Modulation instability, Akhmediev Breathers and continuous wave supercontinuum generation

Dark solitons and their bound states in a nonlinear fiber with second- and fourth-order dispersion

We study the excitations of dark solitons in a nonlinear optical fiber with the second- and fourth-order dispersion, and find the emergence of striped dark solitons (SDSs) and some multi-dark-soliton bound states. The SDSs can exhibit time-domain oscillating structures on a plane wave, and they have two types: the ones with or without the total phase step, while the multi-dark-soliton bound states exhibit different numbers of amplitude humps. By the modified linear stability analysis, we regard the SDSs as the results of the competition between periodicity and localization, and analytically give their existence condition, oscillation frequency, and propagation stability, which show good agreements with numerical results. We also provide a possible interpretation of the formation of the existing striped bright solitons (SBSs), and find that SBS will become the pure-quartic soliton when its periodicity and localization carry equal weight. Our results provide the theoretical support for the experimental observation of striped solitons in nonlinear fibers, and our method can also guide the discovery of striped solitons in other physical systems.

Resonant Akhmediev breathers

Modulation instability is a phenomenon in which a minor disturbance within a carrier wave gradually amplifies over time, leading to the formation of a series of compressed waves with higher amplitudes. In terms of frequency analysis, this process results in the generation of new frequencies on both sides of the original carrier wave frequency. We study the impact of fourth-order dispersion on this modulation instability in the context of nonlinear optics that lead to the formation of a series of pulses in the form of Akhmediev breather. The Akhmediev breather, a solution to the nonlinear Schrödinger equation, precisely elucidates how modulation instability produces a sequence of periodic pulses. We observe that when weak fourth-order dispersion is present, significant resonant radiation occurs, characterized by two modulation frequencies originating from different spectral bands. As an Akhmediev breather evolves, these modulation frequencies interact, resulting in a resonant amplification of spectral sidebands on either side of the breather. When fourth-order dispersion is of intermediate strength, the spectral bandwidth of the Akhmediev breather diminishes due to less pronounced resonant interactions, while stronger dispersion causes the merging of the two modulation frequency bands into a single band. Throughout these interactions, we witness a complex energy exchange process among the phase-matched frequency components. Moreover, we provide a precise explanation for the disappearance of the Akhmediev breather under weak fourth-order dispersion and its resurgence with stronger values. Our study demonstrates that Akhmediev breathers, under the influence of fourth-order dispersion, possess the capability to generate infinitely many intricate yet coherent patterns in the temporal domain.

Modulation instability with high-order dispersion: fundamental limitations of pattern formation

We theoretically and numerically investigate modulation instability in the presence of even, high-order dispersion, focusing on general trends rather than on specific results for a particular dispersion order. We show that high-order dispersion leads to increasingly poor phase matching between the three central waves (i.e. the pump and the ±1 sidebands) and the higher sideband orders, inhibiting in effect four-wave mixing frequency generation. For sufficiently large dispersion orders, the problem in effect can reduce to a three-wave system. Our predictions are in excellent agreement with numerical simulations and show that high-order dispersion imposes a fundamental limit on modulation instability dynamics.

Analysis of interaction dynamics and rogue wave localization in modulation instability using data-driven dominant balance

We analyze the dynamics of modulation instability in optical fiber (or any other nonlinear Schrödinger equation system) using the machine-learning technique of data-driven dominant balance. We aim to automate the identification of which particular physical processes drive propagation in different regimes, a task usually performed using intuition and comparison with asymptotic limits. We first apply the method to interpret known analytic results describing Akhmediev breather, Kuznetsov-Ma, and Peregrine soliton (rogue wave) structures, and show how we can automatically distinguish regions of dominant nonlinear propagation from regions where nonlinearity and dispersion combine to drive the observed spatio-temporal localization. Using numerical simulations, we then apply the technique to the more complex case of noise-driven spontaneous modulation instability, and show that we can readily isolate different regimes of dominant physical interactions, even within the dynamics of chaotic propagation.

Stabilization of modulation instability by control field in semiconductor quantum wells

This article explores the modulation instability of a continuous or quasi-continuous weak probe pulse in a three-level asymmetric double quantum wells under an electromagnetically induced transparency regime, controlled by a strong laser beam. The dynamics of modulation instability reveals that the instability gain as well as its bandwidth is greatly influenced by control field Rabi frequency. The probe pulse is found to be almost stable against modulation instability for higher values of control field Rabi frequency. The results of this investigation may potentially apply for oscillation free generation of supercontinuum in quantum well nanostructures.

Fundamental and second-order dark soliton solutions of two- and three-component Manakov equations in the defocusing regime

We present exact multiparameter families of soliton solutions for two- and three-component Manakov equations in the defocusing regime. Existence diagrams for such solutions in the space of parameters are presented. Fundamental soliton solutions exist only in finite areas on the plane of parameters. Within these areas, the solutions demonstrate rich spatiotemporal dynamics. The complexity increases in the case of three-component solutions. The fundamental solutions are dark solitons with complex oscillating patterns in the individual wave components. At the boundaries of existence, the solutions are transformed into plain (nonoscillating) vector dark solitons. The superposition of two dark solitons in the solution adds more frequencies in the patterns of oscillating dynamics. These solutions admit degeneracy when the eigenvalues of fundamental solitons in the superposition coincide.

Excitation of mirror symmetry higher-order rational soliton in modulation stability regimes on continuous wave background

We study the relationship between the structures of the nonlinear localized waves and the distribution characteristics of the modulation stability regime in a nonlinear fiber with both third-order and fourth-order effects. On the background frequency and background amplitude plane, the modulation stability region consists of two symmetric curves on the left and right and a point on the symmetry axis. We find that the higher-order excitation characteristics are obviously different at different positions in the modulation stability region. Their excitation characteristics are closely related to the modulation instability distribution characteristics of the system. It is shown that asymmetric high-order rational solitons are excited at the left and right stable curves, and the symmetric one is excited at the stable points. Interestingly, the asymmetric higher-order rational solitons on the left and right sides are mirror-symmetrical to each other, which coincides with the symmetry of the modulation instability distribution. These results can deepen our understanding of the relationship between nonlinear excitation and modulation instability and enrich our knowledge about higher-order nonlinear excitations.

Spatiotemporal doubly periodic waves in a phase-mismatched second-harmonic generation

In this Letter, we present an analytical and numerical investigation to characterize the formation of quadratic doubly periodic waves originating from coherent modulation instability in a dispersive quadratic medium in the regime of cascading second-harmonic generation. To the best of our knowledge, such an endeavor has not been undertaken before, despite the growing relevance of doubly periodic solutions as the precursor of highly localized wave structures. Unlike the case with cubic nonlinearity, the periodicity of quadratic nonlinear waves can also be controlled by the wave-vector mismatch in addition to the initial input condition. Our results may impact widely on the formation, excitation, and control of extreme rogue waves and the description of modulation instability in a quadratic optical medium.

Extreme wave excitation from localized phase-shift perturbations

The modulation instability is a focusing mechanism responsible for the formation of strong wave localizations not only on the water surface, but also in a variety of nonlinear dispersive media. Such dynamics is initiated from the injection of sidebands, which translate into an amplitude modulation of the wave field. The nonlinear stage of unstable wave evolution can be described by exact solutions of the nonlinear Schrödinger equation (NLSE). In that case, the amplitude modulation of such coherent extreme wave structures is connected to a particular phase-shift seed in the carrier wave. In this Letter, we show that phase-shift localization applied to the background, excluding any amplitude modulation excitation, can indeed trigger extreme events. Such rogue waves can be for instance generated by considering the parametrization of fundamental breathers, and thus by seeding only the local phase-shift information to the regular carrier wave. Our wave tank experiments show an excellent agreement with the expected NLSE hydrodynamics and confirm that even though delayed in their evolution, breather-type extreme waves can be generated from a purely regular wave train. Such a focusing mechanism awaits experimental confirmation in other nonlinear media, such optics, plasma, and Bose-Einstein condensates.

Diverse Forms of Breathers and Rogue Wave Solutions for the Complex Cubic Quintic Ginzburg Landau Equation with Intrapulse Raman Scattering

This manuscript consist of diverse forms of lump: lump one stripe, lump two stripe, generalized breathers, Akhmediev breather, multiwave, M-shaped rational and rogue wave solutions for the complex cubic quintic Ginzburg Landau (CQGL) equation with intrapulse Raman scattering (IRS) via appropriate transformations approach. Furthermore, it includes homoclinic, Ma and Kuznetsov-Ma breather and their relating rogue waves and some interactional solutions, including an interactional approach with the help of the double exponential function. We have elaborated the kink cross-rational (KCR) solutions and periodic cross-rational (KCR) solutions with their graphical slots. We have also constituted some of our solutions in distinct dimensions by means of 3D and contours profiles to anticipate the wave propagation. Parameter domains are delineated in which these exact localized soliton solutions exit in the proposed model.

Amplification and Generation of Frequency-Modulated Soliton Pulses in Nonuniform Active Fiber Configurations

We report on the theoretical and numerical analysis of the nonlinear Schrödinger equation describing the dynamical evolution of frequency-modulated (FM) optical signals propagating through the fiber configuration comprising active fibers with the anomalous dispersion nonuniformly distributed over the fiber length. In our consideration, a single active fiber section including segments with initially increasing and then decreasing dispersion is used for amplification and compression of an external FM pulse resulting in an increase of ~6 orders of magnitude in the pulse peak power and a 100-fold narrowing of the pulse duration down to a few picoseconds. Moreover, we demonstrate that, with a ~1 mW weakly modulated continuous wave input signal, the fiber configuration comprising two active fiber sections with different dispersion profiles is able to generate a strongly periodic pulse train, resulting in a pulse repetition rate >100 GHz, a pulse duration ~0.5 ps, and peak power up to ~1 kW. An evolution of optical signals governed by modulation instability in both fiber configurations is explored.

Advances in mid-infrared spectroscopy enabled by supercontinuum laser sources

Supercontinuum sources are all-fiber pulsed laser-driven systems that provide high power spectral densities within ultra-broadband spectral ranges. The tailored process of generating broadband, bright, and spectrally flat supercontinua-through a complex interplay of linear and non-linear processes-has been recently pushed further towards longer wavelengths and has evolved enough to enter the field of mid-infrared (mid-IR) spectroscopy. In this work, we review the current state and perspectives of this technology that offers laser-like emission properties and instantaneous broadband spectral coverage comparable to thermal emitters. We aim to go beyond a literature review. Thus, we first discuss the basic principles of supercontinuum sources and then provide an experimental part focusing on the quantification and analysis of intrinsic emission properties such as typical power spectral densities, brightness levels, spectral stability, and beam quality (to the best of the authors' knowledge, the M² factor for a mid-IR supercontinuum source is characterized for the first time). On this basis, we identify key competitive advantages of these alternative emitters for mid-IR spectroscopy over state-of-the-art technologies such as thermal sources or quantum cascade lasers. The specific features of supercontinuum radiation open up prospects of improving well-established techniques in mid-IR spectroscopy and trigger developments of novel analytical methods and instrumentation. The review concludes with a structured summary of recent advances and applications in various routine mid-IR spectroscopy scenarios that have benefited from the use of supercontinuum sources.

Extreme spectral asymmetry of Akhmediev breathers and Fermi-Pasta-Ulam recurrence in a Manakov system

The dynamics of Fermi-Pasta-Ulam (FPU) recurrence in a Manakov system is studied analytically. Exact Akhmediev breather (AB) solutions for this system are found that cannot be reduced to the ABs of a single-component nonlinear Schrödinger equation. Expansion-contraction cycle of the corresponding spectra with an infinite number of sidebands is calculated analytically using a residue theorem. A distinctive feature of these spectra is the asymmetry between positive and negative spectral modes. A practically important consequence of the spectral asymmetry is a nearly complete energy transfer from the central mode to one of the lowest-order (left or right) sidebands. Numerical simulations started with modulation instability of plane waves confirm the findings based on the exact solutions. It is also shown that the full growth-decay cycle of the AB leads to the nonlinear phase shift between the initial and final states in both components of the Manakov system. This finding shows that the final state of the FPU recurrence described by the vector ABs is not quite the same as the initial state. Our results are applicable and can be observed in a wide range of two-component physical systems such as two-component waves in optical fibers, two-directional waves in crossing seas, and two-component Bose-Einstein condensates.

Exact Analytic Spectra of Asymmetric Modulation Instability in Systems with Self-Steepening Effect

Nonlinear waves become asymmetric when asymmetric physical effects are present within the system. One example is the self-steepening effect. When exactly balanced with dispersion, it leads to a fully integrable system governed by the Chen-Lee-Liu equation. The latter provides a natural basis for the analysis of asymmetric wave dynamics just as nonlinear Schrödinger or Korteweg-de Vries equations provide the basis for analyzing solitons with symmetric profile. In this work, we found periodic wave trains of the Chen-Lee-Liu equation evolved from fully developed modulation instability and analyzed a highly nontrivial spectral evolution of such waves in analytic form that shows strong asymmetry of its components. We present the conceptual basis for finding such spectra that can be used in analyzing asymmetric nonlinear waves in other systems.

Numerical spectral synthesis of breather gas for the focusing nonlinear Schrödinger equation

We numerically realize a breather gas for the focusing nonlinear Schrödinger equation. This is done by building a random ensemble of N∼50 breathers via the Darboux transform recursive scheme in high-precision arithmetics. Three types of breather gases are synthesized according to the three prototypical spectral configurations corresponding the Akhmediev, Kuznetsov-Ma, and Peregrine breathers as elementary quasiparticles of the respective gases. The interaction properties of the constructed breather gases are investigated by propagating through them a "trial" generic (Tajiri-Watanabe) breather and comparing the mean propagation velocity with the predictions of the recently developed spectral kinetic theory [El and Tovbis, Phys. Rev. E 101, 052207 (2020)2470-004510.1103/PhysRevE.101.052207].

Onset times of long-lived rogue waves in an optical waveguide array

We study the effects of increasing modulation instability and disorder on the onset times of rogue waves in waveguide arrays as described by the discrete unstable nonlinear Schrödinger equation (UNLSE). We analytically determine regions of instability, where rogue waves are likely to occur in the UNLSE, and then use numerical techniques to study the time evolution of these systems. Only for small modulation instability is the effect of fluctuations prominent on the onset times; otherwise, we find that large modulation instability dominates the onset time behavior.

Modified linear stability analysis for quantitative dynamics of a perturbed plane wave

We develop linear stability analysis (LSA) to quantitatively predict the dynamics of a perturbed plane wave in nonlinear systems. We take a nonintegrable fiber model with purely fourth-order dispersion as an example to demonstrate this method's effectiveness. For a Gaussian-type initial perturbation with cosine-type modulation on a plane wave, its propagation velocities, periodicity, and localization are predicted successfully, and the range of application is discussed. Importantly, the modulation-instability-induced growth of localized perturbation is proved different from the one of purely periodic perturbation and requires the modification of gain value for more accurate prediction. The method offers a needful supplement and improvement for LSA and paves a way to study the dynamics of a perturbed plane wave in more practical nonlinear systems.

Dynamics of optical rogue wave generation in dispersion oscillating fibers

We present an in-depth investigation of optical rogue waves (ORWs) during picosecond supercontinuum generations in photonic crystal fibers with periodic landscapes of group velocity dispersion and nonlinearity, namely dispersion oscillating fibers (DOFs). Specifically, it is shown via ensembles of numerical simulations that during supercontinuum generations, the rogue nature of extreme and rare events formed in uniform fibers can be effectively manipulated in DOFs. This is also verified by comparing single evolution dynamics in different dispersion longitudinal profiles. For investigating the influence of slow dynamics of ORW generation in DOFs, we increase the propagating distance and find out MI gain is still the major factor that influences the generation of ORWs. In addition, analytical results associated with simulations indicate the rogue manipulations in DOFs are attributed to the adjustable modulation-instability-gain due to periodic dispersion variation along fiber length. Finally, unlike MI in uniform fiber, MI gain side lobes result from quasi-phase-matching (QPM) relation in DOFs provide additional degree of freedom to control generations of ORWs. We believe our results will provide not only a novel insight of understanding ORW dynamics in presence of dispersion modulations, but also a new way of harnessing rogue waves in oceanology.

Observation of doubly periodic solutions of the nonlinear Schrödinger equation in optical fibers

We report the first, to the best of our knowledge, experimental observation of doubly periodic first-order solutions of the nonlinear Schrödinger equation in optical fibers. We confirm, experimentally, the existence of A-type and B-type solutions. This is done by using the initial conditions that consist of a strong pump and two weak sidebands. The evolution of power and phase of the main spectral components is recorded using heterodyne time-domain reflectometry. Another important part of our experiment is active loss compensation. We reach a good agreement between theory and experiment.

Machine learning analysis of rogue solitons in supercontinuum generation

Supercontinuum generation is a highly nonlinear process that exhibits unstable and chaotic characteristics when developing from long pump pulses injected into the anomalous dispersion regime of an optical fiber. A particular feature associated with this regime is the long-tailed “rogue wave”-like statistics of the spectral intensity on the long-wavelength edge of the supercontinuum, linked to the generation of a small number of “rogue solitons” with extreme red-shifts. Whilst the statistical properties of rogue solitons can be conveniently measured in the spectral domain using the real-time dispersive Fourier transform technique, we cannot use this technique to determine any corresponding temporal properties since it only records the spectral intensity and one loses information about the spectral phase. And direct temporal characterization using methods such as the time-lens has resolution of typically 100’s of fs, precluding the measurement of solitons which possess typically much shorter durations. Here, we solve this problem by using machine learning. Specifically, we show how supervised learning can train a neural network to predict the peak power, duration, and temporal walk-off with respect to the pump pulse position of solitons at the edge of a supercontinuum spectrum from only the supercontinuum spectral intensity without phase information. Remarkably, the network accurately predicts soliton characteristics for a wide range of scenarios, from the onset of spectral broadening dominated by pure modulation instability to near octave-spanning supercontinuum with distinct rogue solitons.

Supercontinuum generation in nanostructured core gradient index fibers

We report on near-infrared supercontinuum generation in a submeter-long single-mode, nanostructured core fiber. The fiber core is composed of few thousand pure silica and germanium-doped silica glass nanorods with diameter of 200 nm each. The nanorods’ distribution is calculated based on the Maxwell–Garnett effective medium approach to mimic effective parabolic refractive index distribution in the fiber core. The standard stack-and-draw method was used to scale down the fiber structure and obtain subwavelength nanorods in the core. Size and distribution of individual nanorods are essential to determine modal and dispersion properties of the fiber without assistance of air holes in the fiber cladding. We study supercontinuum generation performance in this nanostructured core fiber pumping with low-cost microchip laser operating at 1550 nm with 1 ns pulse length and pulse energy of 0.4 µJ. A modulation instability-driven supercontinuum is generated in the fiber, covering a wavelength span of 1400–2300 nm. Due to possibility of dispersion engineering and all-solid structure the nanostructured fibers offer new possibilities for development of low-cost all-fiber supercontinuum light sources for the near-infrared range and cascaded ultrabroadband supercontinuum all-fiber systems.

Experimental realization of Fermi-Pasta-Ulam-Tsingou recurrence in a long-haul optical fiber transmission system

The integrable nonlinear Schrödinger equation (NLSE) is a fundamental model of nonlinear science which also has important consequences in engineering. The powerful framework of the periodic inverse scattering transform (IST) provides a description of the nonlinear phenomena modulational instability and Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in terms of exact solutions. It associates the complex nonlinear dynamics with invariant nonlinear spectral degrees of freedom that may be used to encode information. While optical fiber is an ideal testing ground of its predictions, maintaining integrability over sufficiently long distances to observe recurrence, as well as synthesizing and measuring the field in both amplitude and phase on the picosecond timescales of typical experiments is challenging. Here we report on the experimental realization of FPUT recurrence in terms of an exact space-time-periodic solution of the integrable NLSE in a testbed for optical communication experiments. The complex-valued initial condition is constructed by means of the finite-gap integration method, modulated onto the optical carrier driven by an arbitrary waveform generator and launched into a recirculating fiber loop with periodic amplification. The measurement with an intradyne coherent receiver after a predetermined number of revolutions provides a non-invasive full-field characterization of the space-time dynamics. The recurrent space-time evolution is in close agreement with theoretical predictions over a distance of 9000 km. Nonlinear spectral analysis reveals an invariant nonlinear spectrum. The space-time scale exceeds that of previous experiments on FPUT recurrence in fiber by three orders of magnitude.

Super-regular breathers in nonlinear systems with self-steepening effect

A family of super-regular (SR) breather solutions in systems with self-steepening effect and in the case of either normal or anomalous dispersion is derived analytically. Derivation is based on the Darboux transformation with a quadratic spectral parameter. In contrast to the SR breather solutions in t-symmetric systems such as the nonlinear Schrödinger equation, the new breathers found in the present work evolve asymmetrically even if started from symmetric initial conditions. The initial stage of this process is modulation instability. Numerical simulations confirm the excitation of the SR breathers when started from the approximate initial conditions leading at first to modulation instability. Our results offer the possibility of experimental observations of SR breather dynamics in systems with self-steepening effects, such as optical frequency-doubling crystals or magnetized plasmas.

Experimental characterization of recurrences and separatrix crossing in modulational instability

We experimentally investigate two cycles of Fermi-Pasta-Ulam-Tsingou recurrence in optical fibers. Using three waves input, we characterize the distance of maximum compression points against the sideband amplitude and relative phase, outlining the qualitative changes of the dynamics due to separatrix crossing. Experimental results are in good agreement with numerical simulations and analytical predictions.

Subharmonic Entrainment of Kerr Breather Solitons

We predict subharmonic entrainment of breather-soliton oscillations to a periodic perturbation at the round-trip time T_{R} in Kerr-nonlinear optical resonators; an integer ratio T_{b}/T_{R}=N≫1 results for breathing period T_{b}. Rigid entrainment is observed with intermediate finesse (F∼30-40) for N up to 20, and we propose a way to realize higher entrainment ratios at higher finesse. This nontrivial synchronization across the widely separated timescales of the photon lifetime and round-trip time points to a new direction for research in this field and may find application, for example, in the measurement of a pulse train repetition rate that is electronically inaccessible.

Supercontinuum noise reduction by fiber undertapering

We demonstrate that the Relative Intensity Noise (RIN) of a supercontinuum source can be significantly reduced using the new concept of undertapering, where the fiber is tapered to a diameter that is smaller than the diameter that gives the shortest blue edge, which is typically regarded as the optimum. We show that undertapering allows to control the second zero dispersion wavelength and use it as a soliton barrier to stop the redshifting solitons at a pre-defined wavelength, and thereby strongly reduce the RIN. We demonstrate how undertapering can reduce the spectrally averaged RIN in the optical coherence tomography bands, 500-800nm and 1150-1450nm, by more than a factor two.

Predicting ocean rogue waves from point measurements: An experimental study for unidirectional waves

Rogue waves are strong localizations of the wave field that can develop in different branches of physics and engineering, such as water or electromagnetic waves. Here, we experimentally quantify the prediction potentials of a comprehensive rogue-wave reduced-order precursor tool that has been recently developed to predict extreme events due to spatially localized modulation instability. The laboratory tests have been conducted in two different water wave facilities and they involve unidirectional water waves; in both cases we show that the deterministic and spontaneous emergence of extreme events is well predicted through the reported scheme. Due to the interdisciplinary character of the approach, similar studies may be motivated in other nonlinear dispersive media, such as nonlinear optics, plasma, and solids, governed by similar equations, allowing the early stage of extreme wave detection.

Full-field characterization of breather dynamics over the whole length of an optical fiber

Full-field longitudinal characterization of picosecond pulse train formation in optical fibers is reported. The spatio-temporal evolution is obtained via fast and non-invasive distributed measurements in phase and intensity of the main spectral components of the pulses. To illustrate the performance of the setup, we report, to the best of our knowledge, the first time-domain experimental observation of the symmetry breaking of Fermi-Pasta-Ulam recurrences. The experimental results are in good agreement with numerical simulations.

Simple method for estimating the fractional Raman contribution

We propose a novel and simple method for estimating the fractional Raman contribution, f_R, based on an analysis of a full model of modulation instability (MI) in waveguides. An analytical expression relating f_R to the MI peak gain beyond the cutoff power is explicitly derived, allowing for an accurate estimation of f_R from a single measurement of the Raman gain spectrum.

Excitation conditions of several fundamental nonlinear waves on continuous-wave background

We study the excitation conditions of antidark solitons and nonrational W-shaped solitons in a nonlinear fiber with both third-order and fourth-order effects. We show that the relative phase can be used to distinguish antidark solitons and nonrational W-shaped solitons. The excitation conditions of these well-known fundamental nonlinear waves (on a continuous-wave background) can be clarified clearly by the relative phase and three previously reported parameters (background frequency, perturbation frequency, and perturbation energy). Moreover, the numerical simulations from the nonideal initial states also support these theoretical results. These results provide an important complement for the studies on relationship between modulation instability and nonlinear wave excitations, and are helpful for controllable nonlinear excitations in experiments.

Manipulating Airy pulse in the regime of optical event horizon

A fiber analogue of the optical event horizon is proposed via an Airy-soliton collision. The characteristics of an Airy pulse are demonstrated in a photonic crystal fiber. The robust Airy wave packet is capable of being manipulated by a strong soliton in the frame of the Airy-soliton horizon, leading to various distinctive interaction scenarios. The Airy pulse realizes the temporal reversing and still maintains its features even after undergoing a complete wavelength conversion process. Different comb-like spectra are created in the presence of Raman effect by changing the duration of the Airy pulse. The rebounds of Airy pulse in a twin-soliton trapping process is investigated. Meanwhile, a "maximally compressed" Akhmediev Breather is employed to collide with a soliton to compare with the Airy-soliton horizon case. The result can be found in the potential applications of pulse reshaping and temporal-spectral imaging for optical communication and signal processing.

Modulation instability in higher-order nonlinear Schrödinger equations

We investigate the dynamics of modulation instability (MI) and the corresponding breather solutions to the extended nonlinear Schrödinger equation that describes the full scale growth-decay cycle of MI. As an example, we study modulation instability in connection with the fourth-order equation in detail. The higher-order equations have free parameters that can be used to control the growth-decay cycle of the MI; that is, the growth rate curves, the time of evolution, the maximal amplitude, and the spectral content of the Akhmediev Breather strongly depend on these coefficients.

Nonlinear spectral analysis of Peregrine solitons observed in optics and in hydrodynamic experiments

The data recorded in optical fiber and in hydrodynamic experiments reported the pioneering observation of nonlinear waves with spatiotemporal localization similar to the Peregrine soliton are examined by using nonlinear spectral analysis. Our approach is based on the integrable nature of the one-dimensional focusing nonlinear Schrödinger equation (1D-NLSE) that governs at leading order the propagation of the optical and hydrodynamic waves in the two experiments. Nonlinear spectral analysis provides certain spectral portraits of the analyzed structures that are composed of bands lying in the complex plane. The spectral portraits can be interpreted within the framework of the so-called finite gap theory (or periodic inverse scattering transform). In particular, the number N of bands composing the nonlinear spectrum determines the genus g=N-1 of the solution that can be viewed as a measure of complexity of the space-time evolution of the considered solution. Within this setting the ideal, rational Peregrine soliton represents a special, degenerate genus 2 solution. While the fitting procedures previously employed show that the experimentally observed structures are quite well approximated by the Peregrine solitons, nonlinear spectral analysis of the breathers observed both in the optical fiber and in the water tank experiments reveals that they exhibit spectral portraits associated with more general, genus 4 finite-gap NLSE solutions. Moreover, the nonlinear spectral analysis shows that the nonlinear spectrum of the breathers observed in the experiments slowly changes with the propagation distance, thus confirming the influence of unavoidable perturbative higher-order effects or dissipation in the experiments.

Growth rate of modulation instability driven by superregular breathers

We report an exact link between Zakharov-Gelash super-regular (SR) breathers (formed by a pair of quasi-Akhmediev breathers) with interesting different nonlinear propagation characteristics and modulation instability (MI). This shows that the absolute difference of group velocities of SR breathers coincides exactly with the linear MI growth rate. This link holds for a series of nonlinear Schrödinger equations with infinite-order terms. For the particular case of SR breathers with opposite group velocities, the growth rate of SR breathers is consistent with that of each quasi-Akhmediev breather along the propagation direction. Numerical simulations reveal the robustness of different SR breathers generated from various non-ideal single and multiple initial excitations. Our results provide insight into the MI nature described by SR breathers and could be helpful for controllable SR breather excitations in related nonlinear systems.

Experimental observation of the emergence of Peregrine-like events in focusing dam break flows

Simple photonic fiber-based workbenches have been able to emulate well-known nonlinear wave dynamics occurring in deep or shallow water conditions. Here, by investigating the nonlinear reshaping of a flat-top pulse upon propagation in an anomalous dispersive optical fiber, we observe that typical signatures of focusing dam break flows and Peregrine-like breather events can locally coexist in spontaneous pattern formations. The experimental measurements are in good agreement with our numerical predictions.

Controllable optical rogue waves via nonlinearity management

Using a similarity transformation, we obtain analytical solutions to a class of nonlinear Schrödinger (NLS) equations with variable coefficients in inhomogeneous Kerr media, which are related to the optical rogue waves of the standard NLS equation. We discuss the dynamics of such optical rogue waves via nonlinearity management, i.e., by selecting the appropriate nonlinearity coefficients and integration constants, and presenting the solutions. In addition, we investigate higher-order rogue waves by suitably adjusting the nonlinearity coefficient and the rogue wave parameters, which could help in realizing complex but controllable optical rogue waves in properly engineered fibers and other photonic materials.

Mechanism of Kuznetsov-Ma breathers

We discuss how to understand the dynamical process of Kuznetsov-Ma breather, based on some basic physical mechanisms. It is shown that the dynamical process of Kuznetsov-Ma breather involves at least two distinctive mechanisms: modulational instability and the interference effects between a bright soliton and a plane-wave background. Our analysis indicates that modulational instability plays dominant roles in the mechanism of Kuznetsov-Ma breather admitting weak perturbations, and the interference effect plays a dominant role for the Kuznetsov-Ma breather admitting strong perturbations. For intermediate cases, the two mechanisms are both greatly involved. These characters provide a possible way to understand the evolution of strong perturbations on a plane-wave background.

Breather solutions of a fourth-order nonlinear Schrödinger equation in the degenerate, soliton, and rogue wave limits

We present one- and two-breather solutions of the fourth-order nonlinear Schrödinger equation. With several parameters to play with, the solution may take a variety of forms. We consider most of these cases including the general form and limiting cases when the modulation frequencies are 0 or coincide. The zero-frequency limit produces a combination of breather-soliton structures on a constant background. The case of equal modulation frequencies produces a degenerate solution that requires a special technique for deriving. A zero-frequency limit of this degenerate solution produces a rational second-order rogue wave solution with a stretching factor involved. Taking, in addition, the zero limit of the stretching factor transforms the second-order rogue waves into a soliton. Adding a differential shift in the degenerate solution results in structural changes in the wave profile. Moreover, the zero-frequency limit of the degenerate solution with differential shift results in a rogue wave triplet. The zero limit of the stretching factor in this solution, in turn, transforms the triplet into a singlet plus a low-amplitude soliton on the background. A large value of the differential shift parameter converts the triplet into a pure singlet.

Several localized waves induced by linear interference between a nonlinear plane wave and bright solitons

We investigate linear interference effects between a nonlinear plane wave and bright solitons, which are admitted by a pair-transition coupled two-component Bose-Einstein condensate. We demonstrate that the interference effects can induce several localized waves possessing distinctive wave structures, mainly including anti-dark solitons, W-shaped solitons, multi-peak solitons, Kuznetsov-Ma like breathers, and multi-peak breathers. Specifically, the explicit conditions for them are clarified by a phase diagram based on the linear interference properties. Furthermore, the interactions between these localized waves are discussed. The detailed analysis indicates that the soliton-soliton interaction induced phase shift brings the collision between these localized waves which can be inelastic for solitons involving collision and can be elastic for breathers. These characters come from the fact that the profile of solitons depends on the relative phase between bright solitons and a plane wave, and the profile of breathers does not depend on the relative phase. These results would motivate more discussions on linear interference between other nonlinear waves. Specifically, the solitons or breathers obtained here are not related to modulational instability. The underlying reasons are discussed in detail. In addition, possibilities to observe these localized waves are discussed in a two species Bose-Einstein condensate.

Abnormal chiral events in a semiconductor laser with coherent injection

We study experimentally and theoretically the dynamics of a spatially extended (along the propagation direction) oscillatory medium with coherent forcing. We observe abnormally high events, responsible for a different statistics of intensity and pulse height, in a regime where solitons and roll patterns are unstable. We focus on the formation of these high-peak events and their connection to the phase dynamics. Each abnormal event can be associated with a change in the slope of the phase time trace. Furthermore, the coexistence of ±2π phase rotations inside the cavity can be associated to the observation of abnormal events, similarly to recent predictions in bidimensional vortex turbulence.

Generation mechanisms of fundamental rogue wave spatial-temporal structure

We discuss the generation mechanism of fundamental rogue wave structures in N-component coupled systems, based on analytical solutions of the nonlinear Schrödinger equation and modulational instability analysis. Our analysis discloses that the pattern of a fundamental rogue wave is determined by the evolution energy and growth rate of the resonant perturbation that is responsible for forming the rogue wave. This finding allows one to predict the rogue wave pattern without the need to solve the N-component coupled nonlinear Schrödinger equation. Furthermore, our results show that N-component coupled nonlinear Schrödinger systems may possess N different fundamental rogue wave patterns at most. These results can be extended to evaluate the type and number of fundamental rogue wave structure in other coupled nonlinear systems.

Superregular breathers in a complex modified Korteweg-de Vries system

We study superregular (SR) breathers (i.e., the quasi-Akhmediev breather collision with a certain phase shift) in a complex modified Korteweg-de Vries equation. We demonstrate that such SR waves can exhibit intriguing nonlinear structures, including the half-transition and full-suppression modes, which have no analogues in the standard nonlinear Schrödinger equation. In contrast to the standard SR breather formed by pairs of quasi-Akhmediev breathers, the half-transition mode describes a mix of quasi-Akhmediev and quasi-periodic waves, whereas the full-suppression mode shows a non-amplifying nonlinear dynamics of localized small perturbations associated with the vanishing growth rate of modulation instability. Interestingly, we show analytically and numerically that these different SR modes can be evolved from an identical localized small perturbation. In particular, our results demonstrate an excellent compatibility relation between SR modes and the linear stability analysis.

Nonconservative higher-order hydrodynamic modulation instability

The modulation instability (MI) is a universal mechanism that is responsible for the disintegration of weakly nonlinear narrow-banded wave fields and the emergence of localized extreme events in dispersive media. The instability dynamics is naturally triggered, when unstable energy sidebands located around the main energy peak are excited and then follow an exponential growth law. As a consequence of four wave mixing effect, these primary sidebands generate an infinite number of additional sidebands, forming a triangular sideband cascade. After saturation, it is expected that the system experiences a return to initial conditions followed by a spectral recurrence dynamics. Much complex nonlinear wave field motion is expected, when the secondary or successive sideband pair that is created is also located in the finite instability gain range around the main carrier frequency peak. This latter process is referred to as higher-order MI. We report a numerical and experimental study that confirms observation of higher-order MI dynamics in water waves. Furthermore, we show that the presence of weak dissipation may counterintuitively enhance wave focusing in the second recurrent cycle of wave amplification. The interdisciplinary weakly nonlinear approach in addressing the evolution of unstable nonlinear waves dynamics may find significant resonance in other nonlinear dispersive media in physics, such as optics, solids, superfluids, and plasma.

Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation

We report experimental confirmation of the universal emergence of the Peregrine soliton predicted to occur during pulse propagation in the semiclassical limit of the focusing nonlinear Schrödinger equation. Using an optical fiber based system, measurements of temporal focusing of high power pulses reveal both intensity and phase signatures of the Peregrine soliton during the initial nonlinear evolution stage. Experimental and numerical results are in very good agreement, and show that the universal mechanism that yields the Peregrine soliton structure is highly robust and can be observed over a broad range of parameters.

Breather-to-soliton transformation rules in the hierarchy of nonlinear Schrödinger equations

We study the exact first-order soliton and breather solutions of the integrable nonlinear Schrödinger equations hierarchy up to fifth order. We reveal the underlying physical mechanism which transforms a breather into a soliton. Furthermore, we show how the dynamics of the Akhmediev breathers which exist on a constant background as a result of modulation instability, is connected with solitons on a zero background. We also demonstrate that, while a first-order rogue wave can be directly transformed into a soliton, higher-order rogue wave solutions become rational two-soliton solutions with complex collisional structure on a background. Our results will have practical implications in supercontinuum generation, turbulence, and similar other complex nonlinear scenarios.

Freak oscillation in a dusty plasma

The freak oscillation in one-dimensional dusty plasma is studied numerically by particle-in-cell method. Using a perturbation method, the basic set of fluid equations is reduced to a nonlinear Schrödinger equation (NLSE). The rational solution of the NLSE is presented, which is proposed as an effective tool for studying the rogue waves in dusty plasma. Additionally, the application scope of the analytical solution of the rogue wave described by the NLSE is given.

Kerr frequency combs and triangular spectra

Nonlinear externally driven optical cavities are known to generate periodic patterns. They grow from the linearly unstable background states due to modulation instability. These periodic solutions are also known as Kerr frequency combs, which have a variety of applications in metrology. The stationary state of periodic wave trains can be explained theoretically only in weakly nonlinear regimes near the onset of the instability using the order parameter description. However, in both weakly and strongly nonlinear dissipative regimes, only numerical solutions can be found. No analytic solutions are known so far except for the homogeneous continuous wave solution. Here, we derive an analytical expression for the intracavity fully nonlinear dissipative periodic wave train profiles that provides good agreement with the results of numerical simulations. Our approach is based on empirical knowledge of the triangular shape of the frequency comb spectrum.

Nondestructive distributed measurement of supercontinuum generation along highly nonlinear optical fibers

Supercontinuum generation (SCG) in optical fibers arises from the spectral broadening of an intense light, which results from the interplay of both linear and nonlinear optical effects. In this Letter, a nondestructive optical time domain reflectometry method is proposed for the first time, to the best of our knowledge, to measure the spatial (longitudinal) evolution of the SC induced along an optical fiber. The method was experimentally tested on highly nonlinear fibers. The experimental results are in a good agreement with the optical spectra measured at the fiber outputs.