational solutions for the fifth-order Sawada-Kotera equation, which represents the motion of long waves under gravity in shallow water, using several ansatz transformations. A three-wave approach is used to identify multi-wave solitons. Additionally, novel forms of exact solutions are constructed using the homoclinic breathers technique. The kink-cross rational and periodic-cross rational solitons are investigated using appropriate transformations. We develop M-shaped solitons and demonstrate their behavior by selecting suitable parameter values. Furthermore, the interactions between kink waves and M-shaped solitons are also studied. The obtained solutions are determined in 3D, 2D, and contour profiles, where the free parameters involved in the solutions are assigned specific values. Their physical relevance is explored to highlight the inner context of tangible incidents in the natural domain. Since these recently discovered solutions contain a few arbitrary constants, they can be used to explain the variation in qualitative characteristics of wave phenomena.

Study of Multi-solitons, Breather Soliton Structures with (r, q) Distributed Ions and Electrons

Study of Multi-solitons, Breather Soliton Structures with (r, q) Distributed Ions and Electrons

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.

Nuclear acoustic envelope soliton in a relativistically degenerate magneto-rotating stellar plasma

The study explores a model called Relativistic Degenerate Magneto-Rotating Quantum Plasma (RDMRQP), which comprises a static heavy nucleus, an inertial non-degenerate light nucleus, and warm non-relativistic or ultra-relativistic electrons. The focus is on observing the emergence of Nucleus-Acoustic Envelope Solitons (NAESs). Using the reductive perturbation method, a Nonlinear Schrödinger equation (NLSE) is derived to characterize the properties of NAESs. We have analysed the Peregrine breather soliton solution. The investigation reveals that the temperature of warm degenerate species, plasma system’s rotational speed, and the presence of heavy nucleus species can alter the fundamental features (height and width) of NAESs in the WDMRQP system. The study emphasizes the existence of only positive NA wave potential. Additionally, a phase plane analysis is conducted to gain a deeper understanding of the parametric dependencies. Through detailed mathematical and numerical analysis, the study avoids overloading with complex mathematics while demonstrating parametric dependence via phase portrait analysis. The research augments the envelop soliton model with breather mode solutions and discusses modulation instability using the Benjamin-Feir Index, highlighting the significance of solitons in star formation. Envelop solitons, stable waves within stars and proto-stars, influence energy transport and stellar evolution, playing a crucial role in the accretion process and formation of stable structures. Key findings include the effects of various parameters on NAESs’ generation and propagation in which non-relativistic and ultra-relativistic electrons support NAESs, with amplitude and width influenced by temperature, rotational frequency, inclination angle, and the presence of a static heavy nucleus. The research is applicable to hot white dwarfs and neutron stars, suggesting further exploration of quantum effects and non-planar or arbitrary amplitude NAESs.

Pattern transformation and control of generalized multi-peak breathing solitons induced by transverse cross modulation

Based on the nonlocal nonlinear Schrödinger equation, the pattern transformation and control of transverse cross-modulated sine-Gaussian (TCMSG) breathing solitons during transmission are studied. Several expressions have been derived, including the transmission, soliton width, phase wavefront curvature, and so on. The study demonstrates that the coefficient of transverse cross modulation term controls the pattern transformation of the TCMSG breathing solitons. TCMSG breathing solitons can form generalized spatial solitons and breathers during transmission. The variation of the soliton width extrema and their change rates with the transverse cross modulation term coefficient is investigated. The influence of the initial incident power and the transverse cross modulation term coefficient on the soliton width change rate and phase wavefront curvature extrema is studied.

Variable dust charge generates multi solitons, breather soliton structures in Saturn’s ring

In this study, we have investigated nonlinear wave structures that are localized along with the exact stationary wave solutions with oscillating, i.e. the breather structures in an unmagnetized dusty plasma with variable dust charge concentration with two temperatures of ions. We have derived GE from the normalized governing equations employing the reductive perturbative technique (RPT). The GE is the extended Korteweg–de Vries (KdV) equation that includes the united effect of quadratic and cubic nonlinearities. Employing the Hirota bilinear method (HBM), multi-soliton solutions and the breather soliton solutions have been obtained. Breathers (oscillating wave packets) play a significant role in hydrodynamics and optics, and their interaction can change the dynamics of the wave fields. It is also important to propagate finite-amplitude waves in the astrophysical atmosphere, plasma dynamics, ocean, optic fibers, signal processing, etc. In the circumstances of Saturn’s ring, it is noticed that the breather soliton structures are modified significantly with variations in dust charge concentration, ion temperature ratio, and other physical parameters.

Study of multi solitons, breather soliton structures in the earth's magnetotail region

The investigation of wave packets with varying magnitudes, which predominantly carry energy across different physical mediums, proves to be highly intriguing and unveils numerous nonlinear characteristics within Earth's magnetotail region and magnetosphere, as confirmed by extensive space observations. Nonlinear features in wave dynamics are elucidated through diverse nonlinear structures, among which breather structures hold prominence and manifest across various wave fields. We have considered an unmagnetized collisionless homogeneous hydrodynamical governing equations set in an electron-ion plasma comprising hot and cold ions. In this study, we have explored 1-soliton, 2-soliton, and breather structures in the framework of the Gardner equation (GE). We have derived the GE from the normalized set of governing equations employing the reductive perturbation technique (RPT). The GE, an extension of the Korteweg-de Vries (KdV) equation, encompasses the combined effects of quadratic and cubic nonlinearities. By employing the Hirota bilinear method (HBM), we have obtained multi-soliton solutions and breather soliton solutions. We have noticed the variations in electron temperature ratios and density concentration ratios substantially impact and reshape breather soliton structures. Such alterations in breather structures, influenced by different electron temperatures and concentration ratios, hold potential significance in understanding energy transport within Earth's magnetotail region. The alteration of shapes and soliton structures may also be investigated in the dynamics of wave fields upon interaction such as hydrodynamics, optics, optic fibers, signal processing, etc.

From breather soliton molecules to chaos in a laser cavity: the scenario of intermittent transitions.

Intermittency is widely observed in various nonlinear dynamical systems as an intriguing transient dynamic far from equilibrium. The internal dynamics formed by a pair of interacting optical solitons are often analogized to typical nonlinear systems. However, whether intermittency exists within the intramolecular motion remains to be investigated. Here, we study the intermittent dynamics of soliton molecules in ultrafast lasers, employing balanced optical cross-correlation techniques with sub-femtosecond temporal resolution. We demonstrate the occurrence of the bursting phase of intense variations of pulse separation within regular breather rhythms. In addition, we discover the intermittent transitions route to chaotic soliton molecules, facilitated by gain control. A series of analysis methods are used to assess the chaotic signals, providing compelling experimental evidence that soliton molecules can be analogized to their matter molecule counterparts. Our experimental findings shed light on the non-equilibrium intramolecular dynamics, providing insight into the transition of the attractors within interacting dissipative solitons in laser and fiber resonators.

Electromagnetic breathing dromion-like structures in an anisotropic ferromagnetic medium

The influence of Gilbert damping on the propagation of electromagnetic waves (EMWs) in an anisotropic ferromagnetic medium is investigated theoretically. The interaction of the magnetic field component of the electromagnetic wave with the magnetization of a ferromagnetic medium has been studied by solving the associated Maxwell’s equations coupled with the Landau–Lifshitz–Gilbert (LLG) equation. When small perturbations are made on the magnetization of the ferromagnetic medium and magnetic field along the direction of propagation of electromagnetic wave by using the reductive perturbation method, the associated nonlinear dynamics is governed by a time-dependent damped derivative nonlinear Schrödinger (TDDNLS) equation. The Lagrangian density function is constructed by using the variational method to solve the TDDNLS equation to understand the dynamics of the system under consideration. The propagation of EMW in a ferromagnetic medium with inherent Gilbert damping admits very interesting nonlinear dynamical structures. These structures include Gilbert damping-managing symmetrically breathing solitons, localized erupting electromagnetic breathing dromion-like modes of excitations, breathing dromion-like soliton, decaying dromion-like modes and an unexpected creation-annihilation mode of excitations in the form of growing-decaying dromion-like modes.

Coherent and Incoherent Breathing Solitons in a Bidirectional Ultrafast Thulium Fiber Laser

Mode‐locked lasers are prone to exhibit a wide range of dynamics, including the coherent dissipative soliton with high robustness or the possible emission of incoherent pulse bursts. Herein, it is proposed to expand the notion of dissipative breathing solitons to incoherent localized formation, which is illustrated experimentally by reporting original coherent and incoherent breathing solitons generated in a 2 μm bidirectional ultrafast fiber laser. A real‐time spectral characterization of the breathing soliton dynamics with different coherence is provided, highlighting salient features of the self‐organization. The switching between coherent breathing soliton and noise‐like pulse is also experimentally demonstrated, corresponding to the multiple transient decoherence and coherence recovery. High behavior similarity exists in bidirectional solitons evolution owing to the common gain and loss modulation. Numerical simulations using a discrete laser model validate the experimental findings. This research generalizes previous observations of breathing solitons made in the coherent case and opens up new possibilities for generating breathers in various dissipative systems.

Formation of solitary waves solutions and dynamic visualization of the nonlinear schrödinger equation with efficient techniques

This article investigates the non-linear generalized geophysical KdV equation, which describes shallow water waves in an ocean. The proposed generalized projective Riccati equation method and modified auxiliary equation method extract a more efficient and broad range of soliton solutions. These include U-shaped, W-shaped, singular, periodic, bright, dark, kink-type, breather soliton, multi-singular soliton, singular soliton with high amplitude, multiple periodic, multiple lump wave soliton, and flat kink-type soliton solutions. The travelling wave patterns of the model are graphically presented with suitable parameter values using the modern software Maple and Wolfram Mathematica. The visual representation of the solutions in 3D, 2D, and contour surfaces enhances understanding of parameter impact. Sensitivity and modulation instability analyses were performed to offer insights into the dynamics of the examined model. The observed dynamics of the proposed model were presented, revealing quasi-periodic chaotic, periodic systems, and quasi-periodic behaviour. This analysis confirms the effectiveness and reliability of the method employed, demonstrating its applicability in discovering travelling wave solitons for a wide range of nonlinear evolution equations.

Generalized cosine-Gaussian breathing solitons with transverse cross modulation in nonlocal nonlinear media

Based on the nonlocal nonlinear Schrödinger equation, the propagation characteristics of transverse cross-modulated cosine-Gaussian (TCMCG) breathing solitons in strongly nonlocal media are studied. A series of mathematical expressions are provided to describe the propagation dynamics characteristics of the TCMCG breathing solitons, including the soliton width, soliton width change rate, soliton width extreme values, change rates of soliton width extreme values, phase wavefront curvature and phase wavefront curvature extreme values, etc. The research results indicate that by modulating TCMCG breathing solitons, generalized high-order spatial solitons can be formed with periodic changes in light intensity modes but unchanged in beam width. The cosine function parameter has a significant impact on the propagation dynamics characteristics, which appears in the initial expression and plays a cross modulation role in the transverse spatial light intensity. The effect of the initial incident power on soliton width extreme values and phase wavefront curvature extreme values is also discussed.

Effect of polarization force on Gardner multi solitons and breather solitons traits in opposite polarity dusty plasma

Effect of polarization force on Gardner multi solitons and breather solitons traits in opposite polarity dusty plasma

Study on soliton behaviors of the complex nonlinear Fokas–Lenells equation utilizing different methods

In this paper, we study the Fokas–Lenells equation, which is a derivation form of the nonlinear Schrödinger equation and can be used to describe nonlinearity in the propagation of optical pulses. To seek solutions for this nonlinearity, the Logistic method has been proposed in order to drive acceptable and understandable results regarding physical meaning. This method is defined based on the summation of an ordinary differential equation. Moreover, for further study, the Bernoulli [Formula: see text]-expansion method is used by considering hypothetical solutions as a function of the Bernoulli equation. Subsequently, solutions of the Fokas–Lenells equation are successfully acquired, displaying efficient results in hyperbolic, trigonometric, and exponential equations. The importance of these results appears in the definition of the wave function under the consideration of the appropriate coefficients. Hence, computer simulation is used for better understanding of the generated results. The dynamic behaviors of these solutions are demonstrated in 3D graphs, contour maps, and line plots, under applying various parameters, solutions of the waves show different soliton behaviors, including bright-dark, bright, and breather solitons. The results indicate that the exerted methods are novel, reliable, and effective approaches which can be employed in a wide range of nonlinear differential equations. These methods and their begotten results are far from the complexities of mathematical structures and therefore go beyond previous efforts in the literature.

Real-time measurements of breathing dissipative soliton pairs in mode-locked fiber lasers

The emergence of breathing solitons, as well as the breathing soliton molecules undergoing periodic evolution in peak power and spectral width has attracted growing research attentions in dissipative systems. Beyond the intense pulse interactions within tightly-bound states with picoseconds interval, the long-range interplay between pulses separated by nanosecond-scale starts to be investigated in recent years, thus filling the long-standing academic gap between the bound solitons and independent multiple pulses. In this work, we report on the real-time observations of breathing dissipative soliton pairs with nanoseconds’ separations in a fiber laser operating under the threshold of stationary mode locking. Compared with conventional soliton counterpart, here the breathing soliton pair exhibit even more fascinating inter-pulse nonlinear dynamics due to the intriguing self-excited periodic evolutions. The shot-to-shot measurements of pulse intensity profiles, associated with dispersive Fourier transform clearly uncover not only the different breathing phase synchronization and pulse separations among different pulses but also the hysteresis behaviors of the breathing ratio by continuously varying the laser parameters. Moreover, we observed various transient stages in this unstable lasing regime, including the transition between stable single soliton and breathing soliton pair, as well as their long-range breather interaction. These results reveal the breathing multi-pulse state and the instantaneous non-equilibrium dynamics involving the breather pairs, enriching the framework of breathing solitons and opening a novel possibility to explore the internal motions of breathing soliton complex in nonlinear systems.

Nonclassical symmetries, optimal classification, and dynamical behavior of similarity solutions of (3+1)-dimensional Burgers equation

This paper is devoted to the study of symmetry group classification, and optimal subalgebras of the three-dimensional Burgers equation, which describes nonlinear wave motion exhibits nonlinear effects and enhanced dispersion in the context of nonlinear dynamics. The equation undergoes a Painlevé analysis to assess its integrability properties to verify the possibility that it passes the Painlevé test, followed by the derivation of nonclassical symmetries. Utilizing the invariance property of Lie groups, desirable infinitesimal symmetries of Lie algebra have been outlined for the equation. Relying on the invariance characteristic of adjoint transformation along with the newly generated similarity variables, an intensive and systematically organized approach is employed to achieve a one-dimensional optimal system. The entire set of group invariant solutions for each of the associated subalgebras has been established. The dynamical behavior of derived exact solutions is examined using numerical simulations, and numerous intriguing occurrences are discovered, such as flat sheet, periodic wave, doubly soliton, multisoliton, bright-dark soliton, breather soliton, line soliton type, which offer valuable insights into the dynamics of the system and can predict the behavior of waves in real-world scenarios.

The dissipative Talbot soliton fiber laser.

Talbot effect, characterized by the replication of a periodic optical field in a specific plane, is governed by diffraction and dispersion in the spatial and temporal domains, respectively. In mode-locked lasers, Talbot effect is rarely linked with soliton dynamics since the longitudinal mode spacing and cavity dispersion are far away from the self-imaging condition. We report switchable breathing and stable dissipative Talbot solitons in a multicolor mode-locked fiber laser by manipulating the frequency difference of neighboring spectra. The temporal Talbot effect dominates the laser emission state-in the breathing state when the integer self-imaging distance deviates from the cavity length and in the steady state when it equals the cavity length. A refined Talbot theory including dispersion and nonlinearity is proposed to accurately depict this evolution behavior. These findings pave an effective way to control the operation in dissipative optical systems and open branches in the study of nonlinear physics.

Periodic transmission theory of circularly symmetric multi-ring solitons in nonlinear Schrödinger equation

In this paper, the evolution characteristics of periodic transmission of circularly symmetric multi-ring solitons in optical nonlocal materials based on nonlinear Schrödinger equation are investigated in detail. The transmission expression of circularly symmetric multi-ring solitons has been derived. It was found that the number and size of rings in these solitons can be controlled by initial parameters. The transmission of circularly symmetric multi-ring solitons is similar to that of high-order temporal solitons in nonlinear fibers, exhibiting periodic variations. When the input energy is a specific value, the statistical width of circularly symmetric multi-ring solitons remains constant during transmission, otherwise it exhibits periodic changes, which can be considered as generalized breathing solitons. The influence of various parameters on the transmission characteristics has been analyzed in detail, and some important transmission characteristics have been intuitively demonstrated through numerical simulation.