We investigate two types of nonlinear interaction between collinear femtosecond laser pulses with power slightly above the critical for self-focusing $P_{cr}$. In the first case we study energy exchange between filaments. The model describes this process through degenerate four-photon parametric mixing (FPPM) scheme and requests initial phase difference between the waves. When there are no initial phase difference between the pulses, the FPPM process does not work. In this case it is obtained the second type of interaction as merging between two, three or four filaments in a single filament with higher power. It is found that in the second case the interflow between the filaments has potential of interaction due to cross-phase modulation (CPM).
We present exact rational solution for a modified nonlinear Schr$\ddot{o}$dinger equation that takes into account quintic nonlinearity and nonlinear dispersion corrections to the cubic nonlinearity, which could be used to describe rogue wave in nonlinear fibers. We find the rogue wave with these higher order effects has identical shape with the well-known one in nonlinear Schr$\ddot{o}$dinger equation. However, the quintic nonlinear term and nonlinear dispersion effect affect the velocity of rogue wave, and the evolution of its phase.
We study discrete rogue waves in an array of nonlinear waveguides. We show that very small degree of disorder due to experimental imperfection has a deep effect on the formation of discrete rogue waves. We predict long-living discrete rogue wave solution of the discrete nonlinear Schrodinger equation.
Akhmediev and Kuznetsov-Ma breathers are rogue wave solutions of the nonlinear Schrödinger equation (NLSE). Talbot effect (TE) is an image recurrence phenomenon in the diffraction of light waves. We report the nonlinear TE of rogue waves in a cubic medium. It is different from the linear TE, in that the wave propagates in a NL medium and is an eigenmode of NLSE. Periodic rogue waves impinging on a NL medium exhibit recurrent behavior, but only at the TE length and at the half-TE length with a π-phase shift; the fractional TE is absent. The NL TE is the result of the NL interference of the lob…
Weak phase noise present on an optical field can be amplified by a self-focusing nonlinearity and form intense "rogue wave" features. Here, we study the effect of the coherence length (or grain size) of this phase noise on the likelihood of rogue wave formation in the presence of a self-focusing nonlinearity. We show that while the likelihood of rogue wave formation increases with laser power when the coherence length is only slightly smaller that the beam diameter, the likelihood is minimally affected by change in laser power when the coherence length is significantly smaller than the beam di…
We considered the modulational instability of continuous-wave backgrounds, and the related generation and evolution of deterministic rogue waves in the recently introduced parity-time (PT)-symmetric system of linearly-coupled nonlinear Schrödinger equations, which describes a Kerr-nonlinear optical coupler with mutually balanced gain and loss in its cores. Besides the linear coupling, the overlapping cores are coupled through cross-phase-modulation term too. While the rogue waves, built according to the pattern of the Peregrine soliton, are (quite naturally) unstable, we demonstrate that the f…
In this work, we have studied the peregrine rogue wave dynamics, with a solitons on finite background (SFB) ansatz, in the recently proposed (Phys. Rev. Lett. 110 (2013) 064105) continuous nonlinear Schrodinger system with parity-time symmetric Kerr nonlinearity. We have found that the continuous nonlinear Schrodinger system with PT-symmetric nonlinearity also admits Peregrine Soliton solution. Motivated by the fact that Peregrine solitons are regarded as prototypical solutions of rogue waves, we have studied Peregrine rogue wave dynamics in the c-PTNLSE model. Upon numerical computation, we o…
We predict the existence of linear discrete rogue waves governed by the discrete nonlinear Schrodinger equation. We discuss that Josephson effect is the underlying reason for the formation of such waves.
In analogy with ocean waves running up towards the beach, shoaling of prechirped optical pulses may occur in the normal group-velocity dispersion regime of optical fibers. We present exact Riemann wave solutions of the optical shallow water equations and show that they agree remarkably well with the numerical solutions of the nonlinear Schrödinger equation, at least up to the point where a vertical pulse front develops. We also reveal that extreme wave events or optical tsunamis may be generated in dispersion tapered fibers in the presence of higher-order dispersion.
We introduce two-dimensional (2D) linear and nonlinear Talbot effects. They are produced by propagating periodic 2D diffraction patterns and can be visualized as 3D stacks of Talbot carpets. The nonlinear Talbot effect originates from 2D rogue waves and forms in a bulk 3D nonlinear medium. The recurrences of an input rogue wave are observed at the Talbot length and at the half-Talbot length, with a πphase shift; no other recurrences are observed. Different from the nonlinear Talbot effect, the linear effect displays the usual fractional Talbot images as well. We also find that the smaller the …
In addition to deep-water rogue waves which develop from the modulation instability of an optical CW, wave propagation in optical fibers may also produce shallow water rogue waves. These extreme wave events are generated in the modulationally stable normal dispersion regime. A suitable phase or frequency modulation of a CW laser leads to chirp-free and flat-top pulses or flaticons which exhibit a stable self-similar evolution. Upon collision, flaticons at different carrier frequencies, which may also occur in wavelength division multiplexed transmission systems, merge into a single, high-inten…
Rogue waves (RWs) are unexpectedly strong excitations emerging from an otherwise tranquil background. The nonlinear Schrödinger equation (NLSE), a ubiquitous model with wide applications to fluid mechanics, optics and plasmas, exhibits RWs only in the regime of modulation instability (MI) of the background. For system of multiple waveguides, the governing coupled NLSEs can produce regimes of MI and RWs, even if each component has dispersion and cubic nonlinearity of opposite signs. A similar effect will be demonstrated for a system of coupled derivative NLSEs (DNLSEs), where the special featur…
In the present work, a nonlocal nonlinear Schrödinger (NLS) model is studied by means of a recent technique that identifies solutions of partial differential equations, by considering them as fixed points in {\it space-time}. This methodology allows to perform a continuation of well-known solutions of the local NLS model to the nonlocal case. Four different examples of this type are presented, namely (a) the rogue wave in the form of the Peregrine soliton, (b) the generalization thereof in the form of the Kuznetsov-Ma breather, as well as two spatio-temporally periodic solutions in the form of…
Rogue waves are solitary waves with extreme amplitudes, which appear to be a ubiquitous phenomenon in nonlinear wave propagation, with the requirement for a nonlinearity being their only unifying characteristics. While many mechanisms have been demonstrated to explain the appearance of rogue waves in a specific system, there is no known generic mechanism or general set of criteria shown to rule their appearance. Presupposing only the existence of a nonlinear Schrödinger-type equation together with a concave dispersion profile around a zero dispersion wavelength we demonstrate that solitons may…
We present both theoretical description and experimental observation of the modulation instability process and related rogue breathers in the case of stationary periodic background waves, namely cnoidal and dnoidal envelopes. Despite being well-known solutions of the nonlinear Schrodinger equation, the stability of such background waves has remained unexplored experimentally until now, unlike the fundamental plane wave. By means of two experimental setups, namely, in nonlinear optics and hydrodynamics, we report on quantitative measurements of spontaneous modulation instability gain seeded by …
We study on the relations between modulational instability and several well-known nonlinear excitations in a nonlinear fiber, such as bright soliton, nonlinear continuous wave, Akhmediev breather, Peregrine rogue wave, and Kuznetsov-Ma breather. We present a quantitative correspondence between them based on the dominant frequency and propagation constant of each perturbation on a continuous wave background. Especially, we find rogue wave comes from modulational instability under the "resonance" perturbation with continuous wave background. These results will deepen our understanding on rogue w…
Ocean rogue waves (RW) -huge solitary waves- have for long triggered the interest of scientists. RWs emerge in a complex environment and it is still dubious the importance of linear versus nonlinear processes. Recent works have demonstrated that RWs appear in various other physical systems such as microwaves, nonlinear crystals, cold atoms, etc. In this work we investigate optical wave propagation in strongly scattering random lattices embedded in the bulk of transparent glasses. In the linear regime we observe the appearance of RWs that depend solely on the scattering properties of the medium…
We advance a statistical theory of extreme event emergence in random nonlinear wave systems with self-similar intermediate asymptotics. We show, within the framework of a generic (1 + 1)D nonlinear Schrodinger equation with linear gain, that extreme events and even rogue waves in weakly nonlinear, statistical open systems emerge as parabolic-shape giant fluctuations in the self-similar asymptotic propagation regime. We analytically demonstrate the self-similar structure of the non-Gaussian statistics of emergent rogue waves and validate our results with numerical simulations. Our results shed …
The defocusing nonlinear Schrödinger (NLS) equation has no the modulational instability, and was not found to possess the rogue wave (RW) phenomenon up to now. In this paper, we firstly investigate some novel nonlinear wave structures in the defocusing NLS equation with real-valued time-dependent and time-independent potentials such that the stable new RWs and W-shaped solitons are found, respectively. Moreover, the interactions of two or three RWs are explored such that the RWs with higher amplitudes are generated in the defocusing NLS equation with real-valued time-dependent potentials. Fina…
We show that rogue waves can be triggered naturally when a stable wave train enters a region of an opposing current flow. We demonstrate that the maximum amplitude of the rogue wave depends on the ratio between the current velocity, $ U_0 $, and the wave group velocity, $ c_g $. We also reveal that an opposing current can force the development of rogue waves in random wave fields, resulting in a substantial change of the statistical properties of the surface elevation. The present results can be directly adopted in any field of physics in which the focusing Nonlinear Schrodinger equation with …
We investigate the Zeno dynamics of the optical rogue waves. Considering their usage in modeling rogue wave dynamics, we analyze the Zeno dynamics of the Akhmediev breathers, Peregrine and Akhmediev-Peregrine soliton solutions of the nonlinear Schrodinger equation. We show that frequent measurements of the wave inhibits its movement in the observation domain for each of these solutions. We analyze the spectra of the rogue waves under Zeno dynamics. We also analyze the effect of observation frequency on the rogue wave profile and on the probability of lingering of the wave in the observation do…
In this paper, a simple and constructive method is presented to find the generalized perturbation (n,M)-fold Darboux transformations (DTs) of the modified nonlinear Schrodinger (MNLS) equation in terms of fractional forms of determinants. In particular, we apply the generalized perturbation (1,N-1)-fold DTs to find its explicit multi-rogue-wave solutions. The wave structures of these rogue-wave solutions of the MNLS equation are discussed in detail for different parameters, which display abundant interesting wave structures, including the triangle and pentagon, etc. and may be useful to study …
A nonlinear self-focusing material can amplify random small-amplitude phase modulations present in an optical beam, leading to the formation of amplitude singularities commonly referred to as optical caustics. By imposing polarization structuring on the beam, we demonstrate the suppression of amplitude singularities caused by nonlinear self-phase modulation. Our results are the first to indicate that polarization-structured beams can suppress nonlinear caustic formation in a saturable self-focusing medium and add to the growing understanding of catastrophic self-focusing effects in beams conta…
We study localized nonlinear waves on a plane wave background in the multiple self-induced transparency (SIT) system, which describes an important enhancement of the amplification and control of optical waves compared to the single SIT system. A hierarchy of exact multiparametric rational solutions in a compact determinant representation are presented. We demonstrate that, this family of solutions contains known rogue wave solution and unusual W-shaped soliton solution, which strictly corresponds to the linear stability analysis that involves modulation instability and stability regimes in the…
The efforts to understand the physics of rogue waves have motivated the study of mechanisms that produce rare, extreme events, often through analogous optical setups. As many studies have reported nonlinear generation mechanisms, recent work has explored whether optical rogue events can be produced in linear systems. Here we report the observation of linear rogue events with tunable height, generated from light imprinted with a non-Markovian wavefront. Moreover, if the non-Markovian wavefront is allowed to propagate through a nonlinear medium, extraordinarily long-tailed intensity distribution…
Rogue waves are intense and unexpected wavepackets ubiquitous in complex systems. In optics, they are promising as robust and noise-resistant beams for probing and manipulating the underlying material. Localizing large optical power is crucial especially in biomedical systems, where, however, extremely intense beams have not yet been observed. We here discover that tumor-cell spheroids manifest optical rogue waves when illuminated by randomly modulated laser beams. The intensity of light transmitted through bio-printed three-dimensional tumor models follows a signature Weibull statistical dist…
We report a theoretical study showing that rogue waves can emerge in whispering gallery mode resonators as the result of the chaotic interplay between Kerr nonlinearity and anomalous group-velocity dispersion. The nonlinear dynamics of the propagation of light in a whispering gallery-mode resonator is investigated using the Lugiato-Lefever equation, and we evidence a range of parameters where rare and extreme events associated with a non-gaussian statistics of the field maxima are observed.
We analytically study rogue-wave (RW) solutions and rational solitons of an integrable fifth-order nonlinear Schrödinger (FONLS) equation with three free parameters. It includes, as particular cases, the usual NLS, Hirota, and Lakshmanan-Porsezian-Daniel (LPD) equations. We present continuous-wave (CW) solutions and conditions for their modulation instability in the framework of this model. Applying the Darboux transformation to the CW input, novel first- and second-order RW solutions of the FONLS equation are analytically found. In particular, trajectories of motion of peaks and depressions o…
We study the existence and properties of rogue wave solutions in different nonlinear wave evolution models that are commonly used in optics and hydrodynamics. In particular, we consider Fokas-Lenells equation, the defocusing vector nolinear Schrödinger equation, and the long-wave-short-wave resonance equation. We show that rogue wave solutions in all of these models exist in the subset of parameters where modulation instability is present, if and only if the unstable sideband spectrum also contains cw or zero-frequency perturbations as a limiting case (baseband instability). We numerically con…
Caustics are natural phenomena in which nature concentrates the energy of waves. Although, they are known mostly in optics, caustics are intrinsic to all wave phenomena. For example, studies show that fluctuations in the profile of an ocean floor can generate random caustics and focus the energy of tsunami waves. Caustics share many similarities to rogue waves, as they both exhibit heavy-tailed distribution, i.e. an overpopulation of large events. Linear Schrödinger-type equations are usually used to explain the wave dynamics of caustics. However, in that the wave amplitude increases dramatica…
In this paper, by considering the potential application in two mode nonlinear waves in nonlinear fibers under a certain case, we define a coupled nonlinear Schrödinger equation(called Frobenius NLS equation) including its Lax pair. Afterwards, we construct the Darboux transformations of the Frobenius NLS equation. New solutions can be obtained from known seed solutions by the Dardoux transformations. Soliton solutions are generated from trivial seed solutions. Also we derive breather solutions $q,r$ of the Frobenius NLS equation obtained from periodic seed solutions. Interesting enough, we fin…
The higher-order dispersive and nonlinear effects (alias {\it the perturbation terms}) like the third-order dispersion, the self-steepening, and the self-frequency shift play important roles in the study of the ultra-short optical pulse propagation. We consider optical rogue wave solutions and interactions for the generalized higher-order nonlinear Schrödinger (NLS) equation with space- and time-modulated parameters. A proper transformation is presented to reduce the generalized higher-order NLS equation to the integrable Hirota equation with constant coefficients. This transformation allows u…
The extreme events are investigated for an $n$-component nonlinear Schrödinger ($n$-NLS) system in the focusing Kerr-like nonlinear media, which appears in many physical fields. We report and discuss the novel multi-parametric families of vector rational rogue wave (RW) solutions featuring the parity-time (PT) symmetry, which are characterized by non-identical boundary conditions for the components, and consistent with the degeneracy of $n$ branches of Benjamin-Feir instability. Explicit examples of PT-symmetric vector RWs are presented. Some parameter constraints can make some components gene…
In this work, based on the recently proposed (Phys. Rev. Lett. 110 (2013) 064105) continuous nonlocal nonlinear Schrödinger system with parity-time symmetric Kerr nonlinearity (PTNLSE), a numerical investigation has been carried out for two first order Peregrine solitons as the initial ansatz. Peregrine soliton, as an exact solution to the PTNLSE, evokes a very potent question: what effects does the interaction of two first order Peregrine solitons have on the overall optical field dynamics. Upon numerical computation, we observe the appearance of Kuznetsov-Ma (KM) soliton trains in the unbrok…
Rogue waves in birefringent optical fibers are analyzed within the framework of the coupled nonlinear Schrödinger (CNLS) system. The generation of rogue waves is frequently associated with modulation instability (MI). It is commonly expected that since the CNLS equations exhibit larger growth rates they should also produce more rogue events than their scalar counterparts. This is found to occur only when both equations are focusing. When at least one component is defocusing, the CNLS equations may still exhibit larger growth rates, compared to the scalar system, but that does not necessarily r…
Material characteristics and input-field specifics limit the controllability of nonlinear electromagnetic-field interactions. As these nonlinear interactions could be exploited to create strongly localized bright and dark waves, such as nonlinear surface polaritons, ameliorating this limitation is important. We present our approach to amelioration, which is based on a surface-polaritonic waveguide reconfiguration that enables excitation, propagation and coherent control of coupled dark rogue waves having orthogonal polarizations. Our control mechanism is achieved by finely tuning laser-field i…
Spontaneous emergence of self-organized patterns and their bifurcations towards a regime of complex dynamics in non-equilibrium dissipative systems is a paradigm of phase transition. Indeed, the behavior of these patterns in the highly nonlinear regime remains less explored, even in recent high-quality-factor resonators such as Kerr-nonlinear optical ones. Here, we investigate theoretically and experimentally the alteration of the resulting Kerr frequency combs from the weakly to the highly nonlinear regime, in the frameworks of spatiotemporal chaos, and dissipative phase transitions. We revea…
Institute for Complex Systems, National Research Council (ISC-CNR)（国家研究委员会复杂系统研究所）；Department of Physics, University Sapienza（罗马大学物理系）
We study artificial neural networks with nonlinear waves as a computing reservoir. We discuss universality and the conditions to learn a dataset in terms of output channels and nonlinearity. A feed-forward three-layer model, with an encoding input layer, a wave layer, and a decoding readout, behaves as a conventional neural network in approximating mathematical functions, real-world datasets, and universal Boolean gates. The rank of the transmission matrix has a fundamental role in assessing the learning abilities of the wave. For a given set of training points, a threshold nonlinearity for un…
We present a numerical study of the evolution dynamics of ``optical rogue waves'', statistically-rare extreme red-shifted soliton pulses arising from supercontinuum generation in photonic crystal fiber [D. R. Solli et al. Nature Vol. 450, 1054-1058 (2007)]. Our specific aim is to use nonlinear Schrodinger equation simulations to identify ways in which the rogue wave dynamics can be actively controlled, and we demonstrate that rogue wave generation can be enhanced by an order of magnitude through a small modulation across the input pulse envelope and effectively suppressed through the use of a …
We mainly investigate a coupled system of the generalized nonlinear Schrödinger equation and the Maxwell-Bloch equations which describes the wave propagation in an erbium-doped nonlinear fiber with higher-order effects including the forth-order dispersion and quintic non-Kerr nonlinearity. We derive the one-fold Darbox transformation of this system and construct the determinant representation of the $n$-fold Darboux transformation. Then the determinant representation of the $n$th new solutions $(E^{[n]},\, p^{[n]},\, η^{[n]})$ which were generated from the known seed solutions $(E, \, p, \, η)…
Over the past decade, the rogue wave debate has stimulated the comparison of predictions and observations among different branches of wave physics, particularly between hydrodynamics and optics, in situations where analogous dynamical behaviors can be identified, thanks to the use of common universal models. Although the scalar nonlinear Schroedinger equation (NLSE) has constantly played a central role for rogue wave investigations, moving beyond the standard NLSE model is relevant and needful for describing more general classes of physical systems and applications. In this direction, the coup…
